Results by chapter Computing the Riemann Hypothesis

The results of Computing the Riemann Hypothesis that are new in the book, that come from the authors’ own papers, or that correct one of those papers, as the book states them: 1012 entries (701 proved, 49 computed, 71 conditional, 186 open, 5 counterexample). Results of other authors that the book presents are credited in the chapter summaries.

Status: proved; computed (established by a computation that the book reports and the entry names, such as an exact census or a fit to one, and not proved for every case); conditional (proved assuming the named statements, which the book does not prove); open (stated as an open question); counterexample (a proved counterexample); withdrawn (found wrong after release; the entry says what was wrong and which version corrects it).

Volume I, Chapter 2. Analytic representations of the Riemann Xi function

Rigidity of the clock, a GGC if and only if RH

Volume I, Proposition 2.1.1, page 18 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s - 1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed zeta function, an entire function whose zeros are the nontrivial zeros of \(\zeta\), and put \(G(u)=\xi(\frac12+\sqrt{u})/\xi(\frac12)\), an entire function of \(u\). The clock is the function \(1/G\). A random variable \(H\ge 0\) is a generalised gamma convolution (GGC) if \(\mathbb{E}\,e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), with a drift \(d\ge 0\) and a positive measure \(U\) on \((0,\infty)\), the Thorin measure, such that \(\int_{(0,1)}|\log z|\,U(dz)+\int_{(1,\infty)}z^{-1}U(dz)<\infty\); the drift and the Thorin measure are unique. Then \(1/G\) is the Laplace transform of a GGC if and only if the Riemann Hypothesis (RH) holds, that is, if and only if every zero of \(\xi\) has real part \(\frac12\). In that case the drift is zero, the Thorin measure is \(U_{\mathrm{RH}}=\sum_{\gamma>0}\delta_{\gamma^2}\), where \(\frac12+i\gamma\) runs over the zeros of \(\xi\) with \(\gamma>0\), counted with multiplicity, and the clock variable is \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential random variables \(e_\gamma\).

Earlier editions: Proposition 1.2.1, page 10 (edition 1).

Source: ch/ch02.tex, label prop:rigid.

Shifted clock is a GGC to the right of every zero

Volume I, Theorem 2.1.3, page 22 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function, and for real \(\alpha\) and real \(r\) put \(v_\alpha(r)=\operatorname{Re}(\xi'/\xi)(\alpha+ir)\). If \(\alpha>\operatorname{Re}\rho\) for every zero \(\rho\) of \(\xi\), in particular for every \(\alpha\ge 1\), then \(u\mapsto\xi(\alpha)/\xi(\alpha+\sqrt{u})\) is the Laplace transform of a generalised gamma convolution \(T_\alpha\) (a random variable \(H\ge 0\) with \(\mathbb{E}\,e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), a drift \(d\ge 0\) and a positive Thorin measure \(U\)) whose Thorin measure has density \(v_\alpha(\sqrt{z})/(2\pi\sqrt{z})\) with respect to \(dz\) on \((0,\infty)\), and \(v_\alpha>0\). The function \(u\mapsto\xi(\alpha)/\xi(\alpha+\sqrt{u})\) is the Laplace transform of a generalised gamma convolution if and only if \(\alpha\ge\sup_\rho\operatorname{Re}\rho\).

Earlier editions: Theorem 1.2.2, page 14 (edition 1).

Source: ch/ch02.tex, label thm:poisson.

Positivity of the Thorin density on one line, open question

Volume I, Remark 2.1.5, page 23 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed zeta function and, for real \(\alpha>\frac12\) and real \(r\), put \(v_\alpha(r)=\operatorname{Re}(\xi'/\xi)(\alpha+ir)\), so that \(v_\alpha(r)/\pi\) is the Thorin density at resolution \(\alpha - \frac12\). If some zero of \(\xi\) has real part greater than \(\alpha\), the shifted clock \(\xi(\alpha)/\xi(\alpha+\sqrt{u})\) is not the Laplace transform of a generalised gamma convolution. The question is whether, without assuming the Riemann Hypothesis, \(v_\alpha(r)\ge 0\) for all real \(r\) implies that \(\xi\) has no zero with real part greater than \(\alpha\). The book states that this is not known for \(\xi\). What is known is that if \(v_\alpha\ge 0\) for every \(\alpha>\sigma_0\) then no zero has real part greater than \(\sigma_0\) (for \(\sigma_0=\frac12\) this is Lagarias’s criterion for RH), and that for an explicit entire function with the order, symmetry and reality of \(\xi\) the answer is negative.

Earlier editions: Remark 1.2.4, page 15 (edition 1).

Source: ch/ch02.tex, label rem:ch2:oneline.

Poisson structure of the Thorin density at resolution epsilon

Volume I, Proposition 2.1.4, page 23 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) with nontrivial zeros \(\rho=\beta+i\gamma\), let \(P_a(x)=a/(\pi(a^2+x^2))\) be the Poisson kernel with \(P_{-a}=-P_a\), let \(\alpha=\frac12+\varepsilon\) with \(\varepsilon>0\), and let \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)\) be the density in \(\theta=\sqrt z\) of the Thorin measure \(\mu_\varepsilon\) at resolution \(\varepsilon\) (the measure for which \(\xi(\alpha)/\xi(\alpha+\sqrt u)\) is a Laplace transform when no zero lies right of \(\alpha\)). For \(\theta\ge0\) such that \(\frac12+\varepsilon+i\theta\) is not a zero, \(\rho_\varepsilon(\theta)=\sum_{\gamma>0,\ \operatorname{Re}\rho=1/2}[P_\varepsilon(\theta-\gamma)+P_\varepsilon(\theta+\gamma)]+\sum_{\text{off line }\rho=\frac12\pm\delta+i\gamma,\ \gamma>0}\big([P_{\varepsilon+\delta}+P_{\varepsilon-\delta}](\theta-\gamma)+[P_{\varepsilon+\delta}+P_{\varepsilon-\delta}](\theta+\gamma)\big)\), where the second sum counts each off line pair \(\{\frac12+\delta+i\gamma,\frac12-\delta+i\gamma\}\) with \(0<\delta<\frac12\) once. Equivalently \(\rho_\varepsilon(\theta)\) is the sum of \(P_{\alpha-\beta}(\theta-\gamma)+P_{\alpha-\beta}(\theta+\gamma)\) over the zeros with \(\gamma>0\). In particular under RH, \(\mu_\varepsilon=P_\varepsilon*U^\pm_{\mathrm{RH}}\) restricted to \(\theta>0\) with \(U^\pm_{\mathrm{RH}}=\sum_{\gamma>0}(\delta_\gamma+\delta_{-\gamma})\). The series converges absolutely, and uniformly for \(\theta\) in compact sets avoiding the zeros.

Source: ch/ch02.tex, label prop:struct.

Three thinnings of the stable Levy density by Smirnov, Kolmogorov and Kuiper bridge laws

Volume I, Proposition 2.1.7, page 25 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(x>0\) let \(\nu_x(t)=x/(2\sqrt\pi\,t^{3/2})\), let \(D^+\), \(K\), \(V\) be the Smirnov, Kolmogorov and Kuiper variables (the supremum, the maximal absolute value and the range of a standard Brownian bridge), and for \(s\ge0\) put \(\Psi_S(s^2)=\int_0^\infty(1-e^{-s^2t})\,\nu_x(t)\,P(\sqrt{2t}\,S<\frac x2)\,dt\) for \(S\in\{D^+,K,V\}\). Then (i) \(\Psi_{D^+}(s^2)=e^{-sx}+sx-1\), with the Thorin density of the sine squared lemma. (ii) \(\Psi_K(s^2)=2\log\cosh\frac{sx}2=2\sum_{m\ge1}\log(1+s^2/c_m)\) with \(c_m=(2m-1)^2\pi^2/x^2\), the exponent of a generalised gamma convolution (GGC) with atomic Thorin measure \(2\sum_m\delta_{c_m}\), the sum of two independent exit times of \((-\frac x2,\frac x2)\). (iii) \(\Psi_V(s^2)=sx\coth\frac{sx}2-2=4\sum_{k\ge1}s^2/(s^2+c'_k)\) with \(c'_k=4\pi^2k^2/x^2\), and it is not the exponent of a GGC. The three closed forms agree with quadrature to fifteen digits.

Source: ch/ch02.tex, label prop:ch2:thin.

Signed Laplace mixture for a finite product of Cauchy type characteristic functions

Volume I, Proposition 9.1.1, page 133 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha_1,\dots,\alpha_n\) be distinct with \(\operatorname{Re}\alpha_j>0\) and \(\alpha_j^2\ne\alpha_m^2\) for \(j\ne m\), and let \(\psi(t)=\prod_j(1+t^2/\alpha_j^2)^{-1}\). Then \(\psi(t)=\sum_jC_j\frac{\alpha_j^2}{\alpha_j^2+t^2}\) with \(C_j=\prod_{m\ne j}\frac{\alpha_m^2}{\alpha_m^2-\alpha_j^2}\) and \(\sum_jC_j=1\), and the inverse Fourier transform of \(\psi\) is the signed Laplace mixture \(p(x)=\sum_jC_j\frac{\alpha_j}2e^{-\alpha_j|x|}\).

Source: ch/c04.tex, label xi:prop:mixture.

The lowest zero of zeta as the exponential decay rate of the density of the partner of the clock

Volume I, Theorem 9.1.2, page 134 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Xi(z)=\xi(\frac12+iz)\), let \(T\) be the clock with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\), let \(Z\) be a standard normal variable independent of \(T\), and let \(p\) be the even density of \(Z\sqrt{2T}\), whose characteristic function is \(\psi(t)=\xi(\frac12)/\xi(\frac12-t)\). Put \(\sigma=\sup\{a\ge0:\int_{\mathbb{R}}p(x)e^{a|x|}dx<\infty\}\). Then \(0<\sigma<\infty\), \(\Xi\) has no zero with \(|\operatorname{Re}z|<\sigma\), and \(\Xi(\sigma)=0\). Hence \(\zeta\) has a zero \(\frac12+i\sigma\) on the critical line and no zero of ordinate of modulus below \(\sigma\). The statement is unconditional.

Source: ch/c04.tex, label xi:thm:landau.

Positive exponential tail of the signed Laplace mixture when the lowest atom is real

Volume I, Lemma 9.1.3, page 134 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha_1,\dots,\alpha_n\) be distinct with \(\operatorname{Re}\alpha_j>0\) and \(\alpha_j^2\ne\alpha_m^2\) for \(j\ne m\), let \(\psi(t)=\prod_j(1+t^2/\alpha_j^2)^{-1}\) with signed Laplace mixture density \(p(x)=\sum_jC_j\frac{\alpha_j}2e^{-\alpha_j|x|}\), \(C_j=\prod_{m\ne j}\alpha_m^2/(\alpha_m^2-\alpha_j^2)\). Suppose \(\alpha_1=\gamma_1>0\) is real, the nonreal \(\alpha_m\) occur in conjugate pairs, and \(\operatorname{Re}\alpha_m>\gamma_1\) for \(m\ne1\). Then \(C_1>0\) and \(p(x)\sim C_1\frac{\gamma_1}2e^{-\gamma_1x}>0\) as \(x\to\infty\).

Source: ch/c04.tex, label xi:lem:tail.

A nonreal zero of Xi destroys complete monotonicity of the exponential sum over the zeros

Volume I, Proposition 9.1.4, page 135 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi(z)=\xi(\frac12+iz)\) for \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), and let \(\mathcal Q(x)=\sum_je^{-\alpha_jx}\) for \(x>0\), over the zeros \(\alpha_j\) of \(\Xi\) with \(\operatorname{Re}\alpha_j>0\), with multiplicity. If \(\Xi\) has a zero \(\alpha_0\) with \(\operatorname{Im}\alpha_0\ne0\), then \(\mathcal Q\) is not completely monotone on \((0,\infty)\). Under RH all \(\alpha_j\) are real and positive and \(\mathcal Q\) is completely monotone termwise, \((-1)^n\mathcal Q^{(n)}(x)=\sum_j\alpha_j^ne^{-\alpha_jx}\ge0\). The proof is that complete monotonicity would give by Bernstein’s theorem a positive measure \(m\) with \(\mathcal Q(x)=\int e^{-xs}m(ds)\), so the Laplace transform of \(\mathcal Q(\cdot+1)\) would be analytic on \(\mathbb{C}\setminus(-\infty,0]\), whereas it equals \(\sum_je^{-\alpha_j}/(s+\alpha_j)\), which has a pole at \(-\alpha_0\notin(-\infty,0]\).

Source: ch/c04.tex, label xi:prop:sharp.

Volume I, Chapter 3. Probabilistic representations of Xi

The clock is not a Brownian first passage time to an independent random level

Volume I, Proposition 3.1.1, page 27 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s completed zeta function, and let \(T\) be the clock variable, the nonnegative random variable whose Laplace transform is \(\mathbb{E}e^{-uT}=\xi(\tfrac12)/\xi(\tfrac12+\sqrt{u})\) for \(u\ge0\) (it exists with no hypothesis on the zeros of \(\xi\)). For a drift \(c\ge0\) and a level \(a>0\), the first passage time of standard Brownian motion with drift \(c\) (unit variance per unit time) to the level \(a\) has the inverse Gaussian law, with Laplace transform \(\exp\{a(c-\sqrt{c^2+2u})\}\). Then for no drift \(c\ge0\) and no probability law \(F\) on \((0,\infty)\) is \(T\) equal in law to the first passage time of Brownian motion with drift \(c\) to an independent random level \(A\) with law \(F\). For a drift \(c<0\) the first passage time is infinite with probability \(1-\mathbb{E}e^{2cA}>0\), so equality is impossible as well.

Earlier editions: Proposition 8.2.1, page 219 (edition 1).

Source: ch/ch19.tex, label prop:nogo.

Under a power tilt the Gamma(2,1) modes of the P'olya variable are negatively correlated, so no ferromagnet has them as spins

Volume I, Proposition 3.1.2, page 31 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Gamma_n\), \(n\ge1\), be independent random variables of law Gamma\((2,1)\), let \(B_n=\Gamma_n/n^2\) and \(V=\sum_{n\ge1}B_n\), and for \(0<q\le1\) let \(\mathbb{P}_q\) have density \(V^q/\mathbb{E}V^q\) with respect to the original law. Then \(\operatorname{Cov}_q(B_i,B_j)<0\) for all \(i\ne j\). Write \(\Gamma_n=\frac12\sum_{r=1}^4Z_{nr}^2\) with \(Z_{nr}\) independent standard normal variables (the Karhunen and Lo`eve modes of a four dimensional Brownian bridge \(b\), for which \(\int_0^1|b_t|^2\,dt=2V/\pi^2\)). Then \(\operatorname{Cov}_q(Z_{ir}^2,Z_{js}^2)=\frac14\operatorname{Cov}_q(\Gamma_i,\Gamma_j)<0\) for \(i\ne j\). Hence no ferromagnet, and no weak limit of finite ferromagnets, with even single site measures and nonnegative pair couplings, even after adjoining and integrating out further even spins, has the \(Z_{nr}\) as spins and the law \(\mathbb{P}_q\); in particular this holds for \(q=\frac14\), the tilt under which \(\frac12\log(V/\pi)\) is the P'olya variable of the completed Riemann zeta function.

Source: ch/ch19.tex, label prop:ch19:ginibre.

Maślanka coefficients as Laguerre moments of Williams’ variable

Volume I, Proposition 3.1.4, page 32 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(W=\pi^2S_2/2=\sum_{n\ge1}\Gamma_n/n^2\) with independent Gamma(2) variables \(\Gamma_n\) of unit scale, so that \(\mathbb{E}e^{-\lambda W}=(\pi\sqrt\lambda/\sinh\pi\sqrt\lambda)^2\), and let \(L^{(1)}_k\) be the generalized Laguerre polynomials. Let \(A_k=\sum_{j=0}^k(-1)^j\binom kj(2j+1)\zeta(2j+2)\), the coefficients in Maślanka’s expansion \((\sigma-1)\zeta(\sigma)=\sum_{k\ge0}A_k(1-\frac\sigma2)_k/k!\). Then (i) \(A_k=\mathbb{E}[WL^{(1)}_k(W)]/(2(k+1))\); (ii) \(\sum_{k\ge0}A_ky^k=(1-\mathbb{E}e^{-Wy/(1-y)})/(2y)\) for \(0<|y|<1\); (iii) for \(0<\operatorname{Re}\sigma<2\), \((\sigma-1)\zeta(\sigma)=\frac{\sin(\pi\sigma/2)}{2\pi}\int_0^\infty r^{-\sigma/2-1}(1-\mathbb{E}e^{-rW})dr\), and for real \(0<\sigma<2\) this equals \(\mathbb{E}[(1-\mathbb{E}[e^{-WR}\mid\mathcal B])/(2(1-\mathcal B))]\) with \(\mathcal B\) a beta variable of parameters \(\frac\sigma2\) and \(1-\frac\sigma2\) independent of \(W\) and \(R=(1-\mathcal B)/\mathcal B\), so that expanding (ii) at \(y=1-\mathcal B\) gives Maślanka’s series, whose rigorous proof is due to Báez-Duarte (2010). The coefficients determine the law of \(W\) but transfer no positivity to the Thorin measure of the clock. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Proposition 8.2.2, page 224 (edition 1).

Source: ch/ch19.tex, label prop:ch19:maslanka.

Volume I, Chapter 4. Infinite divisibility, GGC, EGGC and the Bondesson hierarchy

The variable Z sqrt(2T) of the van Dantzig pair is self decomposable with no hypothesis, with an explicit canonical function

Volume I, Result 4.1.2, page 37 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(T\) be the clock, with Lévy measure \(W(v)\,dv/v\), \(W(v)=\sum_je^{-a_jv}\), \(a_j=-(\rho_j-\frac12)^2\), let \(Z\) be a standard normal variable independent of \(T\), so that \(\mathbb{E}e^{i\theta Z\sqrt{2T}}=\mathbb{E}e^{-\theta^2T}\). With no hypothesis \(Z\sqrt{2T}\) is self decomposable: subordinating the Lévy measure of \(T\) to the heat kernel gives the Lévy density \(k(|x|)/|x|\) with \(k(r)=\int_0^\infty e^{-1/4v}(4\pi v)^{-1/2}W(r^2v)\,dv/v=\sum_je^{-r\sqrt{a_j}}\) (the first passage identity for each term, with the principal root), and \(k\) decreases because \(W\) does. Since the characteristic function determines the law of \(T\), \(Z\sqrt{2T}\) is an extended generalised gamma convolution if and only if \(T\) is a GGC, that is if and only if RH holds.

Earlier editions: Result 1.3.2, page 19 (edition 1).

Source: ch/ch03.tex, label tx3:res:sdz.

Volume I, Chapter 5. The reciprocal Xi function

Möbius series for the density of the clock with no hypothesis, with terms g_n a convolution of one fixed function with a first passage kernel

Volume I, Proposition 5.2.1, page 48 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Write \(\xi=\eta\zeta\) with \(\eta(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\) and \(\phi_\eta(u)=\xi(\frac12)/\eta(\frac12+\sqrt u)\). Let \(c>\frac14\), \(g_1\) the inverse Laplace transform of \(\phi_\eta\) along \(\operatorname{Re}u=c\), \(q_\lambda(x)=\frac{\lambda}{2\sqrt\pi}x^{-3/2}e^{-\lambda^2/4x}\) for \(x>0\), both extended by \(0\) to \(x\le0\). Then for every \(t>0\), \(f(t)=\sum_{n\ge1}\frac{\mu(n)}{\sqrt n}g_n(t)\) with \(g_n=g_1*q_{\log n}\) and \(|g_n(t)|\le\sup_{0<x\le t}|g_1(x)|\operatorname{erfc}\big(\frac{\log n}{2\sqrt t}\big)\), so the series converges absolutely, faster than any power of \(n\). Moreover \(g_1(t)=2\xi(\frac12)e^{t/4}+O(e^{\varepsilon t})\) as \(t\to\infty\) for every \(\varepsilon>0\), so \(g_1\) is not a density and for large \(t\) the series rests on \(\sum_n\mu(n)/n=0\). Numerically (Talbot inversion at 30 digits) the terms with \(n\le5\) reproduce \(f\) to at least \(20\) digits at \(t=0.008,\dots,0.04\).

Earlier editions: Proposition 8.1.1, page 212 (edition 1).

Source: ch/ch18.tex, label tx3:prop:mobius.

Lower tail of the clock: P(T at most y) is at most exp of minus (log 1/y) squared over 16 y, so the density vanishes to infinite order at the origin

Volume I, Result 5.2.1, page 49 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(T\) be the clock, \(\mathbb{E}e^{-uT}=1/G(u)\), \(G(u)=\xi(\frac12+\sqrt u)/\xi(\frac12)\). With no hypothesis, \(P(T\le y)\le\exp\{-(1+o(1))(\log\frac1y)^2/16y\}\) as \(y\downarrow0\), so the density of \(T\) vanishes to infinite order at the origin. The proof is Chernoff’s bound \(P(T\le y)\le e^{uy}/G(u)\), with \(\log G(u)\sim\frac{\sqrt u}4\log u\) by Stirling’s formula for \(\xi\), optimised at \(\sqrt u\approx\log(1/y)/4y\). The saddlepoint exponent has the same leading term but converges slowly: at \(y=0.006\) it gives \(-272\) against \(-33.0\) for the exact \(\log f\).

Earlier editions: Result 8.1.1, page 213 (edition 1).

Source: ch/ch18.tex, label tx3:res:lowertail.

RH holds if and only if the density of the clock is a Pólya frequency function of infinite order

Volume I, Proposition 5.2.2, page 49 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. RH holds if and only if the density \(f\) of the clock \(T\) is a Pólya frequency function of infinite order, that is \(\det[f(x_i-y_j)]_{i,j\le r}\ge0\) for every \(r\) and all \(x_1<\cdots<x_r\), \(y_1<\cdots<y_r\). Under RH \(f\) is the density of \(\sum_\gamma e_\gamma/\gamma^2\), exponential densities are Pólya frequency functions, and the class is closed under convolution and weak limits. Conversely, by Schoenberg’s theorem the bilateral Laplace transform of an integrable Pólya frequency function of infinite order is \(1/\Psi\) with \(\Psi(u)=Ce^{-cu^2+bu}\prod_k(1+\lambda_ku)e^{-\lambda_ku}\), \(\lambda_k\) real; here the transform is \(1/G\), so \(G(u)=\xi(\frac12+\sqrt u)/\xi(\frac12)\) has only real zeros \(u=-a_k\), which forces \(\rho-\frac12\) real or purely imaginary, and \(\xi\) has no real zeros, so every \(\rho\) is on the critical line. The book adds, as necessary conditions tested in floating point and not proved, \((\log f)''<0\) on \(91\) points of \([0.006,0.08]\) with maximum \(-0.026\).

Earlier editions: Proposition 8.1.2, page 213 (edition 1).

Source: ch/ch18.tex, label tx2:prop:pf.

The density of the reciprocal xi law is the limit of grouped residue sums, with no hypothesis on the zeros

Volume I, Theorem 5.3.1, page 55 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. There are radii \(R_n\to\infty\) with \(R_n\in[n,2n]\) such that for every \(y>0\), \(f(y)=\lim_{n\to\infty}\sum_{|u_\rho|<R_n}\operatorname{Res}_{u=u_\rho}e^{uy}/G(u)\), with no hypothesis on the zeros. A simple zero contributes \(e^{u_\rho y}/G'(u_\rho)\), a zero of multiplicity \(m\) contributes \(e^{u_\rho y}P_\rho(y)\) with \(P_\rho\) a polynomial of degree \(m-1\), and a pair of zeros \(\frac12\pm\delta+i\gamma\) off the line contributes \(2\operatorname{Re}(e^{u_\rho y}/G'(u_\rho))\). For \(y\) in a compact subset of \((0,\infty)\) the error after the \(n\)th group is at most \(R_n\exp(-\frac12R_ny+C\sqrt{R_n}\log R_n)+\exp(-c\sqrt{R_n}\log R_n)\) for constants \(c,C>0\).

Earlier editions: Theorem 24.1.1, page 854 (edition 1).

Source: ch/ht1.tex, label ht:thm:grouped.

Residue expansion: absolute convergence of the density series given a lower bound for the derivative of xi at the zeros

Volume I, Theorem 5.3.2, page 56 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the zeros of the Riemann zeta function are simple and \(|\xi'(\rho)|\ge e^{-C|\gamma|}\) for some constant \(C\) and all zeros.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. Assume that the zeros of \(\zeta\) are simple and that \(|\xi'(\rho)|\ge e^{-C|\gamma|}\) for some constant \(C\) and all zeros (a hypothesis not known to hold). Then \(f(y)=\sum_{\gamma>0}e^{u_\rho y}/G'(u_\rho)\), \(y>0\), converges absolutely and locally uniformly. Under RH it reads \(f(y)=\sum_{\gamma>0}w_\gamma e^{-\gamma^2y}\) with \(w_\gamma=\gamma^2\prod_{\gamma'\ne\gamma}(1-\gamma^2/\gamma'^2)^{-1}\).

Earlier editions: Theorem 24.1.2, page 854 (edition 1).

Source: ch/ht1.tex, label ht:thm:res.

Upper tail of the density of the reciprocal xi law: two exponential terms and an explicit remainder

Volume I, Proposition 5.3.3, page 57 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. Let \(c_3=\frac12(\gamma_2^2+\gamma_3^2)\). With no hypothesis, \(f(y)=w_1e^{-\gamma_1^2y}+w_2e^{-\gamma_2^2y}+O(e^{-c_3y})\) as \(y\to\infty\), with the same bound for each derivative, where \(w_1=1.01635\times10^4>0\) and \(w_2=-1.17754\times10^6<0\).

Earlier editions: Proposition 24.1.3, page 855 (edition 1).

Source: ch/ht1.tex, label ht:prop:tail.

Kent’s eigenfunction expansion is the residue expansion under RH with simple zeros

Volume I, Proposition 5.4.1, page 57 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. If RH holds and the zeros are simple, the residue series \(\sum_{\gamma>0}e^{u_\rho y}/G'(u_\rho)\) is Kent’s expansion with \(\lambda_n=\gamma_n^2\), and \(f\) is the density of the hitting time of a Krein string with Dirichlet spectrum \(\{\gamma_n^2\}\). Without these hypotheses the grouped expansion is the formal continuation of Kent’s formula to complex eigenvalues: an off line pair contributes \(2\operatorname{Re}(e^{u_\rho y}/G'(u_\rho))\), a multiple zero contributes a polynomial times an exponential, and no string has such a spectrum.

Earlier editions: Proposition 24.2.1, page 856 (edition 1).

Source: ch/ht1.tex, label ht:prop:kent.

The function xi(1/2+it) is not in Krein’s class and the partial fraction series of 1/G diverges under RH

Volume I, Proposition 5.4.2, page 58 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. The function \(\xi(\frac12+it)\) does not belong to Krein’s class, whether or not RH holds. Moreover, under RH the partial fraction series of \(1/G\) over the points \(u_\rho=-\gamma_n^2\) does not converge absolutely.

Earlier editions: Proposition 24.2.2, page 857 (edition 1).

Source: ch/ht1.tex, label ht:prop:krein.

RH holds if and only if the reciprocal xi law is a weak limit of GGC laws

Volume I, Proposition 5.5.1, page 59 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. RH holds if and only if there are GGC laws, for instance hitting times of strings, converging weakly to \(\mu\).

Earlier editions: Proposition 24.3.1, page 858 (edition 1).

Source: ch/ht1.tex, label ht:prop:red.

Truncations of the Hadamard product: GGC if and only if the first N zeros lie on the line, and GGC up to the verified height

Volume I, Proposition 5.7.1, page 61 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. Order the zeros in the upper half plane by height, let \(G_N(u)=\prod_{n\le N}(1-u/u_{\rho_n})\), \(\varphi_N=1/G_N\), \(\varphi=1/G\), and let \(\mu_N\) be the law with Laplace transform \(\varphi_N\). (i) \(\varphi_N\to\varphi\) pointwise on \([0,\infty)\), and for each fixed zero \(1/G_N'(u_\rho)\to1/G'(u_\rho)\), with correction factor \(\prod_{n>N}(1-u_\rho/u_{\rho_n})^{-1}\approx\exp(u_\rho\sum_{n>N}u_{\rho_n}^{-1})\), which under RH equals \(\exp(\gamma^2\sum_{n>N}\gamma_n^{-2})>1\). (ii) \(\mu_N\) is a GGC if and only if the first \(N\) zeros lie on the line; in particular it is a GGC, unconditionally, for every \(N\le N_0(H)\), the number of zeros below the verified height \(H=3\cdot10^{12}\) (Platt and Trudgian). (iii) RH holds if and only if every \(\mu_N\) is a GGC, and then \(\mu\) is a GGC by Bondesson’s continuity theorem. (iv) Under RH, for \(u\ge0\), \(0\le\log G(u)-\log G_N(u)\le u\,\tau_1(N)\) with \(\tau_1(N)=\sum_{n>N}\gamma_n^{-2}\approx(\log(T_N/2\pi)+1)/(2\pi T_N)\), where \(T_N\) is the height of the \(N\)th zero.

Earlier editions: Proposition 24.5.1, page 859 (edition 1).

Source: ch/ht1.tex, label ht:prop:trunc.

The density of the reciprocal xi law is strictly log concave on a half line, with no hypothesis

Volume I, Theorem 5.7.2, page 62 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. There is \(Y\) such that \(f\) is strictly log concave on \([Y,\infty)\), with no hypothesis.

Earlier editions: Theorem 24.5.2, page 861 (edition 1).

Source: ch/ht1.tex, label ht:thm:lctail.

Log concavity of the reciprocal xi density near the origin, given the saddle point approximation

Volume I, Proposition 5.7.3, page 62 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the saddle point approximation of \(\log f(y)\) holds as \(y\to0\) together with two derivatives.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. Let \(\psi=\log G\), \(\Lambda_f(y)=\sup_{u>0}(\psi(u)-uy)\) and \(u_y\) the solution of \(\psi'(u_y)=y\). Suppose the saddle point approximation \(\log f(y)\approx-\Lambda_f(y)-\frac12\log(2\pi|\psi''(u_y)|)\) holds as \(y\to0\) together with two derivatives (this is assumed, not proved). Then \(f\) is strictly log concave on \((0,y_0]\) for some \(y_0>0\).

Earlier editions: Proposition 24.5.3, page 861 (edition 1).

Source: ch/ht1.tex, label ht:prop:lc0.

The density of the reciprocal xi law is the sum of its first 120 residue terms up to an explicit exponentially small error

Volume I, Proposition 5.7.4, page 63 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. With no hypothesis, for every \(y>0\) and \(j\ge0\), \(|f^{(j)}(y)-\sum_{n\le120}w_n(-\gamma_n^2)^je^{-\gamma_n^2y}|\le M_je^{-cy}\), where \(c\) is the constant of the book’s contour shift beyond the 120th zero, \(w_n\) and \(\gamma_n\) are the weights and ordinates of the first \(120\) zeros, and \(M_j=\frac1{2\pi}\int_{-\infty}^\infty|{-c}+it|^j/|G(-c+it)|\,\mathrm dt\). Numerically \(\log M_0=206.26\) and \(M_{j+1}/M_j=1.0000185\,c\) for \(j=0,1,2\) (computed).

Earlier editions: Proposition 24.5.4, page 862 (edition 1).

Source: ch/ht1.tex, label ht:prop:explicit.

Log concavity of the reciprocal xi density on [0.008, infinity) from computed constants

Volume I, Theorem 5.7.5, page 64 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. Assume the values of the constants \(M_j\), \(j\le8\), of the explicit truncation bound for \(f\) with its first \(120\) residue terms (computed by quadrature). Then, with no hypothesis on the zeros, \(f\) is strictly log concave on \([Y_*,\infty)\) with \(Y_*=0.008\).

Earlier editions: Theorem 24.5.5, page 863 (edition 1).

Source: ch/ht1.tex, label ht:thm:lc.

If the density of the reciprocal xi law were HCM then RH would hold

Volume I, Proposition 5.7.6, page 65 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. If the density \(f\) of \(\mu\) is HCM (hyperbolically completely monotone), then \(\mu\) is a GGC and RH holds.

Earlier editions: Proposition 24.5.6, page 864 (edition 1).

Source: ch/ht1.tex, label ht:prop:hcm.

No HCM: sums of exponentials are not HCM, and the continued reciprocal xi density has a computed zero in the right half plane

Volume I, Theorem 5.7.7, page 66 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on for part (ii), an explicit remainder bound for the omitted terms on a sector containing \(z_*\).

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. (i) No density of a sum of \(n\ge2\) independent exponential variables with distinct rates is HCM. (ii) With the first \(120\) terms of the residue series for \(f\), the continuation of \(f\) to \(\operatorname{Re}z>0\) vanishes at \(z_*=0.013831874754506+0.020001801712034\,i\) (computed). If the omitted terms are controlled by an explicit remainder bound on a sector containing \(z_*\), this proves that \(f\) is not HCM.

Earlier editions: Theorem 24.5.7, page 865 (edition 1).

Source: ch/ht1.tex, label ht:thm:nohcm.

Partial complete monotonicity of the canonical function of the reciprocal xi law up to the verified height, from the verified zeros

Volume I, Theorem 5.8.1, page 68 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(s_\rho=(\rho-\frac12)^2\) and \(\Theta(x)=\sum_{\gamma>0}e^{s_\rho x}\) the canonical function of the reciprocal xi law. Let \(T=3\cdot10^{12}\), so that every zero with \(0<\gamma\le T\) is on the line (Platt and Trudgian). Then \((-1)^n\Theta^{(n)}(x)>0\) for every \(x>0\) and every integer \(0\le n\le1.70(T-23)^2\), a bound which exceeds \(1.52\times10^{25}\), and \(\Theta\) is completely monotone on \((0,1.70]\). RH holds if and only if the bound can be replaced by \(n<\infty\) for every \(x>0\).

Earlier editions: Theorem 24.15.1, page 895 (edition 1).

Source: ch/ht4.tex, label ht:thm:partialcm.

Equivalent forms of the quasi Riemann hypothesis QRH(alpha) through the prime counting error, the prime Dirichlet series and the base point laws

Volume I, Proposition 5.8.2, page 69 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Write \(\mathrm{QRH}(\alpha)\) for the statement that every nontrivial zero of \(\zeta\) has real part at most \(\alpha\), \(\frac12\le\alpha<1\). Let \(\psi\) be the Chebyshev function, \(\Lambda\) the von Mangoldt function, and \(\mu^{(\beta)}\) the base point law with Laplace transform \(\xi(\beta)/\xi(\beta+\sqrt s)\). For \(\frac12\le\alpha<1\) the following are equivalent. (1) \(\mathrm{QRH}(\alpha)\). (2) \(\psi(x)=x+O(x^{\alpha+\varepsilon})\) for every \(\varepsilon>0\). (3) The Dirichlet series \(D(s)=\sum_{n\ge1}(\Lambda(n)-1)n^{-s}\) converges for \(\operatorname{Re}s>\alpha\), so that \(-\zeta'/\zeta=\zeta+D\) there. (4) \(\operatorname{Re}\xi'/\xi(s)>0\) for \(\operatorname{Re}s>\alpha\). (5) The base point laws \(\mu^{(\beta)}\) are GGCs, that is diffusion hitting times, for every \(\beta>\alpha\).

Earlier editions: Proposition 24.15.2, page 897 (edition 1).

Source: ch/ht4.tex, label ht:prop:qrh.

The function 1/xi(beta+w) is never a Laplace transform of a positive measure in w

Volume I, Lemma 5.8.3, page 70 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function. For no real \(\beta\) is \(w\mapsto1/\xi(\beta+w)\) the one or two sided Laplace transform of a positive measure.

Earlier editions: Lemma 24.15.3, page 898 (edition 1).

Source: ch/ht4.tex, label ht:lem:landau.

Prime built GGC hitting times converge to the base point laws for theta < beta < 2 - theta under QRH(theta)

Volume I, Theorem 5.8.4, page 71 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the quasi Riemann hypothesis QRH(\(\theta\)) for some \(\theta<1\) (every nontrivial zero has real part at most \(\theta\)).

Statement. Assume \(\mathrm{QRH}(\theta)\) for some \(\theta<1\) (every nontrivial zero of \(\zeta\) has real part at most \(\theta\)) and let \(\theta<\beta<2-\theta\). Let \(w(u)=(1-u)_+^3\) with Mellin transform \(W(z)=6/(z(z+1)(z+2)(z+3))\), and for \(\beta>\theta\) let \(d_X(t)=\frac1\pi(\operatorname{Re}A(\beta+it)-\operatorname{Re}\sum_{n\le X}\Lambda(n)n^{-\beta-it}(1-n/X)^3+\operatorname{Re}X^{1-\beta-it}W(1-\beta-it))\), where \(A\) is the archimedean (gamma function) term of the explicit formula as defined in the book, so that \(d_X\) uses only the primes up to \(X\). Then \(|d_X(t)-\pi^{-1}\operatorname{Re}\xi'/\xi(\beta+it)|\ll_\beta X^{\theta-\beta}\log(t+2)\) uniformly in \(t\), and there is \(X_0(\beta,\theta)\) such that for \(X\ge X_0\) the function \(d_X\) is nonnegative. The GGC laws \(\mu^{(\beta)}_X\) with Thorin density \(d_X\) in the variable \(z=t^2\) are then hitting times built from the primes up to \(X\), and \(\mu^{(\beta)}_X\Rightarrow\mu^{(\beta)}\) as \(X\to\infty\), where \(\mu^{(\beta)}\) has Laplace transform \(\xi(\beta)/\xi(\beta+\sqrt s)\). Conversely, if for some \(\beta\) and every \(t\) the values \(d_X(t)\) are nonnegative for all large \(X\), then \(\mathrm{QRH}(\beta)\) holds. The unconditional range \(1<\beta<2\) is not covered.

Earlier editions: Theorem 24.15.4, page 899 (edition 1).

Source: ch/ht4.tex, label ht:thm:primebuilt.

Volume I, Chapter 6. Riemann’s continuation and the generalised gamma convolution question

Riemann’s variable V as a GGC with a signed Gamma mixture density

Volume I, Proposition 6.1.1, page 76 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(V\) be the positive random variable of Riemann’s functional equation, with \(\mathbb E V^{s/2}=2\xi(s)\). (i) (Williams; Biane, Pitman and Yor.) \(V=\sum_{n\ge1}\Gamma_n/(\pi n^2)\) with independent Gamma(2) variables \(\Gamma_n\), so \(V\) is a GGC with Thorin measure \(2\sum_n\delta_{\pi n^2}\). (ii) The density of \(V\) is \(f_V(x)=\sum_{n\ge1}\big[4\,g_2(x;\pi n^2)-6\,g_1(x;\pi n^2)\big]\) for \(x>0\), with \(g_j(x;r)\) the Gamma(\(j\)) density of rate \(r\); this is a signed mixture of Gamma(2) and exponential densities over the rates \(\pi n^2\), each term has mass \(-2\), and the masses sum to \(1\) under zeta regularisation since \(-2\zeta(0)=1\). (iii) For \(\operatorname{Re}s>1\), \(\mathbb E V^{s/2}=\sum_{n\ge1}(\pi n^2)^{-s/2}\big(4\Gamma(2+\tfrac s2)-6\Gamma(1+\tfrac s2)\big)=s(s-1)\pi^{-s/2}\Gamma(\tfrac s2)\zeta(s)=2\xi(s)\). (iv) \(V\) is not a scale mixture of Gamma(\(a\)) laws for any \(a>0\); its density is positive, and the negative terms of the series for \(f_V\) cannot be absorbed into a positive mixture.

Source: ch/nc_riemanncont.tex, label riemanncont:prop:riemann.

A GGC with an entire Laplace transform is constant, so X squared and |X| are not GGC

Volume I, Proposition 6.2.1, page 79 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A GGC whose Laplace transform extends to an entire function is a constant. In particular neither \(X^2\) nor \(|X|\) is a GGC, for the Pólya variable \(X\). So Bondesson’s product theorem does not apply to the factorisation \(\tfrac12X^2\cdot C\) of the law whose Laplace transform is the arithmetic mean \(\mathbb E\operatorname{sech}(\sqrt uX)\).

Source: ch/nc_riemanncont.tex, label riemanncont:prop:entire.

Kuiper variable times a normal is a sum of two logistic variables with xi moments

Volume I, Proposition 6.4.1, page 80 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(V\) be Kuiper’s variable (the range of a standard Brownian bridge), \(N\) a standard normal variable independent of \(V\), and \(L_1,L_2\) independent standard logistic variables. Then \(2VN\overset d=L_1+L_2\), its L'evy density is \(2/(|y|(e^{|y|}-1))\), and \(\mathbb{E}|L_1+L_2|^s=\frac2{\sqrt\pi}(4\pi)^{s/2}\Gamma\big(\frac{s+1}2\big)\xi(s)\).

Source: ch/nc_riemanncont.tex, label riemanncont:prop:twologistic.

Positive Thorin density m at the centre with no hypothesis on the zeros

Volume I, Theorem 6.5.1, page 81 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_k=\tfrac12+\delta_k+i\gamma_k\) be the nontrivial zeros with \(\gamma_k>0\) (conjugate pairs taken together), \(a_k=-(\rho_k-\tfrac12)^2\), \(c_k=\sqrt{a_k}\) with positive real part (so \(c_k=\gamma_k-i\delta_k\)), \(K(x)=\sum_ke^{-c_kx}\) and \(P(s)=\prod_k(1+s/c_k)e^{-s/c_k}\). With no hypothesis on the zeros, let \(m(\theta)=\frac1\pi\int_0^\infty(1-\cos x\theta)K(x)\,dx=\frac1\pi\sum_k\frac{\theta^2}{c_k(c_k^2+\theta^2)}\) for \(\theta>0\). (i) \(m(\theta)>0\) for every \(\theta>0\), and \(u\mapsto P(\sqrt u)\) is the Laplace transform of a GGC whose Thorin measure has density \(m\) in the chart \(\theta=\sqrt z\): \(-\log P(s)=\int_0^\infty\log\big(1+s^2/\theta^2\big)m(\theta)\,d\theta\) for \(s\ge0\). Moreover \(P(s)P(-s)=\xi(\tfrac12+is)/\xi(\tfrac12)\), and \(1/P(s)=\mathbb E e^{-sA}\) for the law \(A\) of the book’s half line proposition. (ii) The function \(M(w)=\pi m(\sqrt w)/w=\sum_k\big(c_k(w+a_k)\big)^{-1}\) extends to a meromorphic function on \(\mathbb C\), with simple poles at the points \(-a_k\) of simple zeros and residues \(1/c_k\) there. (iii) RH holds if and only if \(M\) is a Stieltjes function, that is \(M(w)=\int\nu(dz)/(w+z)\) for \(w>0\) with a positive measure \(\nu\). In that case \(\nu=\sum_{\gamma>0}\gamma^{-1}\delta_{\gamma^2}\), which is \(z^{-1/2}U_{\mathrm{RH}}(dz)\) with \(U_{\mathrm{RH}}=\sum_{\gamma>0}\delta_{\gamma^2}\).

Source: ch/nc_riemanncont.tex, label riemanncont:thm:half.

The de Bruijn tilt as a normal mixture of Esscher tilts, and the failure of the weight e to the minus |u|

Volume I, Proposition 6.6.1, page 83 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi\) be the Pólya density, so that \(\int_{\mathbb R}\Phi(u)\cosh(su)\,du=\xi(\tfrac12+s)\) and \(\int_{\mathbb R}\Phi(u)e^{izu}\,du=\Xi(z)=\xi(\tfrac12+iz)\), and let \(N\) be standard normal. (i) For \(\lambda>0\), \(e^{\lambda u^2}=\mathbb E\cosh(\sqrt{2\lambda}\,Nu)\), and for every \(s,z\in\mathbb C\), \(\int_{\mathbb R}\Phi(u)e^{\lambda u^2}\cosh(su)\,du=\mathbb E\,\xi\big(\tfrac12+s+\sqrt{2\lambda}N\big)\) and \(\int_{\mathbb R}\Phi(u)e^{\lambda u^2}e^{izu}\,du=\mathbb E\,\Xi\big(z+i\sqrt{2\lambda}N\big)\); the de Bruijn tilt is a normal scale mixture of Esscher tilts, a Gaussian average of the basepoint moves \(\alpha=\tfrac12+\sqrt{2\lambda}N\). (ii) For \(b>0\), \(e^{-bu^2}=\mathbb E\cos(\sqrt{2b}\,Nu)\) and \(\int_{\mathbb R}\Phi(u)e^{-bu^2}e^{izu}\,du=\mathbb E\,\Xi(z+\sqrt{2b}N)\), a horizontal Gaussian smoothing of \(\Xi\). (iii) \(e^{-|u|}=\mathbb E\big[(\pi\tau)^{-1/2}e^{-u^2/(4\tau)}\big]\), so \(e^{-|u|}\Phi\) is a positive mixture, over \(b=1/(4\tau)\), of the damped kernels of (ii), and \(F(z)=\int_{\mathbb R}\Phi(u)e^{-|u|}e^{izu}\,du=\frac1\pi\int_{\mathbb R}\frac{\Xi(z-y)}{1+y^2}\,dy\). (iv) \(F(x)=2\Phi(0)/x^2+O(x^{-4})\) as \(x\to\infty\) along the real axis, so \(F\) has only finitely many real zeros.

Source: ch/nc_riemanncont.tex, label riemanncont:prop:tilts.

Question: is the Laplace transform E sech(sqrt(u) X) a GGC transform

Volume I, Problem 6.1, page 86 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is \(u\mapsto\mathbb E\operatorname{sech}(\sqrt u\,X)\), the Laplace transform of \(\tfrac12X^2C\) (with \(X\) the Pólya variable), a GGC transform? Bondesson’s product theorem does not apply to the factorisation \(\tfrac12X^2\cdot C\), since \(X^2\) is not a GGC. An answer would not bear on RH directly, but it would show whether the arithmetic mean side of Jensen’s inequality keeps the Thorin property.

Source: ch/nc_riemanncont.tex, label riemanncont:prob:arith.

Question: a class of mixing laws whose harmonic mean of GGC transforms is a GGC transform

Volume I, Problem 6.2, page 86 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Find a class \(\mathcal C\) of mixing laws for which the harmonic mean \(1/\mathbb E[1/\varphi_X(u)]\) of GGC transforms \(\varphi_x\) is again a GGC transform, and which is not already explained by Pólya’s theorems on real zeros. Any such class containing the Pólya density must use the Euler product, since otherwise the Davenport and Heilbronn function refutes it.

Source: ch/nc_riemanncont.tex, label riemanncont:prob:hm.

Question: a quantitative lower bound for Im M(-x - i eta)

Volume I, Problem 6.3, page 86 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The density \(m\) of the half line theorem is positive with no hypothesis, and its continuation \(M(w)\) has the poles \(-a_k\). Is there a quantitative statement, in the spirit of the book’s strip theorem (part 3), that bounds \(\operatorname{Im}M(-x-i\eta)\) from below on \(x>0\), for some \(\eta>0\), in terms of computable data and a zero free region? Such a bound is a positivity statement at a level below one, so it would be at least as strong as a zero free half plane.

Source: ch/nc_riemanncont.tex, label riemanncont:prob:half.

Question: even weights w for which w times Phi has a Fourier transform with only real zeros

Volume I, Problem 6.4, page 86 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For which even weights \(w\) with \(w(0)=1\), other than \(\mathbb E\cosh(uB)\) (a mixture of cosh factors), does \(w\Phi\) have a Fourier transform with only real zeros under RH? The weight \(e^{-|u|}\) fails, and the same argument covers every weight that is smooth on \([0,\infty)\) with \(w'(0+)\ne0\). Is smoothness at the origin together with \(\log w\) convex sufficient?

Source: ch/nc_riemanncont.tex, label riemanncont:prob:tilt.

Volume I, Chapter 7. The reciprocal Xi function as an extended generalised gamma convolution

The z distribution as a normal variance mean mixture and Laplace sum

Volume I, Proposition 7.1.1, page 88 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\sim z(a,b)\), the law of \(\log(G_a/G_b)\) for independent gamma variables of shapes \(a\) and \(b\). (i) For \(-a<\operatorname{Re}s<b\), \(\mathbb E e^{sX}=B(a+s,b-s)/B(a,b)=\Gamma(a+s)\Gamma(b-s)/(\Gamma(a)\Gamma(b))\). (ii) For the same \(s\), \(\mathbb E e^{sX}=\prod_{k\ge0}\frac{(a+k)(b+k)}{(a+k+s)(b+k-s)}\). (iii) \(X\overset d=\mu W+\sqrt W\,Z\) with \(\mu=\tfrac{a-b}2\), \(Z\) standard normal and \(W\) independent of \(Z\) with \(W=\sum_{k\ge0}\frac{2E_k}{(a+k)(b+k)}\), the \(E_k\) independent standard exponential variables; so \(W\) is a GGC with unit Thorin atoms at the points \(\tfrac12(a+k)(b+k)\), and \(X\) is a normal variance mean mixture. (iv) In the symmetric case \(a=b=c\), \(X\overset d=\sum_{k\ge0}L_k/(c+k)\) with independent standard Laplace variables \(L_k\), and \(X\) is a symmetric EGGC with Thorin measure \(\sum_{k\ge0}\delta_{c+k}\) in the chart \(\theta\); its characteristic function is \(|\Gamma(c+i v)|^2/\Gamma(c)^2\).

Source: ch/nc_zdist.tex, label zdist:prop:mix.

Meixner duality for the symmetric z law

Volume I, Proposition 7.2.1, page 89 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\), the density of \(z(c,c)\) is \(f_{c,c}(x)=(2\cosh\tfrac x2)^{-2c}/B(c,c)\), a constant multiple of the characteristic function of the Meixner law of order \(c\), and the characteristic function of \(z(c,c)\) is \(|\Gamma(c+iv)|^2/\Gamma(c)^2\), a constant multiple of the Meixner density \(m_c(v)\). Both laws are symmetric EGGC laws: \(z(c,c)\) with Thorin measure \(\sum_{k\ge0}\delta_{c+k}\) and the Meixner law with Thorin measure \(2c\sum_{k\ge0}\delta_{(2k+1)\pi}\), both in the chart \(\theta\). For \(c=\tfrac12\) the two coincide up to scale with the hyperbolic secant law.

Source: ch/nc_zdist.tex, label zdist:prop:dual.

Van Dantzig pairs for the symmetric z law exist exactly when c is at least one half

Volume I, Proposition 7.2.2, page 90 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(c>0\) and \(\varphi_2(v)=|\Gamma(c+iv)|^2/\Gamma(c)^2\), the characteristic function of \(z(c,c)\). There is a characteristic function \(\varphi_1\) with \(\varphi_2(\theta)=1/\varphi_1(i\theta)\) for all real \(\theta\) if and only if \(c\ge\tfrac12\). Then \(\varphi_1(v)=\Gamma(c)^2/(\Gamma(c+v)\Gamma(c-v))\); for \(c>\tfrac12\) it is the characteristic function of \(2\Theta\), where \(\Theta\) has density proportional to \(\cos^{2c-2}\theta\) on \((-\tfrac\pi2,\tfrac\pi2)\), and for \(c=\tfrac12\) it is \(\cos(\pi v)\), the characteristic function of a symmetric sign times \(\pi\).

Source: ch/nc_zdist.tex, label zdist:prop:vd.

The archimedean factor of xi squared is a z law, and T plus T tilde does not contain H infinity

Volume I, Proposition 7.3.1, page 91 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\sim z(\tfrac14,\tfrac14)\), \(\xi(\tfrac12)/\xi(\tfrac12+u)=\mathbb E e^{-u^2T}\) with \(\tilde T\) an independent copy of \(T\), \(Y=Z\sqrt{2T}\) with \(Y'\) an independent copy, and \(H_\infty\) the GGC with unit Thorin atoms at \(c_k^2\), \(c_k=2k+\tfrac12\). (i) For real \(|s|<\tfrac12\), \(\frac{\xi(\frac12+s)^2}{\xi(\frac12)^2}=(1-4s^2)^2\,\mathbb E e^{sX/2}\,\frac{\zeta(\frac12+s)\zeta(\frac12-s)}{\zeta(\frac12)^2}\), and on the critical line, for real \(v\), \(\frac{\Xi(v)^2}{\Xi(0)^2}=(1+4v^2)^2\,\mathbb E e^{ivX/2}\,\frac{|\zeta(\frac12+iv)|^2}{\zeta(\frac12)^2}\). (ii) Let \(D_0=T+\tilde T-H_\infty\), with \(T\), \(\tilde T\) and \(H_\infty\) independent. For \(0\le w<\tfrac14\), \(\mathbb E e^{-wD_0}=\frac{\zeta(\frac12)^2}{(1-4w)^2\,\zeta(\frac12+\sqrt w)\,\zeta(\frac12-\sqrt w)}\). (iii) \(\mathbb E D_0=2\mathbb E T-\tfrac14\psi'(\tfrac14)=-8+(\log\zeta)''(\tfrac12)=-4.2531223024\). Consequently there is no law \(Q\) on \([0,\infty)\), independent of \(H_\infty\), with \(T+\tilde T\overset d=H_\infty+Q\), and no random variable \(R\) independent of \(X\) with \(Y+Y'\overset d=X/2+R\).

Source: ch/nc_zdist.tex, label zdist:prop:arch.

Sine squared image of the z law Lévy density and the Gamma ratio as a Cauchy factor times an EGGC

Volume I, Proposition 7.5.1, page 93 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. (i) The law \(z(c,c)\) has Lévy density \(k_c(|x|)/|x|\) with \(k_c(r)=e^{-cr}/(1-e^{-r})=\sum_{n\ge0}e^{-(c+n)r}\), completely monotone, and the sine squared image of \(k_c\), \(\frac1\pi\int_0^\infty2\sin^2(x\theta/2)\,k_c(x)\,dx\), is \(\frac1\pi\sum_n\theta^2/\big((c+n)((c+n)^2+\theta^2)\big)\). (ii) For \(a>0\) and real \(u\), \(e^{\psi(a)|u|}\frac{\Gamma(a)}{\Gamma(a+|u|)}=\exp\Big\{-\frac1\pi\int_0^\infty\log\Big(1+\frac{u^2}{\theta^2}\Big)\big(\operatorname{Re}\psi(a+i\theta)-\psi(a)\big)\,d\theta\Big\}\), and \(\operatorname{Re}\psi(a+i\theta)-\psi(a)>0\) for \(\theta\ne0\). So the left side is the characteristic function of a symmetric EGGC, and \(\Gamma(a)/\Gamma(a+|u|)\) is the product of \(e^{-\psi(a)|u|}\), a Cauchy factor when \(\psi(a)>0\) (that is \(a>1.4616\)), and that EGGC.

Source: ch/nc_zdist.tex, label zdist:prop:sin2.

The heat flow of laws Y_t is a symmetric EGGC exactly for t at least the de Bruijn Newman constant

Volume I, Theorem 7.6.1, page 94 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda\) be the de Bruijn Newman constant, \(H_t(z)=\int_0^\infty e^{tw^2}\Phi_{\mathrm{dBN}}(w)\cosh(zw)\,dw\), \(\xi_t(s)=8H_t(-2i(s-\tfrac12))\) and \(R_t(u)=H_t(0)/H_t(2i\sqrt u)\), so that \(\xi_0=\xi\) and \(R_t(v^2)=\xi_t(\tfrac12)/\xi_t(\tfrac12+v)\) for real \(v\). (i) For every real \(t\), \(H_t(iy)>0\) for real \(y\). If \(R_t\) is the Laplace transform of a law \(T_t\) on \([0,\infty)\), then \(v\mapsto R_t(v^2)\) is the characteristic function of the symmetric law \(Y_t=Z\sqrt{2T_t}\). (ii) For \(t\ge\Lambda\) the law \(T_t\) exists, and \(v\mapsto R_t(v^2)\) is the characteristic function of a symmetric EGGC if and only if \(t\ge\Lambda\), and then \(Y_t\overset d=\sum_kL_k/\theta_k(t)\) over the positive zeros \(\tfrac12\pm i\theta_k(t)\) of \(\xi_t\). In particular \(Y_t\) is an EGGC for every \(t\ge0.2\), with no hypothesis, and \(Y_0=Y\) is an EGGC if and only if RH holds. (iii) With \(u_t\) the mode of the book’s large \(t\) proposition, \(Y_t/u_t\Rightarrow\tfrac2\pi X_{1/2}\) as \(t\to\infty\), where \(X_{1/2}\sim z(\tfrac12,\tfrac12)\); the limit has characteristic function \(1/\cosh(2v)\).

Source: ch/nc_zdist.tex, label zdist:thm:flow.

A limit of symmetric EGGC laws equal to Y forces RH

Volume I, Proposition 7.7.1, page 95 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Y=Z\sqrt{2T}\) with \(\mathbb E e^{-u^2T}=\xi(\tfrac12)/\xi(\tfrac12+u)\). If there are symmetric EGGC laws \(Y_n\) with \(Y_n\Rightarrow Y\), then RH holds. Conversely, under RH the constant family \(Y_n=Y\) is such a family.

Source: ch/nc_zdist.tex, label zdist:prop:closure.

A truncated EGGC law within v to the fourth times epsilon of the law of Y

Volume I, Proposition 7.7.2, page 96 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on All zeros of the zeta function with ordinate at most \(T_0=3\times10^{12}\) lie on the critical line, read as quoted from Platt and Trudgian (PT21); this is a finite computation by others that the book does not repeat.

Statement. Let \(T_0=3\times10^{12}\), the height below which every zero lies on the critical line, and for \(2\le T'\le T_0\) put \(m(T')=\sum_{k:\ \gamma_k>T'}1/a_k\) and \(Y^{(T')}=\sqrt{2m(T')}\,Z_0+\sum_{0<\gamma\le T'}L_\gamma/\gamma\), with \(a_k=-(\rho_k-\tfrac12)^2\) summed over the zeros with \(\gamma_k>T'\), \(Z_0\) standard normal and the \(L_\gamma\) standard Laplace, all independent. Then \(m(T')>0\), \(Y^{(T')}\) is a symmetric EGGC, \(\operatorname{Var}Y^{(T')}=\operatorname{Var}Y=2\mathbb E T\), and for real \(v\) with \(v^2\le T'^2/2\), \(\big|\log\mathbb E e^{ivY}-\log\mathbb E e^{ivY^{(T')}}\big|\le v^4\,\epsilon(T')\) with \(\epsilon(T')=\frac2\pi\Big(\frac{\log T'}{3T'^3}+\frac1{9T'^3}\Big)\).

Source: ch/nc_zdist.tex, label zdist:prop:trunc.

Question: is the BNKS mixing law the GGC of the z law proposition

Volume I, Problem 7.1, page 97 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Read the paper of Barndorff-Nielsen, Kent and Sørensen on the \(z\) laws and determine whether its mixing law for \(z(a,b)\) is the GGC of the book’s mixture proposition, part (iii), and whether its results on the asymmetric laws, \(\mu\ne0\), have an analogue for the tilted Pólya variables \(V_\alpha\), whose characteristic functions \(\xi(\alpha+i\theta)/\xi(\alpha)\) are never infinitely divisible.

Source: ch/nc_zdist.tex, label zdist:prob:bnks.

Question: positive sequences c_k for which the sum of L_k over c_k is a van Dantzig second member

Volume I, Problem 7.2, page 97 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The van Dantzig proposition decides the property for the arithmetic progressions \(c_k=c+k\). Characterise the positive sequences \((c_k)\) with \(\sum c_k^{-2}<\infty\) for which \(\sum_kL_k/c_k\) is the second member of a van Dantzig pair. Under RH the ordinates \(\gamma_k\) are such a sequence, with first member \(\Xi(v)/\Xi(0)\); the condition is that \(\prod_k(1-v^2/c_k^2)\) be a characteristic function.

Source: ch/nc_zdist.tex, label zdist:prob:vd.

Question: an unconditional EGGC family from the z law to Y other than the heat flow

Volume I, Problem 7.3, page 97 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is there a one parameter family of symmetric laws, each an EGGC with no hypothesis, joining \(\tfrac2\pi X_{1/2}\) to \(Y\) other than by the heat flow, along which the EGGC property can be followed by a computation that does not reduce to the reality of the zeros of \(H_t\)? The closure proposition implies that such a family cannot reach \(Y\) unconditionally, so the question is which property of the family degenerates at the endpoint.

Source: ch/nc_zdist.tex, label zdist:prob:interp.

Question: is the Lévy density K_t(y) of Y_t monotone in t

Volume I, Problem 7.4, page 98 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is the Lévy density \(K_t(|y|)/|y|\) of \(Y_t\), with \(K_t(y)=\sum_ke^{-\theta_k(t)y}\) for \(t\ge\Lambda\), monotone in \(t\) for each fixed \(y\)? The same question for the stochastic order of \(T_t\) is a problem of an earlier chapter; for \(Y_t\) it would say that the flow moves Lévy mass monotonically between scales.

Source: ch/nc_zdist.tex, label zdist:prob:flow.

Question: the set of lambda for which the ratio of transforms of T plus T tilde and H lambda is a Laplace transform

Volume I, Problem 7.5, page 98 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The variable \(D_0\) of the archimedean proposition has negative mean, so \(T+\tilde T\) does not contain \(H_\infty\) as an independent summand. For \(0\le\lambda\le1\) let \(H^{(\lambda)}\) be the GGC with Thorin atoms of mass \(\lambda\) at the points \(c_k^2\). For which \(\lambda\) is \(\mathbb E e^{-w(T+\tilde T)}/\mathbb E e^{-wH^{(\lambda)}}\) the Laplace transform of a law on \([0,\infty)\)? The mean gives the necessary condition \(\lambda\le2\mathbb E T/\mathbb E H_\infty=0.01075\), and at \(\lambda=0\) the answer is yes. Is the set of such \(\lambda\) an interval \([0,\lambda^*]\) with \(\lambda^*>0\), and what is \(\lambda^*\)?

Source: ch/nc_zdist.tex, label zdist:prob:remainder.

Volume I, Chapter 8. The heat trace and the hierarchy of positivity

The clock is infinitely divisible and self decomposable, unconditionally

Volume I, Theorem 8.1.1, page 100 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), and let \(W(t)=\sum_je^{-a_jt}\) be the heat trace, where \(a_j=-(\rho_j-\tfrac12)^2\) and \(\rho_j\) runs over one zero of each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\), counted with multiplicity. Unconditionally, that is, using only the verification of RH below height \(1000\) (Platt and Trudgian 2021), \(W(t)>0\) and \(W'(t)<0\) for every \(t>0\). Consequently the clock \(\xi(\tfrac12)/\xi(\tfrac12+\sqrt u)=\exp\bigl(-\int_0^\infty(1-e^{-ut})\,W(t)\,dt/t\bigr)\) is the Laplace transform of an infinitely divisible, self decomposable law on \([0,\infty)\) with Lévy density \(W(t)/t\), and \(\bigl(\Xi(\theta)/\Xi(0),\ \xi(\tfrac12)/\xi(\tfrac12+\theta)\bigr)\) is a van Dantzig pair, that is, a pair \((\varphi_1,\varphi_2)\) of characteristic functions with \(\varphi_2(\theta)=1/\varphi_1(i\theta)\).

Earlier editions: Theorem 5.1.1, page 95 (edition 1).

Source: ch/ch12.tex, label thm:ch12:idsd.

Complete asymptotic expansion of the heat trace W(t) at t = 0 with the coefficients d_n and b_k, and the logarithmic divergence of the clock

Volume I, Result 8.1.1, page 100 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\vartheta\) be the Riemann Siegel theta function, \(\Lambda\) the von Mangoldt function. With no hypothesis \(W(t)=e^{t/4}+\frac1\pi\int_0^\infty e^{-u^2t}\vartheta'(u)\,du-\frac1{\sqrt{4\pi t}}\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}e^{-(\log n)^2/4t}\), the prime terms are \(O(t^{-1/2}e^{-(\log2)^2/4t})\), and for every \(N\), as \(t\downarrow0\), \(W(t)=\frac{\log\frac1t-\gamma_E-4\log2-2\log\pi}{8\sqrt{\pi t}}+\sum_{n=0}^Nd_nt^n+\sum_{k=1}^Nb_kt^{k-1/2}+O(t^{N+1/2})\), with \(d_n=(1-\frac{E_{2n}}8)\frac{4^{-n}}{n!}\), \(b_k=\frac{(1-2k)\Gamma(\frac12-k)}{2\pi}\beta_k\), \(E_{2n}\) the Euler numbers and \(\beta_k=\frac{(1-2^{1-2k})|B_{2k}|}{4k(2k-1)}\) the coefficients of Stirling’s series for \(\vartheta\). After the leading term the series is \(\frac78+\frac{\sqrt t}{48\sqrt\pi}+\frac9{32}t-\frac{7t^{3/2}}{2880\sqrt\pi}+\frac3{256}t^2+\frac{31t^{5/2}}{60480\sqrt\pi}+\frac{69}{3072}t^3+\cdots\), and it diverges. Consequently \(\int_\tau^\infty W(t)\,dt/t=\frac{\log\frac1\tau-\gamma_E-4\log2-2\log\pi-2}{4\sqrt{\pi\tau}}+\frac78\log\frac1\tau+O(1)\) as \(\tau\downarrow0\), so the clock has infinitely many jumps. The two term version is in Polson 2017, Corollary 5.5; the full series is checked numerically through order \(t^{5/2}\) in a 40 digit computation for \(10^{-6}\le t\le10^{-3}\).

Earlier editions: Result 5.1.1, page 94 (edition 1).

Source: ch/ch12.tex, label tx3:res:wzero.

Truncation error of the residues of the reciprocal clock when the product over zeros is cut after K factors

Volume I, Proposition 8.1.3, page 103 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a_j=-(\rho_j-\frac12)^2\) for one zero \(\rho_j\) of each pair \(\{\rho,1-\rho\}\) of nontrivial zeros of the Riemann zeta function, ordered by increasing modulus, assumed simple, so that the \(a_j\) are distinct, and let \(G(u)=\prod_j(1+u/a_j)\) be the clock function \(\xi(\frac12+\sqrt u)/\xi(\frac12)\) with \(\xi\) the completed zeta function. Let \(r_k\) and \(r_k^{(K)}\) be the residues at \(u=-a_k\), \(k\le K\), of \(1/G(u)=\prod_ja_j/(a_j+u)\) and of its product over \(j\le K\). Then \(r_k=a_k\prod_{j\ne k}a_j/(a_j-a_k)\) and, when \(|a_j|>|a_k|\) for \(j>K\), \(\log\frac{r_k^{(K)}}{r_k}=\sum_{j>K}\log\big(1-\frac{a_k}{a_j}\big)=-a_k\sum_{j>K}a_j^{-1}-\frac{a_k^2}{2}\sum_{j>K}a_j^{-2}-\cdots\). Under the Riemann Hypothesis, \(a_j=\gamma_j^2\) for the ordinates \(\gamma_j\) and \(\sum_{j>K}\gamma_j^{-2}\sim\log\gamma_K/(2\pi\gamma_K)\), so the first term is \(-\gamma_k^2\log\gamma_K/(2\pi\gamma_K)\): the truncated residue is close to the true one for \(\gamma_k\le(\gamma_K/\log\gamma_K)^{1/2}\), and not beyond.

Earlier editions: Proposition 5.1.2, page 97 (edition 1).

Source: ch/ch12.tex, label tx:prop:trunc.

Hankel positivity of the heat trace at one time is equivalent to RH

Volume I, Theorem 8.1.4, page 104 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Let \(W(t)=\sum_je^{-a_jt}\) be the heat trace of the completed Riemann zeta function \(\xi\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity) and \(c_k(t)=(-1)^kW^{(k)}(t)=\sum_ja_j^ke^{-a_jt}\). For any fixed \(t>0\) the following are equivalent: both infinite Hankel matrices \([c_{i+j}(t)]_{i,j\ge0}\) and \([c_{i+j+1}(t)]_{i,j\ge0}\) are positive semidefinite; Weil’s quadratic form \(f\mapsto\sum_\rho\hat f(\gamma_\rho)\overline{\hat f(\bar\gamma_\rho)}\), where \(\rho=\tfrac12+i\gamma_\rho\) runs over the nontrivial zeros of \(\zeta\) and \(\hat f(r)=\int f(x)e^{irx}\,dx\), is positive semidefinite on the span of the test functions \(f\) with \(\hat f(r)=r^me^{-r^2t/2}\), \(m=0,1,2,\dots\); the Riemann Hypothesis.

Earlier editions: Theorem 5.1.3, page 98 (edition 1).

Source: ch/ch12.tex, label thm:ch12:hankel.

RH is equivalent to boundedness of the heat trace of the zeros on a vertical line

Volume I, Theorem 8.1.5, page 104 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_j=\frac12+\delta_j+i\gamma_j\) be the nontrivial zeros of the Riemann zeta function, one from each pair \(\{\rho,1-\rho\}\), counted with multiplicity, and put \(a_j=-(\rho_j-\frac12)^2\) and \(W(t)=\sum_je^{-a_jt}\) for \(t>0\), extended to complex \(t=\sigma+iv\) with \(\sigma>0\) (the series converges absolutely there). Then the Riemann Hypothesis holds if and only if \(v\mapsto W(\sigma+iv)\) is bounded on \(\mathbb{R}\), for one, equivalently every, \(\sigma>0\). Under RH the bound is \(|W(\sigma+iv)|\le W(\sigma)\) for every \(\sigma>0\) and every real \(v\).

Earlier editions: Theorem 5.1.4, page 98 (edition 1).

Source: ch/ch12.tex, label tx:thm:vertical.

Positive definite Hankel blocks of the heat trace up to size 3 x 10^7

Volume I, Theorem 8.1.6, page 105 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(W(t)=\sum_je^{-a_jt}\) its heat trace (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity) and \(c_k(t)=(-1)^kW^{(k)}(t)\), so that the two Hankel matrices of the heat trace at time \(t\) are \([c_{i+j}(t)]_{i,j\ge0}\) and \([c_{i+j+1}(t)]_{i,j\ge0}\). For \(N\ge1\) let \(T_N=20000N\sqrt{1+\log N}\). If all zeros of \(\xi\) below height \(T_N\), that is, with ordinate in \((0,T_N)\), lie on the critical line, then the leading \(N\times N\) blocks \([c_{i+j}(t)]_{i,j<N}\) and \([c_{i+j+1}(t)]_{i,j<N}\) are positive definite at every \(t>0\). By the verification of the Riemann Hypothesis to height \(3\times10^{12}\) (Platt and Trudgian 2021) this holds unconditionally for every \(N\le3\times10^7\).

Earlier editions: Theorem 5.1.5, page 100 (edition 1).

Source: ch/ch12.tex, label thm:ch12:blocks.

Derivative signs of the heat trace up to order 10^17

Volume I, Theorem 8.1.7, page 106 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Let \(W(t)=\sum_je^{-a_jt}\) be the heat trace of the completed Riemann zeta function \(\xi\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity). Unconditionally, that is, using only the verification of RH below height \(10^{12}\) (Platt and Trudgian 2021), \((-1)^kW^{(k)}(t)>0\) for every \(t>0\) and every integer \(0\le k\le10^{17}\).

Earlier editions: Theorem 5.1.6, page 100 (edition 1).

Source: ch/ch12.tex, label thm:ch12:cm.

Hankel matrices of Stieltjes moments count the zeros off the critical line (eigenvalue form of the Grommer and Chebotarev criterion)

Volume I, Theorem 8.1.8, page 108 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(W(t)=\sum_je^{-a_jt}\) be the heat trace of the completed Riemann zeta function \(\xi\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity), let \(m_n=\frac1{n!}\int_0^\infty t^nW(t)\,dt=\sum_ja_j^{-n-1}\) be its Stieltjes moments, and for \(N\ge1\) let \(\mathcal H_N=[m_{i+j}]_{i,j<N}\) and \(\mathcal H'_N=[m_{i+j+1}]_{i,j<N}\) be the Hankel matrices, with \(\iota_N\) and \(\iota'_N\) the numbers of negative eigenvalues of \(\mathcal H_N\) and \(\mathcal H'_N\). Let \(K\in\{0,1,2,\dots\}\cup\{\infty\}\) be the number of distinct zeros \(\rho\) of \(\xi\) with \(\operatorname{Re}\rho>\tfrac12\) and \(\operatorname{Im}\rho>0\). Then \(\iota_N\) and \(\iota'_N\) are nondecreasing in \(N\), \(\iota_N\le K\) and \(\iota'_N\le K\) for every \(N\), and \(\iota_N\to K\) and \(\iota'_N\to K\) as \(N\to\infty\). In particular RH holds if and only if \(\iota_N=0\) for every \(N\), and if it fails, \(\iota_N\ge1\) from some \(N\) on; the same holds for \(\iota'_N\).

Earlier editions: Theorem 5.1.7, page 102 (edition 1).

Source: ch/ch12.tex, label thm:ch12:inertia.

Heat trace completely monotone on (0, 1.29]; every order up to t = 0.019

Volume I, Theorem 8.1.10, page 111 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(W(t)=\sum_je^{-a_jt}\) be the heat trace of the completed Riemann zeta function \(\xi\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity). (a) For every integer \(k\ge1\) and every \(t\in(0,0.019]\), \((-1)^kW^{(k)}(t)>0\); the proof uses no information on the zeros. (b) Unconditionally, that is, using only the verification of RH below height \(10^{12}\) (Platt and Trudgian 2021), \(W\) is completely monotone on \((0,1.29]\): \((-1)^kW^{(k)}(t)>0\) for every \(t\in(0,1.29]\) and every integer \(k\ge0\).

Earlier editions: Theorem 5.1.9, page 105 (edition 1).

Source: ch/ch12.tex, label thm:ch12:cmint.

Heat trace derivative along a ray is the Gaussian smoothing of the zeros

Volume I, Proposition 8.1.11, page 115 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(W(t)=\sum_je^{-a_jt}\) its heat trace (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity). Take each \(\rho_j\) with positive imaginary part, write \(\rho_j=\tfrac12+\delta+i\gamma\) with \(\gamma>0\) and \(\delta\) real, and put \(\tilde\gamma=\gamma-i\delta\), so that \(a_j=\tilde\gamma^2\). Let \(N(r)\) be the number of zeros of \(\xi\) with ordinate in \((0,r]\), counted with multiplicity. Let \(u\ge2\), \(t>0\) and \(k=u^2t\), not necessarily an integer, let \(c_k(t)=\sum_ja_j^ke^{-a_jt}\) with \(a_j^k\) the principal power (for an integer \(k\) this is \((-1)^kW^{(k)}(t)\)), and let \(f_0=k(\log u^2-1)\). Then \(e^{-f_0}c_k(t)=\sum_{|\tilde\gamma-u|\le u/2}e^{-2t(\tilde\gamma-u)^2}\,(1+\varepsilon_\rho)+R\), where the sum runs over the zeros with \(|\tilde\gamma-u|\le u/2\), \(\varepsilon_\rho\) is the relative error of the Gaussian bump of the zero \(\rho\), \(R\) is the contribution of the other zeros, and \(|\varepsilon_\rho|\le\exp\bigl\{\tfrac{4t}{3u}|\tilde\gamma-u|^3\bigr\}-1\). With \(\ell_\pm=u\pm\tfrac12\sqrt{u^2-1}\), \(H(r)=u^2\log\frac{r^2+1/4}{u^2}-r^2+\tfrac14+u^2\) and \(\eta=1-u^2/(\ell_+^2+\tfrac14)\), which is positive, \(|R|\le N(\ell_-)\,e^{tH(\ell_-)}+\bigl(\ell_+^2+\frac1{t\eta}\bigr)e^{tH(\ell_+)}\). No hypothesis on the zeros is used.

Earlier editions: Proposition 5.1.10, page 109 (edition 1).

Source: ch/ch12.tex, label prop:ch12:ray.

RH is equivalent to nonnegativity of the Gaussian smoothing of the zeros

Volume I, Proposition 8.1.12, page 116 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function. Write each zero of \(\xi\) with positive imaginary part, counted with multiplicity, as \(\rho=\tfrac12+\delta+i\gamma\) with \(\gamma>0\) and \(\delta\) real, and put \(\tilde\gamma=\gamma-i\delta\). For real \(u\) and \(t>0\) let \(\mathcal G(u,t)=\sum e^{-2t(\tilde\gamma-u)^2}\), the sum over these zeros, the Gaussian smoothing of the zeros at resolution \(1/(2\sqrt t)\) (it is real, because zeros off the critical line occur in pairs with conjugate values of \(\tilde\gamma\)). Then the Riemann Hypothesis holds if and only if \(\mathcal G(u,t)\ge0\) for every real \(u\) and every \(t>0\).

Earlier editions: Proposition 5.1.11, page 110 (edition 1).

Source: ch/ch12.tex, label prop:ch12:gauss.

There is no Gaussian smoothing criterion for RH on a line with real part alpha above one half

Volume I, Proposition 8.1.13, page 116 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis: every nontrivial zero of the Riemann zeta function has real part \(\frac12\).

Statement. Assume the Riemann Hypothesis, and let \(\gamma>0\) run over the ordinates of the zeros \(\frac12+i\gamma\) of the Riemann zeta function, with multiplicity. For \(\frac12<\alpha<1\) let \(\eta=\alpha-\frac12\) and \(\mathcal{G}_\alpha(u,t)=\operatorname{Re}\sum_{\gamma>0}e^{-2t(\gamma+i\eta-u)^2}\) for real \(u\) and \(t>0\) (the Gaussian smoothing of the zeros at resolution \(1/(2\sqrt t)\) shifted to the line \(\operatorname{Re}s=\alpha\); it equals the plain smoothing \(\mathcal{G}\) at \(\eta=0\)). Then \(\mathcal{G}_\alpha(u,t)<0\) for some \(u\) and \(t\); in fact, with \(u=\gamma_1+\pi/(4t\eta)\) where \(\gamma_1\) is the lowest ordinate (a simple zero), \(\mathcal{G}_\alpha(u,t)=-e^{2t\eta^2}(1+o(1))<0\) as \(t\to\infty\). So there is no Gaussian criterion for RH on the line \(\operatorname{Re}s=\alpha\).

Earlier editions: Proposition 5.1.12, page 110 (edition 1).

Source: ch/ch12.tex, label tx:prop:offline.

Derivative signs of the heat trace up to order 1.16 x 10^25

Volume I, Theorem 8.1.14, page 118 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(W(t)=\sum_je^{-a_jt}\) be the heat trace of the completed Riemann zeta function \(\xi\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity). Let \(H=3\times10^{12}\), the height to which RH is verified (Platt and Trudgian 2021). Then \((-1)^kW^{(k)}(t)>0\) for every \(t\ge1.29\) and every integer \(k\) with \(0\le k\le t(H-23)^2\). Consequently \((-1)^kW^{(k)}(t)>0\) for every \(t>0\) and every integer \(k\) with \(0\le k\le1.29(H-23)^2\), and \(1.29(H-23)^2>1.16\times10^{25}\).

Earlier editions: Theorem 5.1.13, page 112 (edition 1).

Source: ch/ch12.tex, label thm:ch12:window.

Compound Poisson defect of the off line zeros: the clock plus an independent compound Poisson variable is a GGC

Volume I, Theorem 8.1.15, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(W(t)=\sum_je^{-a_jt}\) its heat trace (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity). The clock \(T\) is the nonnegative random variable with \(\mathbb{E}e^{-uT}=\xi(\tfrac12)/\xi(\tfrac12+\sqrt u)\) for \(u\ge0\); it exists with no hypothesis on the zeros and is infinitely divisible with Lévy density \(W(t)/t\). A generalized gamma convolution (GGC) with Thorin measure \(U\ge0\) and no drift is a law on \([0,\infty)\) with Laplace transform \(\exp\bigl(-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\bigr)\). Group the exponents \(a_k\) into the real ones, \(a_k=\gamma^2>0\), which belong to zeros on the critical line, and the conjugate pairs \(\{a,\bar a\}\) of nonreal exponents, one for each quadruple of zeros \(\tfrac12\pm\delta\pm i\gamma\) off the line, for which \(a=A-iB\) with \(A=\gamma^2-\delta^2\) and \(B=2\delta\gamma\). Put \(U_+=\sum_{a_k\text{ real}}\delta_{a_k}+2\sum_{\text{pairs}}\delta_A\) and \(R(t)=4\sum_{\text{pairs}}e^{-At}\sin^2(Bt/2)\). Unconditionally, that is, using only the verification of RH below height \(H=3\times10^{12}\) (Platt and Trudgian 2021) and the explicit bound for the number of zeros up to a given height of Hasanalizade, Shen and Wong (2022, Corollary 1.2), the following hold. First, \(W(t)=\int_{(0,\infty)}e^{-tz}\,U_+(dz)-R(t)\), where \(U_+\) is a Thorin measure and \(R\ge0\). Let \(T_+\) be the GGC with Thorin measure \(U_+\), and \(Y\) the infinitely divisible variable with Lévy density \(R(t)/t\), independent of \(T\). Then \(T+Y\overset{d}{=}T_+\) and \(\mathbb{E}e^{-uY}=\prod_{\text{pairs}}\frac{(1+u/a)(1+u/\bar a)}{(1+u/A)^2}\). The defect \(Y\) is compound Poisson, with rate \(\lambda=\int_0^\infty\frac{R(t)}t\,dt=\sum_{\text{pairs}}\log\bigl(1+\frac{B^2}{A^2}\bigr)\) and jump density \(R(t)/(\lambda t)\), and \(\lambda<10^{-12}\), so \(\mathbb{P}(Y=0)=e^{-\lambda}>1-10^{-12}\). The following are equivalent: \(R\equiv0\); \(Y=0\); RH holds. In that case \(U_+=\sum_k\delta_{a_k}\), the Thorin measure with one atom at each exponent \(a_k=\gamma_k^2\), and \(T_+=T\). Consequently, unconditionally, the clock plus an independent compound Poisson variable of rate below \(10^{-12}\) is a GGC, and the law of the clock is within total variation distance \(10^{-12}\) of that GGC, since \(T=T+Y\) on the event \(\{Y=0\}\). The clock is itself a GGC exactly when \(Y=0\), which is RH, and \(Y\) comes from the zeros above height \(H\) alone. RH also holds if and only if \(W\) is completely monotone, so \(R\equiv0\) says that \(W\) is the Laplace transform of the positive measure \(U_+\).

Earlier editions: Theorem 5.1.14, page 114 (edition 1).

Source: ch/ch12.tex, label xg:thm:defect.

Problem: derivative signs of the heat trace to order c H squared with c above 1.70

Volume I, Problem 8.1, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(W(t)=\sum_je^{-a_jt}\) be the heat trace of the completed Riemann zeta function \(\xi\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\)). The book proves, from the verification that every zero of \(\zeta\) of height at most \(H=3\times10^{12}\) lies on the critical line, that \((-1)^kW^{(k)}(t)>0\) for every \(t>0\) and every order \(k\le1.70(H-23)^2\). Problem: can this range be increased to \(k\le cH^2\) with an explicit constant \(c>1.70\)? In the model of four zeros of the book (a quadruple just above height \(H\)) the first change of sign occurs at an order of about \(H^2t^*\), with \(t^*\) set by the gaps around the quadruple, which suggests a scale to investigate; this is heuristic, and the model does not show that such a failure occurs for the zeros of \(\zeta\).

Earlier editions: Problem 5.1, page 113 (edition 1).

Source: ch/ch12.tex, label prob:ch12:order.

Heat trace of the zeros of xi is completely monotone on (0, 1.70] and has the right sign at every order up to 1.52e25 (certified)

Volume I, Theorem 8.1.17, page 121 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(W(t)=\sum_je^{-a_jt}\) its heat trace (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\), counted with multiplicity). Let \(H=3\times10^{12}\), the height to which RH has been verified (Platt and Trudgian 2021). Then \(W\) is completely monotone on \((0,1.70]\), that is, \((-1)^kW^{(k)}(t)>0\) for every \(0<t\le1.70\) and every integer \(k\ge0\); and \((-1)^kW^{(k)}(t)>0\) for every \(t>0\) and every integer \(k\) with \(0\le k\le1.70(H-23)^2\), where \(1.70(H-23)^2>1.52\times10^{25}\). The proof extends the book’s earlier results (all orders up to \(1.16\times10^{25}\) at every \(t>0\), and all orders at small times) by running their argument at \(t=1.70\) with the order \(1.16\times10^{25}\) in the place of \(10^{17}\), and evaluates every numerical inequality for the orders above \(1.16\times10^{25}\) in ball arithmetic, with the prime powers up to \(10^9\) (\(50\,851\,223\) of them); all hold, with a margin above \(0.4564\).

Earlier editions: Theorem 5.1.16, page 115 (edition 1).

Source: ch/ch12.tex, label ps:thm:free.

Volume I, Chapter 9. Weak limits, HCM and a Bosch type threshold

RH as a weak limit criterion for generalized gamma convolutions

Volume I, Corollary 9.2.2, page 136 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), so that \(1/G\) is the reciprocal Xi function (the clock). A generalized gamma convolution (GGC) is a law on \([0,\infty)\) whose Laplace transform is \(\exp\bigl(-bx-\int_{(0,\infty)}\log(1+x/t)\,U(dt)\bigr)\) with drift \(b\ge0\) and Thorin measure \(U\ge0\). The Riemann Hypothesis holds if and only if there are GGC random variables \(X_N\) with \(\mathbb{E}e^{-uX_N}\to1/G(u)\) for all \(u\) in a set with an accumulation point in \((0,\infty)\).

Earlier editions: Corollary 4.1.2, page 63 (edition 1).

Source: ch/ch11.tex, label cor:weak:closure.

The density of the partner variable of the clock is not completely monotone

Volume I, Proposition 9.1.7, page 136 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(T\) be the clock with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\), let \(Z\) be a standard normal variable independent of \(T\), and let \(p\) be the even density of \(Y=Z\sqrt{2T}\), whose characteristic function is \(\psi(t)=\xi(\frac12)/\xi(\frac12-t)\). Then \(p\) is not completely monotone on \((0,\infty)\). The proof is that \(\psi\) is positive and decays faster than any power, so \(p\) is \(C^\infty\) and even with \(p''(0)=-\frac1{2\pi}\int_{\mathbb{R}}t^2\psi(t)\,dt<0\), whereas a completely monotone function has \(p''\ge0\).

Source: ch/c04.tex, label xi:prop:notCM.

Is the density of the absolute value of the partner variable hyperbolically completely monotone

Volume I, Problem 9.1, page 136 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(T\) be the clock with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\) for \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(Z\) be a standard normal variable independent of \(T\), and \(Y=Z\sqrt{2T}\) with even density \(p\). Open problem: is the density of \(|Y|\), that is \(2p\) on \((0,\infty)\), hyperbolically completely monotone (HCM)? Known: no finite truncation \(\sum_{j\le N}L_j/\gamma_j\) with \(N\ge2\) of the sum of Laplace variables has an HCM density (proved, using only distinct real rates and nonzero coefficients), and the density of the clock \(T\) is not HCM (computed, and proved under additional hypotheses). For \(|Y|\) itself the question is open.

Source: ch/c04.tex, label xi:prob:hcm.

The reciprocal Xi function is HCM if and only if RH holds

Volume I, Corollary 9.2.3, page 138 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function. A positive continuous function \(f\) on \((0,\infty)\) is hyperbolically completely monotone (HCM) if for every \(u>0\) the function \(v\mapsto f(uv)f(u/v)\) is completely monotone in \(w=v+v^{-1}\) on \([2,\infty)\). (i) For every \(\alpha\ge1\) the function \(\varphi_\alpha(u)=\xi(\alpha)/\xi(\alpha+\sqrt u)\) is HCM, unconditionally. (ii) The reciprocal Xi function \(\varphi_{1/2}(u)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt u)\), the clock, is HCM if and only if the Riemann Hypothesis holds. Part (ii) is noted in versions 21 and 23 of Polson’s arXiv paper 1708.02653; here (i) and (ii) follow from Theorem 3.7 and Proposition 3.4 of Polson (2026) with Bondesson’s theorem.

Earlier editions: Corollary 4.1.3, page 64 (edition 1).

Source: ch/ch11.tex, label cor:hcm.

Deformed reciprocal Xi function is HCM exactly for t at least Lambda

Volume I, Corollary 9.3.5, page 144 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), and \(\Phi(u)=2\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\bigr)\exp(-\pi n^2e^{2u})\) the Pólya kernel, so that \(\Xi(\theta)=\int_{\mathbb{R}}\Phi(u)\cos(\theta u)\,du\). For real \(t\) let \(H_t(z)=\tfrac12\int_0^\infty e^{tu^2}\Phi(2u)\cos(zu)\,du\), so that \(H_0(z)=\tfrac18\Xi(z/2)\), and let \(\varphi_t(s)=H_t(0)/H_t(2i\sqrt s)\) for \(s>0\), so that \(\varphi_0(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\). Let \(\Lambda\) be the de Bruijn Newman constant, defined by the property that \(H_t\) has only real zeros exactly when \(t\ge\Lambda\). Then \(\varphi_t\) is HCM if and only if \(t\ge\Lambda\). Hence \(\varphi_t\) is HCM for \(t\ge0.22\) (by the bound \(\Lambda\le0.22\) of the Polymath project), \(\varphi_t\) is not HCM for \(t<0\) (by the theorem of Rodgers and Tao that \(\Lambda\ge0\)), and RH holds if and only if \(\varphi_t\) is HCM for every \(t>0\).

Earlier editions: Corollary 4.2.5, page 71 (edition 1).

Source: ch/boschA.tex, label bh:cor:dbn.

Question: for which beta is the density of W^beta HCM

Volume I, Question 9.3.7, page 145 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(\Xi(z)=\xi(\tfrac12+iz)\). Let \(W\) be the positive random variable, which exists unconditionally, with \(\mathbb{E}W^{it}=\xi(\tfrac12)/\xi(\tfrac12+t)\) for real \(t\) and \(\mathbb{E}W^z=\Xi(0)/\Xi(z)\) for \(|\operatorname{Re}z|<\gamma_1\), where \(\gamma_1\approx14.1347\) is the smallest positive ordinate of a zero of \(\xi\). For which real \(\beta\) is the density of \(W^\beta\) HCM (hyperbolically completely monotone)? On the Cauchy side, for \(|C|\) with \(C\) a standard Cauchy variable, so that \(\mathbb{E}|C|^z=\sec(\pi z/2)\), the answer is \(|\beta|\ge2\) (Bosch’s theorem).

Earlier editions: Question 4.2.7, page 72 (edition 1).

Source: ch/boschA.tex, label bh:q:beta.

Question: does HCM of W^beta for a single beta imply RH

Volume I, Question 9.3.8, page 145 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(\Xi(z)=\xi(\tfrac12+iz)\), and let \(W\) be the positive random variable, which exists unconditionally, with \(\mathbb{E}W^z=\Xi(0)/\Xi(z)\) for \(|\operatorname{Re}z|<\gamma_1\), where \(\gamma_1\approx14.1347\) is the smallest positive ordinate of a zero of \(\xi\). Does HCM (hyperbolic complete monotonicity) of the density of \(W^\beta\) for a single real \(\beta\) imply the Riemann Hypothesis?

Earlier editions: Question 4.2.8, page 72 (edition 1).

Source: ch/boschA.tex, label bh:q:single.

Elasticity criterion for HCM functions (Bondesson’s representation restated as a Pick condition)

Volume I, Theorem 9.4.2, page 146 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Omega=\mathbb{C}\setminus(-\infty,0]\) and let \(h\) be a positive function on \((0,\infty)\). Call \(h\) HCM if it is continuous and for every \(u>0\) the function \(v\mapsto h(uv)h(u/v)\) is completely monotone in \(w=v+v^{-1}\) on \([2,\infty)\). Then \(h\) is HCM if and only if \(h\) continues to a zero free analytic function on \(\Omega\) and its elasticity \(\varepsilon_h(x)=-xh'(x)/h(x)\) is a Pick function, that is, \(\operatorname{Im}\varepsilon_h(x)\ge0\) for \(\operatorname{Im}x>0\).

Earlier editions: Theorem 4.3.2, page 73 (edition 1).

Source: ch/boschB.tex, label bs:thm:crit.

Bosch type theorem for the reciprocal Xi function

Volume I, Theorem 9.4.5, page 148 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), and let \(\gamma\) run over the zeros of \(\Xi\) with \(\operatorname{Re}\gamma>0\), counted with multiplicity. Let \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)=\prod_\gamma(1+s/\gamma^2)^{-1}\) be the reciprocal Xi function, let \(\delta_*=\max_\gamma|\arg\gamma|\) be the largest angle of these zeros (the maximum is attained if some zero is off the critical line, and \(\delta_*=0\) if and only if the Riemann Hypothesis holds), and put \(\alpha_\xi=2\delta_*/\pi\) and \(p_*=1-\alpha_\xi\). Unconditionally, for real \(p\) the function \(x\mapsto\varphi(x^p)\) on \((0,\infty)\) is HCM (hyperbolically completely monotone) if and only if \(|p|\le p_*=1-\alpha_\xi\). Equivalently, \(\varphi(x^{1/\beta})\) is HCM if and only if \(|\beta|\ge1/(1-\alpha_\xi)\). The Riemann Hypothesis is the statement \(\alpha_\xi=0\), that is, \(p_*=1\).

Earlier editions: Theorem 4.3.5, page 75 (edition 1).

Source: ch/boschB.tex, label bs:thm:power.

Equivalent forms of RH: HCM of all powers, Pick elasticity

Volume I, Corollary 9.4.6, page 149 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) the reciprocal Xi function and \(\varepsilon_\varphi(s)=-s\varphi'(s)/\varphi(s)=\tfrac{\sqrt s}{2}\,\frac{\xi'}{\xi}\bigl(\tfrac12+\sqrt s\bigr)\) its elasticity. The following are equivalent. (i) The Riemann Hypothesis. (ii) \(\varphi\) is HCM. (iii) \(\varphi(x^p)\) is HCM for every \(|p|<1\). (iv) \(\varepsilon_\varphi\) is a Pick function (analytic on the upper half plane with nonnegative imaginary part). (v) \(\operatorname{Im}\bigl[z\,(\xi'/\xi)(\tfrac12+z)\bigr]\ge0\) for all \(z\) with \(\operatorname{Re}z>0\) and \(\operatorname{Im}z>0\).

Earlier editions: Corollary 4.3.6, page 76 (edition 1).

Source: ch/boschB.tex, label bs:cor:rh.

Unconditional HCM range for the reciprocal Xi function composed with a power

Volume I, Corollary 9.4.7, page 149 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\gamma\) the zeros of \(\Xi\) with \(\operatorname{Re}\gamma>0\) counted with multiplicity, \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) the reciprocal Xi function with elasticity \(\varepsilon_\varphi(s)=-s\varphi'(s)/\varphi(s)\), and \(\delta_*=\max_\gamma|\arg\gamma|\). Let \(T_0=3\times10^{12}\). RH has been verified for all zeros with \(0<\operatorname{Re}\gamma\le T_0\) (Platt and Trudgian 2021). Hence \(\delta_*<\arctan(1/(2T_0))\) and \(\varphi(x^p)\) is HCM for all \(|p|\le1-\frac{1}{\pi T_0}\), where \(1-\frac{1}{\pi T_0}>1-1.1\times10^{-13}\). Unconditionally, \(\operatorname{Im}\varepsilon_\varphi(s)\ge0\) on the sector \(0<\arg s<\pi-1/T_0\).

Earlier editions: Corollary 4.3.7, page 76 (edition 1).

Source: ch/boschB.tex, label bs:cor:uncond.

Stieltjes sign of minus the logarithmic derivative of the clock outside a thin parabolic region

Volume I, Theorem 9.4.9, page 150 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\gamma\) the zeros of \(\Xi\) with \(\operatorname{Re}\gamma>0\) counted with multiplicity, \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) the reciprocal Xi function and \(\psi(s)=-\varphi'(s)/\varphi(s)=\sum_\gamma(s+\gamma^2)^{-1}\). A Stieltjes function is a function \(a/s+b+\int_{(0,\infty)}\sigma(dt)/(s+t)\) with \(a,b\ge0\) and \(\sigma\ge0\) a measure satisfying \(\int(1+t)^{-1}\sigma(dt)<\infty\). Let \(T_0=3\times10^{12}\), so that every zero \(\gamma\) with \(0<\operatorname{Re}\gamma\le T_0\) is real (Platt and Trudgian 2021), and let \(\mathcal E_{T_0}=\bigl\{x+iy:\ x<-T_0^2+T_0+\tfrac14,\ y^2<\tfrac12-x\bigr\}\). Split \(\psi=S_{\mathrm{on}}+R_{\mathrm{off}}\), where \(S_{\mathrm{on}}(s)=\sum_{\gamma\ \mathrm{real}}(s+\gamma^2)^{-1}\) is the sum over the zeros on the critical line and \(R_{\mathrm{off}}\) is the same sum over the nonreal \(\gamma\). For a nonreal zero \(\gamma\) let \(D_\gamma\) be the closed disc with diameter the segment from \(-\gamma^2\) to \(-\bar\gamma^2\). Then (i) \(S_{\mathrm{on}}\) is a Stieltjes function, with measure \(\sum_{\gamma\ \mathrm{real}}\delta_{\gamma^2}\); (ii) \(R_{\mathrm{off}}\) is meromorphic on \(\mathbb{C}\) and nonnegative on \([0,\infty)\), for every nonreal \(\gamma\) the disc \(D_\gamma\), and with it the poles \(-\gamma^2\) and \(-\bar\gamma^2\) of \(R_{\mathrm{off}}\) on its boundary, lies in \(\mathcal E_{T_0}\), and \(\operatorname{Im}R_{\mathrm{off}}\le0\) at every point of the upper half plane outside these discs; (iii) \(\psi>0\) on \([0,\infty)\), and \(\operatorname{Im}\psi(s)<0\) at every \(s=x+iy\) with \(y>0\) outside \(\mathcal E_{T_0}\).

Earlier editions: Theorem 4.3.9, page 77 (edition 1).

Source: ch/boschB.tex, label bs:thm:local.

Question: HCM exponent of the deformed reciprocal Xi function for negative time

Volume I, Question 9.4.11, page 151 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\Phi\) the Pólya kernel (with \(\Xi(\theta)=\int_{\mathbb{R}}\Phi(u)\cos(\theta u)\,du\)) and, for real \(t\), \(H_t(z)=\tfrac12\int_0^\infty e^{tu^2}\Phi(2u)\cos(zu)\,du\) the de Bruijn Newman deformation, so that \(H_0(z)=\tfrac18\Xi(z/2)\). Let \(\delta_*(t)\) be the largest angle \(|\arg z|\) over the zeros \(z\) of \(H_t\) with \(\operatorname{Re}z>0\) and \(p_*(t)=1-\tfrac2\pi\delta_*(t)\); whenever the zeros of \(H_t\) satisfy \(|\arg z|<\pi/4\), the function \(x\mapsto H_t(0)/H_t(2i\sqrt{x^p})\) is HCM if and only if \(|p|\le p_*(t)\). Describe \(p_*(t)\) for \(t<0\). By Rodgers and Tao, \(p_*(t)<1\) for every \(t<0\). Assume RH; how fast does \(1-p_*(t)\to0\) as \(t\uparrow0\)?

Earlier editions: Question 4.3.11, page 79 (edition 1).

Source: ch/boschB.tex, label bs:q:four.

Thorin measure of the reciprocal Xi function composed with a power

Volume I, Theorem 9.4.12, page 152 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\gamma\) the zeros of \(\Xi\) with \(\operatorname{Re}\gamma>0\) counted with multiplicity, \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) the reciprocal Xi function with elasticity \(\varepsilon_\varphi(s)=-s\varphi'(s)/\varphi(s)=\tfrac{\sqrt s}{2}\frac{\xi'}{\xi}(\tfrac12+\sqrt s)=\sum_\gamma\frac{s}{s+\gamma^2}\), \(\delta_*=\max_\gamma|\arg\gamma|\) and \(p_*=1-2\delta_*/\pi\). A generalized gamma convolution (GGC) is a law on \([0,\infty)\) whose Laplace transform is \(\exp\bigl(-bs-\int_{(0,\infty)}\log(1+s/t)\,U(dt)\bigr)\) with drift \(b\ge0\) and Thorin measure \(U\ge0\). Let \(0<p\le p_*\). Then \(\varphi(s^p)=\mathbb{E}e^{-sX_p}\) for a GGC variable \(X_p\) with no drift. If \(p<p_*\), its Thorin measure has the density \(u_p(\tau)=\frac{p}{\pi\tau}\operatorname{Im}\varepsilon_\varphi\bigl(\tau^pe^{ip\pi}\bigr)=\frac{p}{2\pi\tau}\operatorname{Im}\Bigl[z\,\frac{\xi'}{\xi}\bigl(\tfrac12+z\bigr)\Bigr]_{z=\tau^{p/2}e^{ip\pi/2}}\ge0\) for \(\tau>0\), which in terms of the zeros is \(u_p(\tau)=\frac{p}{\pi\tau}\sum_\gamma\operatorname{Im}\frac{\tau^pe^{ip\pi}}{\tau^pe^{ip\pi}+\gamma^2}\). If \(p=p_*\), its Thorin measure is \(u_{p_*}(\tau)\,d\tau\) plus a unit atom at \(|\gamma|^{2/p_*}\) for every zero \(\gamma\) with \(\arg\gamma=-\delta_*\), counted with multiplicity, that is, one atom for each conjugate pair of zeros at the extreme angle; when \(p_*=1\) there is an atom at \(\gamma^2\) for every zero, and \(u_1=0\).

Earlier editions: Theorem 4.3.12, page 79 (edition 1).

Source: ch/boschB.tex, label bs:thm:ggc.

Thorin measure of X_p under RH: scaled Lamperti laws (Bosch’s variable)

Volume I, Corollary 9.4.13, page 153 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis.

Statement. Assume the Riemann Hypothesis. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\gamma\) the (real, positive) zeros of \(\Xi\) counted with multiplicity, \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\), and for \(0<p<1\) let \(X_p\) be the generalized gamma convolution (GGC) variable with \(\mathbb{E}e^{-sX_p}=\varphi(s^p)\); a GGC is a law on \([0,\infty)\) whose Laplace transform is \(\exp\bigl(-bs-\int_{(0,\infty)}\log(1+s/t)\,U(dt)\bigr)\) with drift \(b\ge0\) and Thorin measure \(U\ge0\). Let \(L_p=Z_p/\tilde Z_p\) be the ratio of independent positive \(p\) stable variables, whose density \(\frac{\sin p\pi}{\pi}\,\frac{t^{p-1}}{t^{2p}+2t^p\cos p\pi+1}\) on \((0,\infty)\) is Lamperti’s; its power \(L_p^{\,p}\) is Bosch’s variable \(T_\alpha\) with \(\alpha=p\). Writing \(\mathcal L(Y)\) for the law of a random variable \(Y\), the Thorin measure of \(X_p\) is \(U_p=p\sum_\gamma\mathcal L\bigl(\gamma^{2/p}L_p\bigr)\), and \(X_p\stackrel{d}{=}S_pX^{1/p}\), where \(X=\sum_\gamma e_\gamma/\gamma^2\) is the clock variable, with independent standard exponential variables \(e_\gamma\), and \(S_p\) is a positive \(p\) stable variable independent of \(X\) with \(\mathbb{E}e^{-\lambda S_p}=e^{-\lambda^p}\). At \(p=\tfrac12\), \(L_{1/2}\stackrel{d}{=}C^2\) for a standard Cauchy variable \(C\), and \(U_{1/2}=\tfrac12\sum_\gamma\mathcal L(\gamma^4C^2)\). As \(p\uparrow1\), \(L_p\to1\) and \(U_p\to\sum_\gamma\delta_{\gamma^2}\) vaguely, the Thorin measure of \(\varphi\) itself.

Earlier editions: Corollary 4.3.13, page 80 (edition 1).

Source: ch/boschB.tex, label bs:cor:lamperti.

RH is equivalent to the existence of the GGC variable X_p for every p in (0, 1)

Volume I, Corollary 9.4.14, page 153 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(\varphi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) the reciprocal Xi function. A generalized gamma convolution (GGC) is a law on \([0,\infty)\) whose Laplace transform is \(\exp\bigl(-bs-\int_{(0,\infty)}\log(1+s/t)\,U(dt)\bigr)\) with drift \(b\ge0\) and Thorin measure \(U\ge0\). For \(0<p<1\) say that \(X_p\) exists if \(\varphi(s^p)=\mathbb{E}e^{-sX_p}\) for a GGC random variable \(X_p\). Then the Riemann Hypothesis holds if and only if \(X_p\) exists for every \(p\in(0,1)\). Unconditionally, using the verification of the Riemann Hypothesis to height \(T_0=3\times10^{12}\) (Platt and Trudgian 2021), \(X_p\) exists for \(0<p\le1-1/(\pi T_0)\), in particular for \(0<p\le1-1.1\times10^{-13}\), and its Thorin measure then has the density \(u_p(\tau)=\frac{p}{2\pi\tau}\operatorname{Im}\Bigl[z\,\frac{\xi'}{\xi}\bigl(\tfrac12+z\bigr)\Bigr]_{z=\tau^{p/2}e^{ip\pi/2}}\) for \(\tau>0\).

Earlier editions: Corollary 4.3.14, page 81 (edition 1).

Source: ch/boschB.tex, label bs:cor:rhX.

Question: construct X_{1/2} from Brownian motion without the zeros

Volume I, Question 9.4.15, page 154 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function and, for \(0<p<1\), let \(X_p\) be the variable with \(\mathbb{E}e^{-sX_p}=\xi(\tfrac12)/\xi(\tfrac12+s^{p/2})\), a generalized gamma convolution whenever it exists; unconditionally it exists for \(0<p\le1-1.1\times10^{-13}\), and RH holds if and only if it exists for every \(p\in(0,1)\). In particular \(X_{1/2}\) has Laplace transform \(\xi(\tfrac12)/\xi(\tfrac12+s^{1/4})\) and exists without any hypothesis. Construct \(X_{1/2}\) from Brownian motion, Bessel processes or the Biane, Pitman and Yor variables without reference to the zeros. A construction that survived \(p\uparrow1\) would prove RH.

Earlier editions: Question 4.3.15, page 81 (edition 1).

Source: ch/boschB.tex, label bs:q:five.

Question: prove positivity of u_p for all p without locating zeros

Volume I, Question 9.4.16, page 154 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function. For \(0<p<1\) and \(\tau>0\) put \(u_p(\tau)=\frac{p}{2\pi\tau}\operatorname{Im}\Bigl[z\,\frac{\xi'}{\xi}\bigl(\tfrac12+z\bigr)\Bigr]_{z=\tau^{p/2}e^{ip\pi/2}}\), the Thorin density of the reciprocal Xi function composed with \(s^p\); the formula makes sense whatever the zeros are. Prove \(u_p\ge0\) for every \(p\in(0,1)\) directly from this formula, without locating the zeros. (Positivity of \(u_p\) for all \(p\in(0,1)\) is equivalent to the Riemann Hypothesis.)

Earlier editions: Question 4.3.16, page 81 (edition 1).

Source: ch/boschB.tex, label bs:q:one.

Thorin density u_p is positive beyond a finite point, from the primes

Volume I, Proposition 9.4.18, page 155 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(0<p<1\), \(\vartheta=p\pi/2\), and for \(\tau>0\) put \(t=\tau^{p/2}\), \(z=te^{i\vartheta}\), \(w=\tfrac12+z\) and \(u_p(\tau)=\frac{p}{2\pi\tau}\operatorname{Im}\bigl[z\,\frac{\xi'}{\xi}(w)\bigr]\), the Thorin density of the reciprocal Xi function composed with \(s^p\) (the formula makes sense whatever the zeros are). Let \(\operatorname{Re}w>1\) and \(\sigma=\operatorname{Re}w\). Then \(\frac{2\pi\tau}{p}\,u_p(\tau)\ \ge\ m_p(t):=\operatorname{Im}\Bigl[z\Bigl(\frac{1}{2w}+\frac{1}{w-1}+\frac12\log\frac{w}{2\pi}\Bigr)\Bigr]-\frac{t}{6\sigma^2}-t\Bigl(-\frac{\zeta'}{\zeta}\Bigr)(\sigma)\). Every term of \(m_p\) is explicit, no zero of \(\zeta\) enters, and \(m_p(t)=\frac t2\sin\vartheta\,\log t\,(1+o(1))\) as \(t\to\infty\). Consequently for each \(p\in(0,1)\) there is \(\tau_1(p)<\infty\) with \(u_p(\tau)>0\) for all \(\tau>\tau_1(p)\).

Earlier editions: Proposition 4.3.18, page 82 (edition 1).

Source: ch/boschB.tex, label bs:prop:prime.

Cauchy analogue of the clock: eta cubed density with natural boundary, not HCM

Volume I, Theorem 9.4.20, page 156 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X_C=\sum_{k\ge1}e_k/(2k-1)^2\) with independent standard exponential variables \(e_k\), so that \(\mathbb{E}e^{-sX_C}=\operatorname{sech}(\pi\sqrt s/2)\). The density of \(X_C\) is \(f_C(t)=\frac4\pi\sum_{k\ge1}(-1)^{k-1}(2k-1)e^{-(2k-1)^2t}=\frac4\pi\,\eta\Bigl(\frac{4it}{\pi}\Bigr)^3\) for \(t>0\), with \(\eta\) the Dedekind eta function. The line \(\operatorname{Re}t=0\) is a natural boundary of \(f_C\). Hence \(X_C\) is a generalized gamma convolution whose Laplace transform is HCM, but the density of \(X_C\) is not HCM.

Earlier editions: Theorem 4.3.20, page 83 (edition 1).

Source: ch/boschB.tex, label bs:thm:eta.

Question: natural boundary of the clock density

Volume I, Question 9.4.21, page 156 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Assume the Riemann Hypothesis and that all zeros are simple. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\gamma_1<\gamma_2<\cdots\) the positive zeros of \(\Xi\), \(X=\sum_ne_n/\gamma_n^2\) the clock variable with independent standard exponential variables \(e_n\) (its Laplace transform is \(\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\)), and \(A_n=\prod_{m\ne n}\bigl(1-\gamma_n^2/\gamma_m^2\bigr)^{-1}\). Does the density \(f_X(t)=\sum_nA_n\gamma_n^2e^{-\gamma_n^2t}\) of \(X\) have \(\operatorname{Re}t=0\) as a natural boundary? If so, \(X\) is not HCM.

Earlier editions: Question 4.3.21, page 83 (edition 1).

Source: ch/boschB.tex, label bs:q:two.

Natural boundary of the clock density under RH, simple zeros and gap hypotheses

Volume I, Theorem 9.4.22, page 156 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis; all zeros of the zeta function are simple; hypothesis (G), \(\inf_n(\gamma_{n+1}^2-\gamma_n^2)>0\); hypothesis (D), \(\log(1/|\zeta'(\tfrac12+i\gamma_n)|)=o(\gamma_n^2)\).

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), and assume the Riemann Hypothesis and that all zeros are simple; let \(\gamma_1<\gamma_2<\cdots\) be the positive zeros of \(\Xi\) (so \(\tfrac12+i\gamma_n\) are the zeros of \(\xi\) in the upper half plane), and let \(X=\sum_ne_n/\gamma_n^2\) be the clock variable, with independent standard exponential variables \(e_n\) and Laplace transform \(\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\). Assume (G) \(\inf_n(\gamma_{n+1}^2-\gamma_n^2)>0\) and (D) \(\log\bigl(1/|\zeta'(\tfrac12+i\gamma_n)|\bigr)=o(\gamma_n^2)\). Then for \(t>0\) the density of \(X\) is \(f_X(t)=\sum_nc_ne^{-\gamma_n^2t}\) with \(c_n=A_n\gamma_n^2=-\frac{2\gamma_n\Xi(0)}{\Xi'(\gamma_n)}\), where \(A_n=\prod_{m\ne n}(1-\gamma_n^2/\gamma_m^2)^{-1}\), the series has abscissa of convergence \(0\), and \(\operatorname{Re}t=0\) is a natural boundary of \(f_X\). In particular the density of \(X\) is not HCM.

Earlier editions: Theorem 4.3.22, page 84 (edition 1).

Source: ch/boschB.tex, label bs:thm:nb.

Question: for which p are X_p and its Cauchy analogue HCM

Volume I, Question 9.4.24, page 158 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function and, for \(0<p<1\), let \(X_p\) be the variable with \(\mathbb{E}e^{-sX_p}=\xi(\tfrac12)/\xi(\tfrac12+s^{p/2})\), a generalized gamma convolution whenever it exists (unconditionally for \(0<p\le1-1.1\times10^{-13}\), and for every \(p<1\) under RH), and let \(X_{C,p}\) be the variable with Laplace transform \(\operatorname{sech}(\pi s^{p/2}/2)\), the Cauchy family. Determine \(\{p:\ X_p\ \text{is HCM}\}\), and the same set for the Cauchy family \(X_{C,p}\). (A random variable is HCM if it has an HCM density.)

Earlier editions: Question 4.3.24, page 85 (edition 1).

Source: ch/boschB.tex, label bs:q:three.

Lower bound for the density of log W: existence of W (second proof)

Volume I, Theorem 9.4.25, page 158 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\gamma\) the zeros of \(\Xi\) with \(\operatorname{Re}\gamma>0\) counted with multiplicity, \(\chi(t)=\xi(\tfrac12)/\xi(\tfrac12+t)=\prod_\gamma(1+t^2/\gamma^2)^{-1}\) for real \(t\), and \(g(y)=\frac1\pi\int_0^\infty\chi(t)\cos(ty)\,dt\). Unconditionally, by the verification of RH to height \(T_0=3\times10^{12}\) (Platt and Trudgian 2021), \(g(y)\ge0.9998\,g_R(y)>0\) for all real \(y\), where \(g_R\) is the density of \(\sum_{\gamma\ \mathrm{real}}L_\gamma/\gamma\), the sum over the real zeros \(\gamma\) with independent standard Laplace variables \(L_\gamma\) (density \(\tfrac12e^{-|x|}\)). Hence \(\chi\) is a characteristic function and there is a positive random variable \(W\) with \(\mathbb{E}W^{it}=\chi(t)\) and \(\mathbb{E}W^z=\Xi(0)/\Xi(z)\) for \(|\operatorname{Re}z|<\gamma_1\), where \(\gamma_1\approx14.1347\) is the smallest positive ordinate of a zero of \(\xi\).

Earlier editions: Theorem 4.3.25, page 86 (edition 1).

Source: ch/boschB.tex, label bs:thm:gpos.

HCM criterion for powers of the Mellin variable W

Volume I, Proposition 9.4.26, page 160 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\chi(t)=\xi(\tfrac12)/\xi(\tfrac12+t)\) and \(g(y)=\frac1\pi\int_0^\infty\chi(t)\cos(ty)\,dt\), the density of \(\log W\) for the positive random variable \(W\) with \(\mathbb{E}W^{it}=\chi(t)\); \(g\) is an entire function because \(\chi(t)=\exp\bigl(-\tfrac12|t|\log|t|\,(1+o(1))\bigr)\). Let \(\beta\in\mathbb{R}\setminus\{0\}\). Then \(W^\beta\) is HCM (has an HCM density) if and only if \(g\) has no zeros in the strip \(|\operatorname{Im}u|<\pi/|\beta|\) and \(\operatorname{Im}\bigl(-g'(u)/g(u)\bigr)\ge0\) for \(0<\operatorname{Im}u<\pi/|\beta|\).

Earlier editions: Proposition 4.3.26, page 87 (edition 1).

Source: ch/boschB.tex, label bs:prop:Wcrit.

First gap theorem: HCM of W^beta forces |beta| at least gamma_2 minus gamma_1

Volume I, Theorem 9.4.27, page 160 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis; all zeros of the zeta function are simple; \(\log(1/|\zeta'(\rho_n)|)=O(\gamma_n)\), where \(\rho_n=\tfrac12+i\gamma_n\) are the zeros of zeta in the upper half plane.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\chi(t)=\xi(\tfrac12)/\xi(\tfrac12+t)\), and \(g(u)=\frac1\pi\int_0^\infty\chi(t)\cos(tu)\,dt\) (an entire function), the density of \(\log W\) for the positive random variable \(W\) with \(\mathbb{E}W^{it}=\chi(t)\). Assume the Riemann Hypothesis, that all zeros are simple, and that \(\log(1/|\zeta'(\rho_n)|)=O(\gamma_n)\), where \(\rho_n=\tfrac12+i\gamma_n\) with \(0<\gamma_1<\gamma_2<\cdots\) are the zeros of \(\zeta\) in the upper half plane. Then \(u_0=\frac\pi4+\limsup_n\gamma_n^{-1}\log(1/|\zeta'(\rho_n)|)\) is finite, and for \(\operatorname{Re}u>u_0\) one has \(g(u)=-\Xi(0)\sum_n\frac{e^{-\gamma_nu}}{\Xi'(\gamma_n)}=\sum_nb_ne^{-\gamma_nu}\) with \(b_1>0>b_2\). If \(W^\beta\) is HCM (has an HCM density), then \(|\beta|\ge\gamma_2-\gamma_1=6.8873\ldots\)

Earlier editions: Theorem 4.3.27, page 88 (edition 1).

Source: ch/boschB.tex, label bs:thm:gap.

Question: first gap conjecture for HCM powers of W

Volume I, Question 9.4.28, page 161 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(W\) be the positive random variable, which exists unconditionally, with \(\mathbb{E}W^z=\Xi(0)/\Xi(z)\) for \(|\operatorname{Re}z|<\gamma_1\), where \(\Xi(z)=\xi(\tfrac12+iz)\), \(\xi\) is the completed Riemann zeta function and \(0<\gamma_1<\gamma_2\) are the two smallest positive ordinates of zeros of \(\xi\). Is \(W^\beta\) HCM (does it have an HCM density) exactly when \(|\beta|\ge\gamma_2-\gamma_1\)? On the Cauchy side the analogous statement, with frequencies \(2k-1\), is Bosch’s theorem at \(\alpha=1/2\): \(3-1=2\).

Earlier editions: Question 4.3.28, page 88 (edition 1).

Source: ch/boschB.tex, label bs:q:gap.

Refined first gap conjecture for sums of Laplace variables

Volume I, Question 9.4.29, page 162 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(Y=\sum_nL_n/\lambda_n\) (a convergent series) with independent standard Laplace variables \(L_n\) and \(0<\lambda_1<\lambda_2<\cdots\), and let \(W=e^Y\). If the characteristic function of \(Y\) decays faster than every exponential, or if the first gap \(\lambda_2-\lambda_1\) is at least every later gap, is the HCM threshold of \(W^\beta\) exactly \(\lambda_2-\lambda_1\)? In particular, is it \(\gamma_2-\gamma_1\) for the Riemann variable \(W\) with \(\mathbb{E}W^z=\Xi(0)/\Xi(z)\)?

Earlier editions: Question 4.3.29, page 90 (edition 1).

Source: ch/boschB.tex, label bs:q:gaprefined.

Volume I, Chapter 10. Thorin type forms of the reciprocal zeta function

Thorin type density of an Euler factor

Volume I, Proposition 10.1.2, page 170 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(0<q<1\) and \(\lambda>0\), and let \(E(s)=1-q\,e^{-\lambda\sqrt s}\), where \(\sqrt s\) is the principal square root on the cut plane \(\mathbb{C}\setminus(-\infty,0]\). Then \(E\) is analytic and has no zeros on the cut plane, and there \(\log\frac{E(s)}{1-q}=-\int_0^\infty\log\bigl(1+\frac st\bigr)\,u(t)\,dt\) with \(u(t)=-\frac{q\lambda}{2\pi\sqrt t}\,\frac{\cos(\lambda\sqrt t)-q}{1-2q\cos(\lambda\sqrt t)+q^2}=-\frac{\lambda}{2\pi\sqrt t}\sum_{k\ge1}q^k\cos(k\lambda\sqrt t)\).

Earlier editions: Proposition 6.1.2, page 138 (edition 1).

Source: ch/rz1.tex, label rz:prop:euler.

Lai and Vere-Jones identity (the exponential as a Möbius sum of hyperbolic secants)

Volume I, Proposition 10.1.3, page 171 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\mu\) be the Möbius function and \(\chi_4\) the nontrivial Dirichlet character modulo \(4\). For \(\operatorname{Re}x>0\), \(\frac12\operatorname{sech}x=\sum_{m\ \mathrm{odd}}\chi_4(m)\,e^{-mx}\) and \(e^{-x}=\frac12\sum_{m\ge1}\mu(m)\chi_4(m)\operatorname{sech}(mx)\).

Earlier editions: Proposition 6.1.3, page 139 (edition 1).

Source: ch/rz1.tex, label rz:prop:lvj.

Cauchy density as a Möbius mixture of hyperbolic secant laws (Polson note, corrected)

Volume I, Proposition 10.1.4, page 171 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal character modulo \(4\), \(\mu\) the Möbius function, \(\mathrm{HS}_b(x)=(2b)^{-1}\operatorname{sech}(\pi x/2b)\) the hyperbolic secant density (characteristic function \(\operatorname{sech}(b\omega)\)), and \(P_a(x)=a/(\pi(a^2+x^2))\) the Cauchy density. For \(a>0\) and every real \(x\), and for \(z\in\mathbb{C}\) with \(z\ne\pm iN\), \(N\) odd, \(P_a(x)=\frac12\sum_{m\ge1}\mu(m)\chi_4(m)\mathrm{HS}_{am}(x)\) and \(\frac1{1+z^2}=\frac\pi4\sum_{m\ge1}\frac{\mu(m)\chi_4(m)}m\operatorname{sech}(\frac{\pi z}{2m})\), the two being equivalent through \(z=x/a\). The series converge conditionally and not absolutely: the total variation of the right side is infinite, the weights have total value \(\frac12/\beta(0)=1\), where \(\beta=L(\cdot,\chi_4)\), only as the continuation of \(\frac12\sum_m\mu(m)\chi_4(m)m^{-s}\) to \(s=0\) (they are not Abel summable), and \(\sum_m\mu(m)\chi_4(m)\) diverges. Both sides agree to \(10^{-14}\) at \(z=0.7\) and \(2\) (computed).

Source: ch/rz1.tex, label rz:prop:cauchyhs.

Cyclotomic kernels for N equal 3, 4, 6 invert to the exponential and are generalised gamma convolutions (Polson note)

Volume I, Proposition 10.1.5, page 172 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(N=3,4,6\) put \(f_N(z)=1/(2\cosh z-2\cos(2\pi/N))=q/\Phi_N(q)\) with \(q=e^{-z}\) and \(\Phi_N\) the \(N\)th cyclotomic polynomial, so \(f_4=\frac12\operatorname{sech}\). Let \(\mu\) be the Möbius function and \(\chi_{-3}\), \(\chi_4\) the nonprincipal characters modulo \(3\) and \(4\). Then \(f_3=\sum_{k\ge1}\chi_{-3}(k)e^{-kz}\), \(f_4=\sum_{k\ge1}\chi_4(k)e^{-kz}\) and \(f_6=\sum_{k\ge1}\varepsilon_ke^{-kz}\) with Dirichlet series \(L(s,\chi_{-3})(1+2^{1-s})\), and for \(\operatorname{Re}x>0\), with absolute convergence, \(e^{-x}=\sum_{m\ge1}\mu(m)\chi_{-3}(m)f_3(mx)=\sum_{m\ge1}\mu(m)\chi_4(m)f_4(mx)=\sum_{m\ge1}b_mf_6(mx)\), where \(\sum_mb_mm^{-s}=1/((1+2^{1-s})L(s,\chi_{-3}))\). Moreover \(f_N(\sqrt\lambda)/f_N(0)=\prod_{j\ge1,\ j\equiv\pm1\,(N)}(1+\lambda/(2\pi j/N)^2)^{-1}\), so each normalised kernel is a generalised gamma convolution whose Thorin measure is a lattice in the chart \(\theta=\sqrt z\).

Source: ch/rz1.tex, label rz:prop:cyclo.

The Möbius signs are forced, no positive mixture of cyclotomic kernels gives the exponential (Polson note)

Volume I, Proposition 10.1.6, page 172 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With \(f_N(z)=1/(2\cosh z-2\cos(2\pi/N))\), \(N=3,4,6\) (\(f_4=\frac12\operatorname{sech}\)), there is no positive measure \(\rho\) on \((0,\infty)\) with \(e^{-x}=\int f_N(tx)\rho(dt)\) for all \(x>0\), and none with \(1/(1+z^2)=\int\operatorname{sech}(tz)\rho(dt)\) for all \(z>0\). Nor, under the Riemann Hypothesis, is \(\sum_{\gamma>0}\delta_{\gamma^2}\) (the Thorin measure of \(\xi(\frac12)/\xi(\frac12+\sqrt\lambda)\), \(\gamma\) running over the positive ordinates of the zeros of \(\zeta\)) a positive mixture of the lattices of the generalised gamma convolutions with transforms \(f_N(\sqrt\lambda)/f_N(0)\). The proof uses a zero of \(\beta=L(\cdot,\chi_4)\) in \(\operatorname{Re}s>0\), namely \(\frac12+6.0209\ldots i\), and the zero of \(L(s,\chi_{-3})\) at \(\frac12+8.0397\ldots i\) (computed); for the lattice statement it uses the ordinates \(3\gamma_1=42.404\), \(2\gamma_1=28.269\), \(5\gamma_1=70.674\) with \(\gamma_1=14.1347\ldots\), which lie strictly between consecutive zero ordinates.

Source: ch/rz1.tex, label rz:prop:mobsigns.

Vere-Jones representation of the reciprocal zeta function by hyperbolic secants

Volume I, Theorem 10.1.8, page 173 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\mu\) the Möbius function, \(\chi_4\) the nontrivial Dirichlet character modulo \(4\), \(\sqrt s\) the principal square root, and \(T_\lambda\) the exit time of \(\sqrt2B\) from \((-\lambda,\lambda)\) for a standard Brownian motion \(B\) started at \(0\), so that \(\mathbb{E}\,e^{-sT_\lambda}=\operatorname{sech}(\lambda\sqrt s)\). For \(s\) in the region \(H_{1/2}=\{s:\operatorname{Re}\sqrt s>\frac12\}=\{x+iy:\ x>\frac14-y^2\}\), \(\frac{1}{\zeta(\frac12+\sqrt s)}=1+\sum_{N\ge2}A_N\operatorname{sech}(\sqrt s\log N)=1+\sum_{N\ge2}A_N\,\mathbb{E}\,e^{-sT_{\log N}}\), where \(A_N=\frac12\,\mu(m)\chi_4(m)\mu(n)\,n^{-1/2}\) if \(N=n^m\) with \(n\ge2\) squarefree and \(m\) odd, and \(A_N=0\) otherwise. In particular \(A_N=\mu(N)/(2\sqrt N)\) for squarefree \(N\ge2\).

Earlier editions: Theorem 6.1.5, page 139 (edition 1).

Source: ch/rz1.tex, label rz:thm:sech.

Logarithmic twin of the Lai and Vere-Jones identity (the exponential as a sum of logarithms with Dirichlet eta weights)

Volume I, Proposition 10.1.9, page 173 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\mu\) be the Möbius function and \(\eta(z)=\sum_{k\ge1}(-1)^{k+1}k^{-z}=(1-2^{1-z})\zeta(z)\) Dirichlet’s eta function. Define \(d_n\) by \(\sum_nd_nn^{-v}=1/\eta(1+v)\), that is \(d_n=\sum_{2^jm=n}\mu(m)/m\). Then \(|d_n|\le1\) for all \(n\ge1\), and for \(\operatorname{Re}x>0\), \(e^{-x}=\sum_{m\ge1}d_m\log\bigl(1+e^{-mx}\bigr)\).

Supersedes: Proposition 27.6 of Polson (2026) states the same identity with the weaker bound \(|d_n|\le2\).

Earlier editions: Proposition 6.1.6, page 140 (edition 1).

Source: ch/rz1.tex, label rz:prop:eta.

Sawtooth Thorin measure of a Vere-Jones block

Volume I, Lemma 10.1.10, page 174 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\lambda>0\), let \(s\) lie in the cut plane \(\mathbb{C}\setminus(-\infty,0]\) with \(\sqrt s\) the principal square root, and put \(\sigma_k(\lambda)=\bigl((2k+1)\pi/\lambda\bigr)^2\) for \(k\ge0\). Then \(\log\bigl(1+e^{-\lambda\sqrt s}\bigr)=\log2-\int\log\bigl(1+\frac st\bigr)\,U_\lambda(dt)\) with \(U_\lambda(dt)=\frac{\lambda}{4\pi\sqrt t}\,dt-\sum_{k\ge0}\delta_{\sigma_k(\lambda)}\), where \(\delta_a\) is the unit point mass at \(a\). Equivalently \(U_\lambda(dt)=-d_t\Sigma_\lambda(\sqrt t)\), where \(\Sigma_\lambda(\tau)=\#\{k\ge0:(2k+1)\pi/\lambda\le\tau\}-\lambda\tau/(2\pi)\) is a sawtooth function with \(|\Sigma_\lambda|\le\frac12\). So the reciprocal block \(2/(1+e^{-\lambda\sqrt s})\) has the signed Thorin type measure \(-U_\lambda=\sum_{k\ge0}\delta_{\sigma_k(\lambda)}-\frac{\lambda}{4\pi\sqrt t}\,dt\).

Earlier editions: Lemma 6.1.7, page 140 (edition 1).

Source: ch/rz1.tex, label rz:lem:saw.

Vere-Jones representation of log zeta by sawtooth blocks

Volume I, Theorem 10.1.11, page 174 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\Lambda\) be the von Mangoldt function and \(\mu\) the Möbius function, and put \(a_n=\Lambda(n)/(\sqrt n\log n)\), \(d_m=\sum_{2^jk=m}\mu(k)/k\) (so that \(\sum_m d_mm^{-v}=1/\eta(1+v)\) for Dirichlet’s eta function \(\eta\), and \(|d_m|\le1\)), \(B_N=\sum_{n^m=N}d_m\,a_n\) (the sum over \(n\ge2\) and \(m\ge1\) with \(n^m=N\)) and \(\sigma_k(\lambda)=\bigl((2k+1)\pi/\lambda\bigr)^2\). For \(s\) in the region \(H_{1/2}=\{s:\operatorname{Re}\sqrt s>\frac12\}\), with \(\sqrt s\) the principal square root, \(\log\zeta\bigl(\tfrac12+\sqrt s\bigr)=\sum_{N\ge2}B_N\log\bigl(1+N^{-\sqrt s}\bigr)\) and \(\frac{d}{ds}\log\zeta\bigl(\tfrac12+\sqrt s\bigr)=\sum_{N\ge2}B_N\Bigl[\sum_{k\ge0}\frac{1}{s+\sigma_k(\log N)}-\frac{\log N}{4\sqrt s}\Bigr]\). The first values are \(B_p=p^{-1/2}\) for primes \(p\), \(B_4=\frac14+\frac1{2\sqrt2}\), \(B_8=-\frac1{6\sqrt2}\) and \(B_9=\frac16+\frac{1}{2\sqrt3}\).

Earlier editions: Theorem 6.1.8, page 141 (edition 1).

Source: ch/rz1.tex, label rz:thm:saw.

Thorin type measure of the normalised reciprocal zeta function under RH

Volume I, Theorem 10.1.13, page 175 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: conditional on The Riemann Hypothesis: every nontrivial zero of the Riemann zeta function has real part \(\frac12\).

Statement. Assume the Riemann Hypothesis. Let \(\zeta\) be the Riemann zeta function, \(\varphi(s)=\zeta(\frac12)/\{(1-4s)\,\zeta(\frac12+\sqrt s)\}\) (so \(\varphi(0)=1\)), \(\theta(\tau)=\operatorname{Im}\log\Gamma(\frac14+\frac{i\tau}2)-\frac\tau2\log\pi\) the Riemann Siegel theta function, and \(N(T)\) the number of zeros \(\frac12+i\gamma\) of \(\zeta\) with \(0<\gamma\le T\), counted with multiplicity, as are the sums over \(\gamma\) below. Then for \(s\) in the cut plane \(\mathbb{C}\setminus(-\infty,0]\), \(\log\varphi(s)=-\int_{(0,\infty)}\log\bigl(1+\tfrac st\bigr)\,dM(t)\) with \(M(t)=N(\sqrt t)-\theta(\sqrt t)/\pi\), that is \(dM(t)=\sum_{\gamma>0}\delta_{\gamma^2}-\frac{\theta'(\sqrt t)}{2\pi\sqrt t}\,dt\) and \(-\frac{\varphi'}{\varphi}(s)=\int\frac{dM(t)}{s+t}\). By the Riemann von Mangoldt formula \(M(t)=1+S(\sqrt t)\) for \(\sqrt t\) not an ordinate, where \(S(T)=\pi^{-1}\arg\zeta(\frac12+iT)\) is the argument function and \(S(0^+)=-1\). In particular \(dM\) is not a positive measure, so \(\varphi\) is not the Laplace transform of a GGC even under RH.

Earlier editions: Theorem 6.1.10, page 142 (edition 1).

Source: ch/rz1.tex, label rz:thm:thorin.

The normalised reciprocal of xi on the cut plane is analytic and zero free if and only if RH holds

Volume I, Proposition 10.1.12, page 175 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(z)=\frac12z(z-1)\pi^{-z/2}\Gamma(z/2)\zeta(z)\), \(\mathbb{C}\setminus(-\infty,0]\) the cut plane with \(\sqrt s\) the principal root, \(R(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\), \(G(s)=\Gamma(\frac14+\frac{\sqrt s}2)\pi^{-\sqrt s/2}/\Gamma(\frac14)\) and \(\varphi=RG\), which equals \(\zeta(\frac12)/((1-4s)\zeta(\frac12+\sqrt s))\). Then \(\varphi\) is meromorphic on the cut plane, and it is analytic and zero free there if and only if the Riemann Hypothesis holds. On \((0,\infty)\) it is analytic and positive, continuous at \(0\) with \(\varphi(0)=1\), and \(-\varphi'/\varphi\) is analytic in a neighbourhood of \((0,\infty)\).

Source: ch/rz1.tex, label rz:prop:rh.

Canonical function of the normalised reciprocal zeta function, prime and zero forms

Volume I, Theorem 10.1.14, page 177 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\Lambda\) the von Mangoldt function, \(\varphi(s)=\zeta(\frac12)/\{(1-4s)\,\zeta(\frac12+\sqrt s)\}\) the normalised reciprocal zeta function, and \(\theta(\tau)=\operatorname{Im}\log\Gamma(\frac14+\frac{i\tau}2)-\frac\tau2\log\pi\) the Riemann Siegel theta function. Define, with no hypothesis, \(w(x)=e^{x/4}-P(x)\) with \(P(x)=\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\,e^{-(\log n)^2/4x}\) for \(x>0\). Then \(-\varphi'/\varphi(s)=\int_0^\infty e^{-sx}w(x)\,dx\) for real \(s>\frac14\), and for every \(x>0\), \(w(x)=W(x)-W_0(x)\) with \(W(x)=\frac12\sum_\rho e^{x(\rho-\frac12)^2}\) and \(W_0(x)=\frac1\pi\int_0^\infty e^{-x\tau^2}\theta'(\tau)\,d\tau\), the sum running over all nontrivial zeros \(\rho\) of \(\zeta\) with multiplicity. Under RH, \(W(x)=\sum_{\gamma>0}e^{-x\gamma^2}\) and \(w(x)=\int e^{-tx}\,dM(t)\), where \(dM(t)=\sum_{\gamma>0}\delta_{\gamma^2}-\frac{\theta'(\sqrt t)}{2\pi\sqrt t}\,dt\) and \(\gamma>0\) runs over the ordinates of the zeros \(\frac12+i\gamma\) (counted with multiplicity). The identity \(w=W-W_0\) is Weil’s explicit formula for the Gaussian test function, in the arithmetic form of the heat trace stated as Theorem 4.4 of Polson (2026). Moreover \(w(0^+)=1\) and \(w(x)\sim c/(2\sqrt{\pi x})\) as \(x\to\infty\), with \(c=-\theta'(0)=\zeta'(\frac12)/\zeta(\frac12)\approx2.68609\).

Earlier editions: Theorem 6.1.11, page 143 (edition 1).

Source: ch/rz1.tex, label rz:thm:w.

Theta function series for the canonical function w

Volume I, Proposition 10.1.16, page 178 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\Lambda\) be the von Mangoldt function, \(\mu\) the Möbius function, \(d_m=\sum_{2^jk=m}\mu(k)/k\), \(B_N=\sum_{n^m=N}d_m\Lambda(n)/(\sqrt n\log n)\) (the sum over \(n\ge2\) and \(m\ge1\) with \(n^m=N\)), \(\sigma_k(\lambda)=\bigl((2k+1)\pi/\lambda\bigr)^2\), and \(h_\lambda(x)=\sum_{k\ge0}e^{-x\sigma_k(\lambda)}-\frac{\lambda}{4\sqrt{\pi x}}=\frac{\lambda}{2\sqrt{\pi x}}\sum_{j\ge1}(-1)^je^{-j^2\lambda^2/4x}\). Let \(w(x)=e^{x/4}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\,e^{-(\log n)^2/4x}\). Then for every \(x>0\), with absolutely convergent sums, \(w(x)=e^{x/4}+\sum_{N\ge2}B_N\,h_{\log N}(x)\).

Earlier editions: Proposition 6.1.13, page 144 (edition 1).

Source: ch/rz1.tex, label rz:prop:theta.

Sawtooth expansion of the argument function after Gaussian averaging, under RH

Volume I, Corollary 10.1.17, page 178 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: conditional on The Riemann Hypothesis: every nontrivial zero of the Riemann zeta function has real part \(\frac12\).

Statement. Assume the Riemann Hypothesis. Let \(S(\tau)=\pi^{-1}\arg\zeta(\frac12+i\tau)\) be the argument function of the Riemann zeta function, let \(B_N\) be the coefficients of the Vere-Jones representation of \(\log\zeta\) (\(B_N=\sum_{n^m=N}d_m\Lambda(n)/(\sqrt n\log n)\) with \(d_m=\sum_{2^jk=m}\mu(k)/k\), \(\Lambda\) the von Mangoldt function and \(\mu\) the Möbius function), let \(\Sigma_\lambda(\tau)=\#\{k\ge0:(2k+1)\pi/\lambda\le\tau\}-\lambda\tau/(2\pi)\) be the sawtooth function, and let \(h_\lambda(x)=\sum_{k\ge0}e^{-x((2k+1)\pi/\lambda)^2}-\frac{\lambda}{4\sqrt{\pi x}}\). Then for every \(x>0\), \(\int_0^\infty S(\tau)\,2\tau e^{-x\tau^2}\,d\tau=\frac{e^{x/4}-1}{x}+\sum_{N\ge2}B_N\int_0^\infty\Sigma_{\log N}(\tau)\,2\tau e^{-x\tau^2}\,d\tau\), and each integral on the right equals \(h_{\log N}(x)/x\).

Earlier editions: Corollary 6.1.14, page 145 (edition 1).

Source: ch/rz1.tex, label rz:cor:Ssaw.

Criterion for RH from the canonical function w

Volume I, Theorem 10.1.18, page 179 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\Lambda\) be the von Mangoldt function and \(w(x)=e^{x/4}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\,e^{-(\log n)^2/4x}\) for \(x>0\). The following are equivalent. (i) The Riemann Hypothesis. (ii) There is a real signed Borel measure \(m\) on \([0,\infty)\) with \(\int|m|(dt)/(1+t)<\infty\) and \(w(x)=\int e^{-tx}m(dt)\) for all \(x>0\). In that case \(m=dM\), where \(dM(t)=\sum_{\gamma>0}\delta_{\gamma^2}-\frac{\theta'(\sqrt t)}{2\pi\sqrt t}\,dt\), the sum is over the ordinates \(\gamma>0\) of the zeros \(\frac12+i\gamma\) of the Riemann zeta function counted with multiplicity, and \(\theta(\tau)=\operatorname{Im}\log\Gamma(\frac14+\frac{i\tau}2)-\frac\tau2\log\pi\) is the Riemann Siegel theta function.

Earlier editions: Theorem 6.1.15, page 145 (edition 1).

Source: ch/rz1.tex, label rz:thm:crit.

Thorin measure of the gamma factor as a renormalised product of Poisson kernels

Volume I, Theorem 10.2.1, page 180 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(G(s)=\Gamma(\frac14+\frac{\sqrt s}2)\,\pi^{-\sqrt s/2}/\Gamma(\frac14)\) be the gamma factor, \(c_n=2n+\frac12\) for \(n\ge0\), \(b=\frac12(\gamma_E+\log\pi)\) with \(\gamma_E\) Euler’s constant, and \(\theta(\tau)=\operatorname{Im}\log\Gamma(\frac14+\frac{i\tau}2)-\frac\tau2\log\pi\) the Riemann Siegel theta function. For \(s\) in the cut plane \(\mathbb{C}\setminus(-\infty,0]\), \(G(s)=e^{-b\sqrt s}\prod_{n\ge0}\frac{e^{\sqrt s/(2n+2)}}{1+\sqrt s/c_n}\), and the Thorin type measure \(U_G\) of \(G\), defined by \(G(s)=\exp\bigl(-\int_{(0,\infty)}\log(1+\frac st)\,U_G(dt)\bigr)\), is \(U_G(dt)=\frac{1}{2\pi\sqrt t}\Bigl\{b+\sum_{n\ge0}\Bigl(\frac{c_n}{t+c_n^2}-\frac{1}{2n+2}\Bigr)\Bigr\}dt=-\frac{\theta'(\sqrt t)}{2\pi\sqrt t}\,dt\). Each factor \((1+\sqrt s/\kappa)^{-1}\) with \(\kappa>0\) is the Laplace transform of a generalized gamma convolution (GGC) with positive Thorin density \(q_\kappa(t)=\frac{\kappa}{2\pi\sqrt t\,(t+\kappa^2)}\), a Poisson kernel in \(\sqrt t\); each factor \(e^{a\sqrt s}\) with \(a>0\) has the negative Thorin density \(-a/(2\pi\sqrt t)\).

Earlier editions: Theorem 6.2.1, page 147 (edition 1).

Source: ch/rz2.tex, label rz:thm:gamma.

Closed form heat trace over the zeros and formal Thorin measure of the reciprocal Xi function

Volume I, Theorem 10.2.3, page 181 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\xi(z)=\frac12z(z-1)\pi^{-z/2}\Gamma(z/2)\zeta(z)\) Riemann’s xi function, \(W(x)=\frac12\sum_\rho e^{x(\rho-\frac12)^2}\) for \(x>0\) (the sum over the nontrivial zeros \(\rho\) of \(\zeta\) with multiplicity), \(\Lambda\) the von Mangoldt function and \(\mu\) the Möbius function. Put \(d_m=\sum_{2^jk=m}\mu(k)/k\), \(B_N=\sum_{n^m=N}d_m\Lambda(n)/(\sqrt n\log n)\), \(\sigma_k(\lambda)=\bigl((2k+1)\pi/\lambda\bigr)^2\), \(h_\lambda(x)=\sum_{k\ge0}e^{-x\sigma_k(\lambda)}-\frac{\lambda}{4\sqrt{\pi x}}\), \(b=\frac12(\gamma_E+\log\pi)\) with \(\gamma_E\) Euler’s constant, and \(c_n=2n+\frac12\). Then for every \(x>0\), with absolutely convergent sums, \(W(x)=e^{x/4}+\sum_{N\ge2}B_N\,h_{\log N}(x)-\frac{b}{2\sqrt{\pi x}}+\sum_{n\ge0}\Bigl\{\frac{1}{(4n+4)\sqrt{\pi x}}-\frac12e^{c_n^2x}\operatorname{erfc}\bigl(c_n\sqrt x\bigr)\Bigr\}\). Equivalently, \(W(x)\) is the Laplace transform, taken term by term, of the formal series \(U_R=\delta_{-1/4}+\sum_{N\ge2}B_N\Bigl\{\sum_{k\ge0}\delta_{\sigma_k(\log N)}-\frac{\log N}{4\pi\sqrt t}\,dt\Bigr\}-\frac{1}{2\pi\sqrt t}\Bigl\{b+\sum_{n\ge0}\Bigl(\frac{c_n}{t+c_n^2}-\frac1{2n+2}\Bigr)\Bigr\}dt\), the formal Thorin measure of \(R(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\). Under RH, \(R\) is a GGC (generalized gamma convolution), and its Thorin measure \(\sum_{\gamma>0}\delta_{\gamma^2}=dN(\sqrt t)\), where \(N(T)\) counts the zeros \(\frac12+i\gamma\) with \(0<\gamma\le T\), is a positive measure whose Laplace transform is also \(W\).

Supersedes: Theorem 27.18 of Polson (2026) states that the series \(U_R\) is the Thorin measure of \(R\) in the weak sense of Laplace transforms on \(x>0\), and that \(U_R=\sum_{\gamma>0}\delta_{\gamma^2}\) under RH.

Earlier editions: Theorem 6.2.3, page 149 (edition 1).

Source: ch/rz2.tex, label rz:thm:RLVJ.

Sine squared lemma (Polson 2017, Lemma 1)

Volume I, Lemma 10.2.4, page 183 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Suppose that \(f\) is a function, \(\alpha\) a real number and \(\mu\) a measure on \((0,\infty)\) with \(\int_0^\infty(1\wedge x)\,e^{-\alpha x}\,\mu(dx)<\infty\) such that \(\log\frac{f(\alpha+s)}{f(\alpha)}-s\frac{f'(\alpha)}{f(\alpha)}=\int_0^\infty\bigl(e^{-sx}-1+sx\bigr)e^{-\alpha x}\,\frac{\mu(dx)}x\) for \(s\ge0\). Then, for \(s\ge0\), \(\log\frac{f(\alpha)}{f(\alpha+s)}+s\frac{f'(\alpha)}{f(\alpha)}=-\int_0^\infty\bigl(1-e^{-\frac12s^2t}\bigr)\frac{\nu_\alpha(t)}t\,dt\) with \(\nu_\alpha(t)=\frac{1}{\sqrt{2\pi}}\int_0^\infty e^{-tz}\Bigl\{\int_0^\infty2\sin^2\bigl(x\sqrt{z/2}\bigr)e^{-\alpha x}\,\mu(dx)\Bigr\}\frac{dz}{\sqrt{\pi z}}\). The map \(\mu\mapsto\nu_\alpha\) is linear and positivity preserving.

Earlier editions: Lemma 6.2.4, page 150 (edition 1).

Source: ch/rz2.tex, label rz:lem:P1.

Sine squared lemma in the variable s = u squared (Polson 2017, Lemma 1)

Volume I, Lemma 10.2.5, page 184 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Let \(\tilde\mu\) be a signed measure on \((0,\infty)\) with \(\int(1\wedge x)\,|\tilde\mu|(dx)<\infty\), and let \(f\) be a function with \(\log\frac{f(u)}{f(0)}-u\frac{f'(0)}{f(0)}=\int_0^\infty\bigl(e^{-ux}-1+ux\bigr)\frac{\tilde\mu(dx)}x\) for real \(u\ge0\). Then for \(s\) in the cut plane \(\mathbb{C}\setminus(-\infty,0]\), \(\log\frac{f(0)}{f(\sqrt s)}+\sqrt s\,\frac{f'(0)}{f(0)}=-\int_0^\infty\log\Bigl(1+\frac st\Bigr)U(t)\,dt\) with \(U(t)=\frac{1}{2\pi\sqrt t}\int_0^\infty2\sin^2\Bigl(\frac{x\sqrt t}2\Bigr)\tilde\mu(dx)\). If \(\tilde\mu\ge0\) then \(U\ge0\), and \(s\mapsto f(0)\,e^{\sqrt s f'(0)/f(0)}/f(\sqrt s)\) is the Laplace transform of a GGC (generalized gamma convolution). Since \(\sqrt s=\int\log(1+s/t)\,\frac{dt}{2\pi\sqrt t}\), the Thorin type measure of \(f(0)/f(\sqrt s)\) itself is \(U(t)\,dt+\frac{f'(0)}{f(0)}\frac{dt}{2\pi\sqrt t}\).

Earlier editions: Lemma 6.2.5, page 151 (edition 1).

Source: ch/rz2.tex, label rz:lem:sin2.

Reciprocal zeta to the right of the line Re z = 1 is a GGC after a stable tilt

Volume I, Proposition 10.2.6, page 185 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function and \(\Lambda\) the von Mangoldt function. For \(\beta>1\) and \(s\) in the cut plane \(\mathbb{C}\setminus(-\infty,0]\), \(\frac{\zeta(\beta)}{\zeta(\beta+\sqrt s)}\,e^{\sqrt s\,\zeta'(\beta)/\zeta(\beta)}=\exp\Bigl(-\int_0^\infty\log\Bigl(1+\frac st\Bigr)U_\beta(t)\,dt\Bigr)\) with \(U_\beta(t)=\frac{1}{2\pi\sqrt t}\sum_{n\ge2}\frac{\Lambda(n)}{n^\beta}\,2\sin^2\Bigl(\frac{\sqrt t\log n}2\Bigr)\ge0\), so the left side is the Laplace transform of a generalized gamma convolution (GGC) with Thorin density \(U_\beta\). Adding the tilt, the Thorin type measure of \(\zeta(\beta)/\zeta(\beta+\sqrt s)\) is \(-\frac{1}{2\pi\sqrt t}\sum_{n\ge2}\Lambda(n)n^{-\beta}\cos(\sqrt t\log n)\,dt\).

Earlier editions: Proposition 6.2.6, page 152 (edition 1).

Source: ch/rz2.tex, label rz:prop:zetaright.

Positive Thorin density of a shifted Vere-Jones block

Volume I, Proposition 10.2.7, page 185 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\lambda>0\), \(\alpha>0\) and \(q=e^{-\lambda\alpha}\), let \(f(u)=1+e^{-\lambda(\alpha+u)}\), for which \(\log\frac{f(u)}{f(0)}-u\frac{f'(0)}{f(0)}=\int_0^\infty\bigl(e^{-ux}-1+ux\bigr)\frac{\tilde\mu_{\lambda,\alpha}(dx)}x\) with the alternating lattice \(\tilde\mu_{\lambda,\alpha}=\lambda\sum_{k\ge1}(-1)^{k+1}q^k\delta_{k\lambda}\), and let \(\sigma_k(\lambda)=\bigl((2k+1)\pi/\lambda\bigr)^2\). The sine squared transform of \(\tilde\mu_{\lambda,\alpha}\) is positive: \(U_{\lambda,\alpha}(t)=\frac{1}{2\pi\sqrt t}\int_0^\infty2\sin^2\bigl(\frac{x\sqrt t}2\bigr)\tilde\mu_{\lambda,\alpha}(dx)=\frac{\lambda}{2\pi\sqrt t}\,\frac{q(1-q)\bigl(1-\cos\lambda\sqrt t\bigr)}{(1+q)\bigl(1+2q\cos\lambda\sqrt t+q^2\bigr)}\ge0\), so \(\frac{1+e^{-\lambda\alpha}}{1+e^{-\lambda(\alpha+\sqrt s)}}\,e^{-\sqrt s\,\lambda q/(1+q)}\) is the Laplace transform of a generalized gamma convolution (GGC) with Thorin density \(U_{\lambda,\alpha}\). As \(\alpha\downarrow0\), \(U_{\lambda,\alpha}(t)\,dt\to\sum_{k\ge0}\delta_{\sigma_k(\lambda)}\) vaguely, the tilt density tends to \(-\frac{\lambda}{4\pi\sqrt t}\), and the Thorin type measure of \(2/(1+e^{-\lambda\sqrt s})\) is recovered as \(\sum_{k\ge0}\delta_{\sigma_k(\lambda)}-\frac{\lambda}{4\pi\sqrt t}\,dt\).

Supersedes: Proposition 28.4 of Polson (2026) states that \(\frac{1+e^{-\lambda(\alpha+\sqrt s)}}{1+e^{-\lambda\alpha}}\,e^{-\sqrt s\,\lambda q/(1+q)}\), the block rather than its reciprocal, is the Laplace transform of a generalized gamma convolution (GGC), and that \(U_{\lambda,\alpha}(t)\,dt\to\sum_{k\ge0}\delta_{\sigma_k(\lambda)}\) weakly as \(\alpha\downarrow0\), where the book has vaguely because the limit has infinite mass.

Earlier editions: Proposition 6.2.7, page 152 (edition 1).

Source: ch/rz2.tex, label rz:prop:block.

Divergence of the cosine Lévy series of the Vere-Jones blocks

Volume I, Proposition 10.2.9, page 187 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. For a measure \(U\) on \((0,\infty)\) put \(K_U(x)=2\int e^{-\sqrt t\,x}\,U(dt)\), let \(U_\lambda(dt)=\frac{\lambda}{4\pi\sqrt t}\,dt-\sum_{k\ge0}\delta_{((2k+1)\pi/\lambda)^2}\) be the sawtooth measure, for which \(K_{U_\lambda}(x)=\frac{\lambda}{\pi x}-\frac{1}{\sinh(\pi x/\lambda)}\), and let \(B_N=\sum_{n^m=N}d_m\Lambda(n)/(\sqrt n\log n)\) with \(d_m=\sum_{2^jk=m}\mu(k)/k\) be the coefficients of the Vere-Jones representation of \(\log\zeta\) (\(\Lambda\) the von Mangoldt function, \(\mu\) the Möbius function). Then for every \(x>0\) the series \(\sum_{N\ge2}B_N\,K_{U_{\log N}}(x)\), summed in order of increasing \(N\), diverges.

Earlier editions: Proposition 6.2.9, page 154 (edition 1).

Source: ch/rz2.tex, label rz:prop:diverge.

The Meixner density of the gamma factor is not a Pólya frequency function of infinite order

Volume I, Proposition 10.2.10, page 187 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(p(\tau)=|\Gamma(\frac14+\frac{i\tau}2)|^2/(2^{3/2}\pi^{3/2})\) be the Meixner density, a probability density on \(\mathbb{R}\) with characteristic function \(\int p(\tau)e^{iv\tau}\,d\tau=\operatorname{sech}(v)^{1/2}\). Then \(p\) is not a Pólya frequency function of infinite order, that is, in the sense of Schoenberg, the kernel \(p(x-y)\) is not totally positive of every order. The proof applies Schoenberg’s theorem: the reciprocal of the characteristic function of a Pólya frequency density of infinite order is an entire function (of the Laguerre Pólya class), whereas here the reciprocal is \(\cosh(v)^{1/2}\), which has branch points at the simple zeros \(v=i(k+\frac12)\pi\) of \(\cosh\) and is not entire.

Earlier editions: Proposition 6.2.10, page 154 (edition 1).

Source: ch/rz2.tex, label rz:prop:meixstruct.

Bernstein function of the gamma factor: theta derivative and Meixner score

Volume I, Proposition 10.2.11, page 188 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\psi=\Gamma'/\Gamma\) be the digamma function, \(\theta(\tau)=\operatorname{Im}\log\Gamma(\frac14+\frac{i\tau}2)-\frac\tau2\log\pi\) the Riemann Siegel theta function, \(p(\tau)=|\Gamma(\frac14+\frac{i\tau}2)|^2/(2^{3/2}\pi^{3/2})\) the Meixner density, \(c_k=2k+\frac12\), and \(\nu(dy)=\frac{e^{y/2}}{\sinh y}\,dy=2\sum_{k\ge0}e^{-c_ky}\,dy\) on \((0,\infty)\). Then \(\Phi_\Gamma(\lambda)=\psi(\frac14+\frac\lambda2)-\psi(\frac14)=\int_0^\infty(1-e^{-\lambda y})\,\nu(dy)\) is a Bernstein function, and its boundary values on the imaginary axis are \(\operatorname{Re}\Phi_\Gamma(i\tau)=2\bigl(\theta'(\tau)-\theta'(0)\bigr)=\int_0^\infty(1-\cos\tau y)\,\nu(dy)\) and \(\operatorname{Im}\Phi_\Gamma(i\tau)=-\frac{d}{d\tau}\log p(\tau)=\int_0^\infty\sin(\tau y)\,\nu(dy)\). Consequently \(2\theta(\tau)+\tau\log\pi\) and \(\log p(\tau)\) are, up to constants, the imaginary and real parts of \(2\log\Gamma(\frac14+\frac{i\tau}2)\), the boundary value of a function analytic in the right half plane, so \(2\theta(\tau)+\tau\log\pi\) is the harmonic conjugate of the Meixner log density. The smooth heat trace is the heat trace of \(\nu\): \(W_0(x)=\frac1\pi\int_0^\infty e^{-x\tau^2}\theta'(\tau)\,d\tau=\frac{\theta'(0)}{2\sqrt{\pi x}}+\frac{1}{4\sqrt{\pi x}}\int_0^\infty\bigl(1-e^{-y^2/4x}\bigr)\,\nu(dy)\).

Supersedes: Proposition 28.16 of Polson (2026) says that the smooth part \(\theta/\pi\) of the zero counting function, rather than \(2\theta(\tau)+\tau\log\pi\), is the harmonic conjugate of the Meixner log density.

Earlier editions: Proposition 6.2.11, page 155 (edition 1).

Source: ch/rz2.tex, label rz:prop:bernstein.

The reciprocal gamma factor is a GGC after a stable tilt

Volume I, Theorem 10.2.12, page 189 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\theta(\tau)=\operatorname{Im}\log\Gamma(\frac14+\frac{i\tau}2)-\frac\tau2\log\pi\) be the Riemann Siegel theta function, \(\psi=\Gamma'/\Gamma\), \(c=-\theta'(0)=\frac12\bigl(\log\pi-\psi(\frac14)\bigr)\approx2.68609\) (which equals \(\zeta'(\frac12)/\zeta(\frac12)\)), \(G(s)=\Gamma(\frac14+\frac{\sqrt s}2)\,\pi^{-\sqrt s/2}/\Gamma(\frac14)\) the gamma factor, and \(c_k=2k+\frac12\). Then for \(s\) in the cut plane \(\mathbb{C}\setminus(-\infty,0]\), \(\frac{e^{-c\sqrt s}}{G(s)}=\frac{\Gamma(\frac14)\,\pi^{\sqrt s/2}\,e^{-c\sqrt s}}{\Gamma(\frac14+\frac{\sqrt s}2)}=\exp\Bigl(-\int_0^\infty\log\Bigl(1+\frac st\Bigr)U_\Gamma(t)\,dt\Bigr)\) with \(U_\Gamma(t)=\frac{\theta'(\sqrt t)-\theta'(0)}{2\pi\sqrt t}=\frac{1}{2\pi\sqrt t}\sum_{k\ge0}\frac{t}{c_k\,(c_k^2+t)}\ge0\). So \(e^{-c\sqrt s}/G(s)\) is the Laplace transform of a generalized gamma convolution (GGC), and its Thorin distribution function is \(\bigl(\theta(\sqrt t)-\theta'(0)\sqrt t\bigr)/\pi\), a convex increasing function of \(\sqrt t\).

Earlier editions: Theorem 6.2.12, page 155 (edition 1).

Source: ch/rz2.tex, label rz:thm:gammaggc.

Factorisation of the Pólya autocorrelation through the Meixner law

Volume I, Proposition 10.2.14, page 191 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(\tau)=\xi(\frac12+i\tau)\), and \(\Phi\) the Pólya kernel, the even function with \(\Xi(\tau)=\int_{\mathbb{R}}\Phi(u)e^{i\tau u}\,du\). Let \(p(\tau)=|\Gamma(\frac14+\frac{i\tau}2)|^2/(2^{3/2}\pi^{3/2})\) be the Meixner density, with characteristic function \(\int p(\tau)e^{iv\tau}d\tau=\operatorname{sech}(v)^{1/2}\), and put \(\mathcal{C}(v)=\int p(\tau)\,|\zeta(\frac12+i\tau)|^2e^{iv\tau}\,d\tau\). Then the integral \(\mathcal{C}(v)\) converges absolutely, and \((\Phi*\Phi)(v)=2^{-3/2}\Bigl(\frac{d^2}{dv^2}-\frac14\Bigr)^2\mathcal{C}(v)\). Formally \(\mathcal{C}=\operatorname{sech}(\cdot)^{1/2}*\mathcal{A}\) with \(\mathcal{A}=\sum_{m,n\ge1}\frac{\delta_{\log(m/n)}}{\sqrt{mn}}\), where \(\operatorname{sech}(v)^{1/2}\) is the characteristic function of the Meixner law and \(\mathcal{A}\), whose Fourier transform is formally \(|\zeta(\frac12+i\tau)|^2\), is the autocorrelation of the measure \(\sum_nn^{-1/2}\delta_{\log n}\). The measure \(\mathcal{A}\) is positive and positive definite, and the series for \(\mathcal{C}\) diverges: truncated at \(m,n\le X\) it equals \(X\int\operatorname{sech}(v-r)^{1/2}e^{-|r|/2}\,dr+O(X^{1/2})\). This growth comes from the pole of \(\zeta\), and the factor \((\frac{d^2}{dv^2}-\frac14)^2\) does not remove it, since \((\frac{d^2}{dv^2}-\frac14)e^{-|v|/2}=-\delta_0\).

Supersedes: Proposition 28.22 of Polson (2026) states the factorisation formally with the constant \(\frac14\pi^{-1/2}2^{3/2}\pi^{3/2}\), and says that the convolution \(\operatorname{sech}(\cdot)^{1/2}*\mathcal{A}\) diverges logarithmically and that the factor \(\bigl(\frac{d^2}{dv^2}-\frac14\bigr)^2\) removes the divergence.

Earlier editions: Proposition 6.2.14, page 158 (edition 1).

Source: ch/rz2.tex, label rz:prop:autocorr.

Splitting of the canonical function of the floor of the reciprocal Xi function, and GGC if and only if RH

Volume I, Theorem 10.2.15, page 192 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi(z)=\frac12z(z-1)\pi^{-z/2}\Gamma(z/2)\zeta(z)\) be Riemann’s xi function, \(c=\zeta'(\frac12)/\zeta(\frac12)=-\theta'(0)\approx2.68609\) with \(\theta\) the Riemann Siegel theta function, and \(W(x)=\frac12\sum_\rho e^{x(\rho-\frac12)^2}\) for \(x>0\), the sum over the nontrivial zeros \(\rho\) of \(\zeta\) with multiplicity. Unconditionally, \(e^{-c\sqrt s}\,\xi(\frac12)/\xi(\frac12+\sqrt s)\) is the Laplace transform of an infinitely divisible law on \((0,\infty)\) with Lévy density \(\frac{W(x)+c/(2\sqrt{\pi x})}{x}\), and \(W(x)+c/(2\sqrt{\pi x})>0\) for all \(x>0\). The following are equivalent: (i) the Riemann Hypothesis; (ii) this law is a generalized gamma convolution (GGC); (iii) \(W(x)+c/(2\sqrt{\pi x})\) is completely monotone; (iv) \(W\) is completely monotone. The proof uses the positivity of \(w(x)=e^{x/4}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}e^{-(\log n)^2/4x}\) for all \(x>0\) (\(\Lambda\) the von Mangoldt function), which rests on the verification of RH to height \(50\), an explicit bound for the zero counting function (Hasanalizade, Shen and Wong 2022) and three interval enclosures. That the floor is infinitely divisible also follows from Theorem A of Polson (2026), which gives \(W>0\) and \(W'<0\).

Earlier editions: Theorem 6.2.15, page 159 (edition 1).

Source: ch/rz2.tex, label rz:thm:floor.

Derivative signs of the canonical function of the floor to order ten billion, a separate proof (Williamson measures)

Volume I, Theorem 10.2.17, page 195 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(k(y)=W(y)+c/(2\sqrt{\pi y})\) with \(W(y)=\frac12\sum_\rho e^{y(\rho-\frac12)^2}\) (the sum over the nontrivial zeros \(\rho\) of the Riemann zeta function \(\zeta\) with multiplicity) and \(c=\zeta'(\frac12)/\zeta(\frac12)\). For every integer \(n\) with \(0\le n\le10^{10}\) and every \(y>0\), \((-1)^n\,k^{(n)}(y)>0\). Consequently, by Williamson’s theorem, for each \(1\le n\le10^{10}\) there is a positive measure \(\Omega_n\) on \((0,\infty)\) with \(k(y)=\int(1-ty)_+^{\,n-1}\,\Omega_n(dt)\). For \(n\ge1\) the proof uses the verification of RH to height \(T_0=3\times10^{12}\) (Platt and Trudgian 2021), an explicit bound for the zero counting function (Hasanalizade, Shen and Wong 2022) and one interval enclosure, and no other information about the zeros. This is a separate proof of a smaller range: the same signs hold for every \(n\le10^{17}\) by Theorem 7.10 of Polson’s arXiv paper 1708.02653 (version 24, 2026), which proves them for \(W\), and the book proves them for every \(n\le1.16\times10^{25}\) by sharper estimates of the heat trace.

Earlier editions: Theorem 6.2.17, page 161 (edition 1).

Source: ch/rz2.tex, label rz:thm:nmono.

Volume I, Chapter 11. Infinite divisibility, necklaces and solved models

The normalised reciprocal zeta function is infinitely divisible

Volume I, Theorem 11.1.1, page 197 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\sqrt s\) the principal square root on \(\mathbb{C}\setminus(-\infty,0]\), \(\Lambda\) the von Mangoldt function and [ (s)=,w(x)=e^{x/4}-_{n},e^{-(n)2/4x}(x>0), ] so that \(\varphi\) is the normalised reciprocal zeta function (the factor \(1-4s\) removes the pole of \(\zeta(\frac12+\sqrt s)\) at \(s=\frac14\)) and \(w\) is its canonical function: \(-\varphi'(s)/\varphi(s)=\int_0^\infty e^{-sx}w(x)\,dx\) for \(s>\frac14\). Then \(w(x)>0\) for every \(x>0\). Consequently, unconditionally (with no hypothesis on the zeros of \(\zeta\)), \(\varphi\) is the Laplace transform of an infinitely divisible probability law on \((0,\infty)\) with zero drift and Lévy density \(w(x)/x\): [ =(-_0^(1-e{-sx}),dx),s>0 . ] Equivalently, the positivity of \(w\) is the Gaussian smoothed Chebyshev bound \(\sum_{n\ge2}\Lambda(n)\,n^{-1/2}e^{-(\log n)^2/4x}<\sqrt{4\pi x}\,e^{x/4}\) for all \(x>0\). The proof uses the verification of the Riemann Hypothesis up to height \(50\), an explicit Riemann von Mangoldt bound for the zero counting function, and three interval enclosures computed in outward rounded interval arithmetic.

Earlier editions: Theorem 7.1.1, page 166 (edition 1).

Source: ch/rz3.tex, label rz:thm:id.

Negative bias of the Gaussian smoothed prime number theorem error

Volume I, Proposition 11.1.2, page 199 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\Lambda\) be the von Mangoldt function, \(\psi_C(y)=\sum_{n\le y}\Lambda(n)\) Chebyshev’s function, \(\Delta(y)=\psi_C(y)-y\) and \(\operatorname{erfc}\) the complementary error function. For \(x>0\) put [ w(x)=e^{x/4}-{n},e^{-(n)2/4x},I(x)=12e^{x/4}(2), ] [ Q(x)=_{1^-}y^{-1/2}e{-(y)^2/4x},d(y), ] where \(w\) is the canonical function of the normalised reciprocal zeta function \(\varphi(s)=\zeta(\frac12)/\bigl((1-4s)\,\zeta(\frac12+\sqrt s)\bigr)\) (so \(-\varphi'/\varphi\) is the Laplace transform of \(w\)) and \(Q_\Delta\), a Stieltjes integral, is the Gaussian smoothed error of the prime number theorem. Then \(w(x)=I(x)-Q_\Delta(x)\) for \(x>0\), and as \(x\to\infty\), \(I(x)\sim(\pi x)^{-1/2}\) and \(w(x)\sim\frac c2(\pi x)^{-1/2}\), where \(c=\frac12\bigl(\log\pi-\psi(\frac14)\bigr)\approx2.68609\) and \(\psi\) is the digamma function. Hence \(Q_\Delta(x)\sim-(\frac c2-1)(\pi x)^{-1/2}\) with \(\frac c2-1\approx0.343\), a definite negative bias. On the other hand, any bound \(|\Delta(y)|\le A\sqrt y\log^2y\) for \(y\ge2\), including the one that the Riemann Hypothesis implies, yields only \(|Q_\Delta(x)|\le A\,\bigl(x/2+O(\sqrt x)\bigr)\) as \(x\to\infty\) after integration by parts.

Earlier editions: Proposition 7.1.2, page 167 (edition 1).

Source: ch/rz3.tex, label rz:prop:bias.

The reciprocal zeta function in the Bondesson hierarchy: not self decomposable, not a GGC

Volume I, Theorem 11.1.3, page 200 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\varphi(s)=\zeta(\frac12)/\bigl((1-4s)\,\zeta(\frac12+\sqrt s)\bigr)\) be the normalised reciprocal zeta function (\(\zeta\) the Riemann zeta function, \(\sqrt s\) the principal square root). For an infinitely divisible law (ID) on \([0,\infty)\) with Lévy density \(k(x)/x\), the class SD of self decomposable laws is given by \(k\) nonincreasing, the Bondesson class BO by the Lévy density \(k(x)/x\) completely monotone, and the class GGC of generalised gamma convolutions by \(k\) completely monotone; these classes are nested as \(\mathrm{GGC}\subset\mathrm{SD}\subset\mathrm{ID}\) and \(\mathrm{GGC}\subset\mathrm{BO}\subset\mathrm{ID}\). Then, with no hypothesis on the zeros of \(\zeta\), the law with Laplace transform \(\varphi\) is in ID but not in SD and not in BO, and it is not a mixture of exponentials (its density is not completely monotone, equivalently \(\varphi\) is not a Stieltjes function). In particular it is not a generalised gamma convolution. The ID clause is the infinite divisibility theorem for \(\varphi\); the SD and GGC clauses follow from the canonical function of \(\varphi\) having positive derivative \(\frac14\) at the origin; the BO clause combines the fact that membership in BO would force the Riemann Hypothesis with Selberg’s central limit theorem for the argument function \(S\) of \(\zeta\), applied under that hypothesis.

Earlier editions: Theorem 7.1.3, page 168 (edition 1).

Source: ch/rz3.tex, label rz:thm:class.

Density of the reciprocal zeta law at the origin

Volume I, Theorem 11.1.4, page 202 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\varphi(s)=\zeta(\frac12)/\bigl((1-4s)\,\zeta(\frac12+\sqrt s)\bigr)\) be the normalised reciprocal zeta function and \(X_\varphi\) a random variable with \(\mathbb{E}e^{-sX_\varphi}=\varphi(s)\) (it is infinitely divisible, by the positivity of the canonical function \(w\) of \(\varphi\)). Here \(w(x)=e^{x/4}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\Lambda(n)\,n^{-1/2}e^{-(\log n)^2/4x}\) with \(\Lambda\) the von Mangoldt function, and \(w(0^+)=1\), \(w'(0^+)=\frac14\). The law of \(X_\varphi\) is absolutely continuous, with a density \(f\) that can be taken continuous on \((0,\infty)\) and satisfies Steutel’s equation \(x f(x)=\int_0^x f(x-y)\,w(y)\,dy\). Then \(f(0^+)=-\zeta(\frac12)/4\approx0.36509\) and \(f'(0^+)=f(0^+)\,w'(0^+)=-\zeta(\frac12)/16\approx0.09127>0\). In particular \(f\) is not monotone, so \(X_\varphi\) is not a mixture of uniform laws on intervals \([0,a]\), and \(f\) is not log convex and not completely monotone.

Earlier editions: Theorem 7.1.4, page 170 (edition 1).

Source: ch/rz3.tex, label rz:thm:shape.

The reciprocal zeta law is not a sum, product or limit of GGC variables

Volume I, Corollary 11.1.6, page 203 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(X_\varphi\) be a random variable whose Laplace transform is the normalised reciprocal zeta function \(\varphi(s)=\zeta(\frac12)/\bigl((1-4s)\,\zeta(\frac12+\sqrt s)\bigr)\). Then, with no hypothesis on the zeros of \(\zeta\), \(X_\varphi\) is not a sum or a product of independent generalised gamma convolution (GGC) variables, nor a weak limit of such. (The class of GGC laws is closed under independent sums and weak limits, and by a theorem of Bondesson under independent products, so any variable assembled from GGC variables by these operations is a GGC, while \(X_\varphi\) is not a GGC.)

Earlier editions: Corollary 7.1.6, page 171 (edition 1).

Source: ch/rz3.tex, label rz:cor:hm.

The GGC class is not closed under the harmonic mean

Volume I, Proposition 11.1.7, page 203 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. The statement (H) says: if \(V_1\) and \(V_2\) are independent generalised gamma convolution (GGC) variables, then so is the harmonic mean \(2V_1V_2/(V_1+V_2)\). (H) is false. If \(E_1\) and \(E_2\) are independent standard exponential variables, then \(H=2E_1E_2/(E_1+E_2)\) has density [ f_H(h)=e^{-h}_0^(1+h{2u})e{-u-h^2/4u},du,h>0, ] and \(H\) is not a GGC.

Earlier editions: Proposition 7.1.7, page 171 (edition 1).

Source: ch/rz3.tex, label rz:prop:hmfalse.

Vere-Jones weights of the zeta function are necklace polynomials

Volume I, Theorem 11.1.8, page 204 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. For \(k\ge1\) let \(M_k(q)=\frac1k\sum_{d\mid k}\mu(k/d)\,q^d\) be the necklace polynomial (\(\mu\) the Möbius function) and, for \(n\ge1\), \(b_n(q)=\sum_{i\ge0,\ 2^i\mid n}M_{n/2^i}(q)\). Then for \(|y|<1\) and \(0<q\le1\) [ =_{n}(1+yⁿ⁾{b_n(q)}, ] so that \(\zeta(\frac12+u)=\prod_p\prod_{n\ge1}\bigl(1+p^{-nu}\bigr)^{b_n(p^{-1/2})}\) for \(\operatorname{Re}u>\frac12\), the product over the primes \(p\). Equivalently, the weights \(B_N\) of the logarithmic Vere-Jones representation \(\log\zeta(\frac12+u)=\sum_{N\ge2}B_N\log\bigl(1+N^{-u}\bigr)\) for \(\operatorname{Re}u>\frac12\), which are defined by \(B_N=\sum_{n^m=N}d_m\,\Lambda(n)/(\sqrt n\,\log n)\) with \(\Lambda\) the von Mangoldt function, \(\sum_{m\ge1}d_mm^{-v}=1/\eta(1+v)\) and \(\eta(s)=(1-2^{1-s})\zeta(s)\), satisfy \(B_N=0\) unless \(N=p^n\) is a prime power, and then \(B_{p^n}=b_n(p^{-1/2})\). At \(q=1\), \(b_n(1)=1\) if \(n\) is a power of \(2\) and \(b_n(1)=0\) otherwise, and the identity becomes Euler’s binary identity \(\frac1{1-y}=\prod_{j\ge0}\bigl(1+y^{2^j}\bigr)\).

Earlier editions: Theorem 7.1.8, page 173 (edition 1).

Source: ch/rz3.tex, label rz:thm:necklace.

Sign rule for the necklace weights of the zeta function

Volume I, Proposition 11.1.9, page 205 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. For a prime \(p\) put \(q=p^{-1/2}\), and let \(b_n(q)=\sum_{i\ge0,\ 2^i\mid n}M_{n/2^i}(q)\), where \(M_k(q)=\frac1k\sum_{d\mid k}\mu(k/d)\,q^d\) is the necklace polynomial and \(\mu\) the Möbius function (these numbers \(b_n(p^{-1/2})\) are the Vere-Jones weights \(B_{p^n}\) of the zeta function at prime powers). Then \(b_{2^a}(q)>0\) for all \(a\ge1\), and for every odd \(n>1\), \(b_n(q)\ne0\) and the sign of \(b_n(q)\) is \((-1)^{\omega(n)}\), where \(\omega(n)\) is the number of distinct prime factors of \(n\). The book proves that \(b_n(q)\ne0\) for every odd \(n>1\); the earlier statement of the sign rule, in Polson’s 2026 book On Hilbert’s 8th Problem, is for the odd \(n>1\) with \(b_n(q)\ne0\) only.

Supersedes: Proposition 29.2 of Polson (2026) states the sign rule only for odd \(n>1\) with \(b_n(q)\ne0\).

Earlier editions: Proposition 7.1.9, page 173 (edition 1).

Source: ch/rz3.tex, label rz:prop:signs.

Lévy representation of the reciprocal zeta function across the critical strip

Volume I, Theorem 11.1.12, page 207 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Fix \(\sigma\in(0,1)\), let \(\zeta\) be the Riemann zeta function, \(\sqrt s\) the principal square root, and put [ (s)=, ] so that \(\varphi_{1/2}\) is the normalised reciprocal zeta function. For \(s>(1-\sigma)^2\), \(-\varphi_\sigma'(s)/\varphi_\sigma(s)=\int_0^\infty e^{-sx}w_\sigma(x)\,dx\), where \(w_\sigma\) has the three equivalent forms \[\begin{align*} w_\sigma(x)&=e^{(1-\sigma)^2x}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\frac{\Lambda(n)}{n^{\sigma}}\,e^{-(\log n)^2/4x}\\ &=e^{(1-\sigma)^2x}+\sum_p\sum_{n\ge1}b_n(p^{-\sigma})\,h_{n\log p}(x)\\ &=\tfrac12e^{(1-\sigma)^2x}\operatorname{erfc}\bigl((1-\sigma)\sqrt x\bigr)+\sum_{\gamma>0}\operatorname{Re}\Bigl[e^{a_\gamma^2x}\operatorname{erfc}(a_\gamma\sqrt x)\Bigr]\\ &\qquad+\frac{\gamma_E+\log\pi}{4\sqrt{\pi x}}+\sum_{k\ge0}\Bigl[\tfrac12e^{a_k^2x}\operatorname{erfc}(a_k\sqrt x)-\frac1{4(k+1)\sqrt{\pi x}}\Bigr]. \end{align*}\] Here \(\Lambda\) is the von Mangoldt function, \(\gamma_E\) is Euler’s constant, \(\operatorname{erfc}\) is the complementary error function, \(b_n(q)=\sum_{i\ge0,\ 2^i\mid n}M_{n/2^i}(q)\) with \(M_k(q)=\frac1k\sum_{d\mid k}\mu(k/d)q^d\) the necklace polynomials, \(h_\lambda(x)=\sum_{k\ge0}e^{-x((2k+1)\pi/\lambda)^2}-\frac{\lambda}{4\sqrt{\pi x}}=\frac{\lambda}{\sqrt{4\pi x}}\sum_{j\ge1}(-1)^je^{-j^2\lambda^2/4x}\) is the theta function of a block, \(a_\gamma=\sigma-\frac12-i\gamma\) and \(a_k=2k+2+\sigma\). The last form is written under the Riemann Hypothesis, with \(\gamma>0\) the ordinates of the zeros \(\frac12+i\gamma\) of \(\zeta\); in general \(\gamma\) runs over \(-i(\rho-\frac12)\) for the zeros \(\rho\). As \(x\to\infty\), \(w_\sigma(x)\sim\dfrac{\zeta'(\sigma)/\zeta(\sigma)}{2\sqrt{\pi x}}\), with \(\zeta'(\sigma)/\zeta(\sigma)>0\) on \((0,1)\). If \(w_\sigma\ge0\), then \(\varphi_\sigma\) is the Laplace transform of an infinitely divisible law on \((0,\infty)\) with Lévy density \(w_\sigma(x)/x\): [ (s)=-_0^(1-e{-sx})x,dx,s>0 . ]

Earlier editions: Theorem 7.1.12, page 176 (edition 1).

Source: ch/rz3.tex, label rz:thm:levystrip.

The shifted reciprocal zeta function is never a GGC on a line of the strip

Volume I, Proposition 11.1.13, page 209 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. For every \(\sigma\in(0,1)\) let \(\varphi_\sigma(s)=\zeta(\sigma)/\bigl((1-s/(1-\sigma)^2)\,\zeta(\sigma+\sqrt s)\bigr)\) (\(\zeta\) the Riemann zeta function, \(\sqrt s\) the principal square root). Then \(\varphi_\sigma\) is not the Laplace transform of a generalised gamma convolution, and if \(\varphi_\sigma\) is the Laplace transform of an infinitely divisible law, that law is not self decomposable. No hypothesis on the zeros of \(\zeta\) and no computation is used: the canonical function \(w_\sigma(x)=e^{(1-\sigma)^2x}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\Lambda(n)\,n^{-\sigma}e^{-(\log n)^2/4x}\) of \(\varphi_\sigma\), with \(-\varphi_\sigma'/\varphi_\sigma(s)=\int_0^\infty e^{-sx}w_\sigma(x)\,dx\) for \(s>(1-\sigma)^2\) and \(\Lambda\) the von Mangoldt function, satisfies \(w_\sigma(x)=1+(1-\sigma)^2x+o(x)\) as \(x\downarrow0\) and so is strictly increasing near the origin. The book proves this for every \(\sigma\in(0,1)\) and includes the clause on self decomposability; the earlier statement, in Polson’s 2026 book On Hilbert’s 8th Problem, proves that \(\varphi_\sigma\) is not a generalised gamma convolution for \(\sigma\le\frac12\) only (for \(\frac12<\sigma<1\) it is verified numerically) and has no such clause.

Supersedes: Proposition 29.5 of Polson (2026) proves that the shifted reciprocal zeta function is not a generalised gamma convolution for \(\sigma\le\frac12\) only (for \(\frac12<\sigma<1\) it is verified numerically and not proved), and has no clause on self decomposability.

Earlier editions: Proposition 7.1.13, page 177 (edition 1).

Source: ch/rz3.tex, label rz:prop:onlyhalf.

Laguerre Pólya class as integer atom GGCs

Volume I, Proposition 11.1.15, page 212 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(G\) be a real even entire function of order less than \(2\) with \(G(0)\ne0\). Then \(G\) is in the Laguerre Pólya class (the locally uniform limits of real polynomials with only real zeros, which for such \(G\) means that all zeros of \(G\) are real) if and only if \(s\mapsto G(0)/G(i\sqrt s)\) is the Laplace transform of a generalised gamma convolution whose Thorin measure is \(\sum_km_k\delta_{\alpha_k^2}\) with integers \(m_k\ge1\), where \(\pm\alpha_k\) are the zeros of \(G\) with multiplicities \(m_k\).

Earlier editions: Proposition 7.1.15, page 180 (edition 1).

Source: ch/rz3.tex, label rz:prop:lpthorin.

Lee and Yang single spin laws as integer atom GGCs

Volume I, Proposition 11.1.16, page 213 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\rho\) be a symmetric probability law on \(\mathbb{R}\) with \(\int e^{bt^2}\rho(dt)<\infty\) for every \(b\). Say that \(\rho\) has the Lee and Yang property if ferromagnetic systems of \(\rho\) spins have partition function zeros only on the imaginary field axis; by a theorem of Newman this holds exactly when \(z\mapsto\int e^{izt}\rho(dt)\) is in the Laguerre Pólya class. Then \(\rho\) has the Lee and Yang property if and only if \(s\mapsto1/\int e^{\sqrt s\,t}\rho(dt)\) is the Laplace transform of a generalised gamma convolution with zero drift whose Thorin measure has integer atoms. The atoms are at the squares \(\alpha_k^2\) of the zeros \(\pm\alpha_k\) of \(\int e^{izt}\rho(dt)\), with the multiplicities of the zeros.

Earlier editions: Proposition 7.1.16, page 181 (edition 1).

Source: ch/rz3.tex, label rz:prop:newmanthorin.

Random harmonic series and the strip of the Pólya kernel

Volume I, Proposition 11.1.17, page 214 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\Phi(t)=2\sum_{n\ge1}\bigl(2\pi^2n^4e^{9t/2}-3\pi n^2e^{5t/2}\bigr)\exp(-\pi n^2e^{2t})\) be the Pólya kernel, an even function whose Fourier transform \(\int_{\mathbb{R}}\Phi(t)\cos(\tau t)\,dt\) is \(\xi(\frac12+i\tau)\), with \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\); it is analytic in the strip \(|\operatorname{Im}t|<\pi/4\) and \(\log\Phi(t)=-\pi e^{2t}+\frac92t+O(1)\) as \(t\to+\infty\). For \(\kappa>-1\) let \(\varepsilon_k\) be independent fair signs \(\pm1\) and \(H_\kappa=\sum_{k\ge1}\varepsilon_k/(2(k+\kappa))\), with characteristic function \(\hat f_\kappa(v)=\prod_{k\ge1}\cos\bigl(v/(2(k+\kappa))\bigr)\) and density \(f_\kappa\). (i) As \(s\to\infty\), \(\log\mathbb{E}e^{sH_\kappa}=\frac s2\log s+A(\kappa)s+o(s)\) with \(A(\kappa)=A(0)-\frac12\bigl(\psi(1+\kappa)+\gamma_E\bigr)\), where \(\psi\) is the digamma function, \(\gamma_E\) is Euler’s constant and \(A(0)=-0.1485757\ldots\). Hence \(\log\Pr(H_\kappa>u)=-\frac12e^{2u-2A(\kappa)-1}(1+o(1))\) as \(u\to\infty\), and the double exponential terms of \(H_\kappa\) and of \(\Phi\) agree exactly when \(A(\kappa)=-\frac12(1+\log2\pi)\), that is at \(\kappa^*=6.61845\ldots\). (ii) \(\limsup_{|v|\to\infty}|v|^{-1}\log|\hat f_\kappa(v)|=-\pi/4\) for every \(\kappa>-1\). Hence \(f_\kappa\) is analytic in the strip \(|\operatorname{Im}t|<\pi/4\), which is exactly the strip of analyticity of \(\Phi\).

Earlier editions: Proposition 7.1.17, page 182 (edition 1).

Source: ch/rz3.tex, label rz:prop:harmonic.

The reciprocal Xi transform is a harmonic mean of Brownian exit time transforms

Volume I, Proposition 11.1.23, page 219 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function. The Pólya variable \(X\) is the symmetric real random variable with density \(\Phi/\xi(\frac12)\), where \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and moment generating function \(\mathbb{E}e^{sX}=\xi(\frac12+s)/\xi(\frac12)\). The clock \(T\) is the nonnegative random variable with \(\mathbb{E}e^{-sT}=\xi(\frac12)/\xi(\frac12+\sqrt s)\) for \(s\ge0\) (it exists with no hypothesis on the zeros). Let \(C\) be the exit time of standard Brownian motion from \((-1,1)\), so that \(\mathbb{E}e^{-\theta C}=1/\cosh\sqrt{2\theta}\), and for \(x\ne0\) let \(L_x(s)=1/\cosh(x\sqrt s)=\mathbb{E}\exp(-\tfrac12x^2sC)\), the Laplace transform of \(\tfrac12x^2C\) (the transform of a generalized gamma convolution with Thorin atoms \(\pi^2(k-\frac12)^2/x^2\), \(k\ge1\)). Then, for \(s\ge0\), \(\mathbb{E}e^{-sT}=1/\mathbb{E}\bigl[1/L_X(s)\bigr]\): the reciprocal Xi transform is the harmonic mean, over the Pólya density, of the Brownian exit time transforms \(L_x\). By Jensen’s inequality, \(\mathbb{E}e^{-sT}\le\mathbb{E}\exp(-\tfrac12sX^2C)\), with \(C\) independent of \(X\).

Earlier editions: Proposition 7.1.23, page 187 (edition 1).

Source: ch/rz3.tex, label xg:prop:harm.

A harmonic mean of two GGC transforms that is not a Laplace transform of a positive measure

Volume I, Example 11.1.24, page 219 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: counterexample, proved.

Statement. A generalized gamma convolution (GGC) is a law on \([0,\infty)\) whose Laplace transform is \(\exp\bigl(-bs-\int_{(0,\infty)}\log(1+s/z)\,U(dz)\bigr)\) with drift \(b\ge0\) and Thorin measure \(U\ge0\); in particular \((1+s)^{-2}\) and \((1+2s)^{-2}\) are GGC transforms, those of Gamma(2) laws with scale \(1\) and \(2\). Their harmonic mean with equal weights, \(\bigl(\tfrac12(1+s)^2+\tfrac12(1+2s)^2\bigr)^{-1}\), equals \(1/(1+3s+\tfrac52s^2)\). It has poles at \(s=-0.6\pm0.2i\), and its inverse Laplace transform is \(2e^{-0.6t}\sin(0.2t)\), which is negative on \((5\pi,10\pi)\). So the harmonic mean is not the Laplace transform of a positive measure, and in particular not a GGC transform: a harmonic mean of GGC transforms need not even be a Laplace transform of a positive measure.

Earlier editions: Example 7.1.24, page 188 (edition 1).

Source: ch/rz3.tex, label xg:ex:harm.

A two point scale mixture of Gamma(2) that is infinitely divisible and not a GGC

Volume I, Example 11.1.25, page 219 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: counterexample, proved.

Statement. A generalized gamma convolution (GGC) is a law on \([0,\infty)\) whose Laplace transform is \(\exp\bigl(-bs-\int_{(0,\infty)}\log(1+s/z)\,U(dz)\bigr)\) with drift \(b\ge0\) and Thorin measure \(U\ge0\); such a Laplace transform has no zeros outside \((-\infty,0]\). Let \(\Gamma_2\) be a Gamma(2) variable (density \(te^{-t}\) on \((0,\infty)\)) and \(Y\ge0\) a random variable independent of \(\Gamma_2\). Every scale mixture \(\Gamma_2Y\) is infinitely divisible (Kristiansen 1994), and it is a GGC if \(Y\) is a GGC, since products of independent GGC variables are GGC (Bondesson 2015). Without a hypothesis on \(Y\) it need not be a GGC: let \(Y\) equal \(1\) or \(\lambda\), with \(\lambda>0\) and \(\lambda\ne1\), with probability \(\frac12\) each. The Laplace transform \(\mathbb{E}(1+sY)^{-2}\) of \(\Gamma_2Y\) has numerator \((1+\lambda s)^2+(1+s)^2\) over the denominator \(2(1+s)^2(1+\lambda s)^2\), and the numerator vanishes at nonreal \(s\). Hence \(\Gamma_2Y\) is infinitely divisible but not a GGC.

Earlier editions: Example 7.1.25, page 188 (edition 1).

Source: ch/rz3.tex, label xg:ex:g2.

The clock is not a Gamma scale mixture, for any shape

Volume I, Theorem 11.1.26, page 220 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(T\) the clock, the nonnegative random variable with \(\mathbb{E}e^{-sT}=\xi(\frac12)/\xi(\frac12+\sqrt s)\) for \(s\ge0\) (it exists with no hypothesis on the zeros). For \(a>0\) let \(\Gamma_a\) be a gamma variable with shape \(a\) and unit scale, with density \(t^{a-1}e^{-t}/\Gamma(a)\) on \((0,\infty)\). Then for no \(a>0\) and no random variable \(Y\ge0\) independent of \(\Gamma_a\) does \(T\) have the law of \(\Gamma_aY\). The proof uses no hypothesis on the zeros of \(\xi\).

Earlier editions: Theorem 7.1.26, page 188 (edition 1).

Source: ch/rz3.tex, label xg:thm:nogamma.

Volume I, Chapter 12. The basepoint one and hitting times at the basepoints

Barnes beta laws (after Ostrovsky): the beta case is a GGC if and only if b1 is a positive integer, and the sech squared Levy density is a Gaussian mixture whose mixing law has Mellin transform xi(2q+2)

Volume I, Remark 12.1.1, page 224 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(C_2=\sum_{k\ge1}\frac2{\pi^2(k-\frac12)^2}(e_k+e_k')\) with independent standard exponentials, so that \(\mathbb{E}e^{-uC_2}=\operatorname{sech}^2\sqrt{2u}\) (Biane, Pitman and Yor). Then \(\mathbb{E}[C_2^q]=\frac{2^{2q+2}-1}{q+1}\big(\frac2\pi\big)^{q+1}\xi(2q+2)\), checked at \(q=-0.3,\frac12,1,2\) to \(10^{-12}\) by quadrature of the Laplace transform, and the Lévy density \(\frac1{4t}\operatorname{sech}^2\frac t4\) of the Barnes beta law (Ostrovsky) satisfies \(\frac1{4t}\operatorname{sech}^2\frac t4=\frac1{4t}\mathbb{E}e^{-t^2C_2/32}\), a mixture of Gaussians in \(t\) whose mixing law has Mellin transform \(\xi(2q+2)\). The positivity comes from the gamma side, and the zeros of \(\xi\) enter only through the Mellin transform of \(C_2\). The Barnes beta laws are due to Ostrovsky; the numerical check is the book’s. Let \(B\) be a beta variable of parameters \(b_0,b_1>0\) (the Barnes beta law \(\beta_{1,1}(a_1=1;b_0,b_1)\) of Ostrovsky). The law of \(-\log B\) has Lévy density \(e^{-b_0t}(1-e^{-b_1t})/(t(1-e^{-t}))\), and \(t\) times it is the Laplace transform of the signed measure \(\sum_{j\ge0}(\delta_{b_0+j}-\delta_{b_0+b_1+j})\). Hence \(-\log B\) is a generalized gamma convolution if and only if \(b_1\) is a positive integer, and then it is a sum of independent exponential variables of rates \(b_0,\dots,b_0+b_1-1\); for other \(b_1\) the negative atoms meet no positive one, and a Laplace transform determines its measure. For \(M=N=2\), \(a=(1,1)\) and \(b=(b_0,\frac12,\frac12)\) the Lévy density is \(\frac{e^{-(b_0-\frac12)t}}{4t}\operatorname{sech}^2\frac t4\), again not that of a GGC, since \(t\) times it is the Laplace transform of the signed measure \(\sum_{j\ge1}(-1)^{j+1}j\delta_{b_0-\frac12+j/2}\). The beta case is credited to a note of Nicholas Polson merged into the book.

Earlier editions: Remark 9.1.1, page 337 (edition 1).

Source: ch/basepoint.tex, label tx3:rem:ostrovsky.

Thorin measure of the shifted clock at basepoint one is positive with no hypothesis

Volume I, Proposition 12.1.2, page 225 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(v_1(\theta)=\operatorname{Re}(\xi'/\xi)(1+i\theta)\). The function \(\varphi_1(u)=\xi(1)/\xi(1+\sqrt u)\) is the Laplace transform of a generalised gamma convolution \(T_1\) with no drift, \(\log\varphi_1(u)=-\int_0^\infty\log(1+u/z)\,U_1(dz)\), and in the variable \(\theta=\sqrt z\) its Thorin measure \(U_1\) has density \(\rho_{1/2}(\theta)=\frac1\pi v_1(\theta)=\frac1\pi\sum_\rho\frac{1-\beta}{(1-\beta)^2+(\theta-\gamma)^2}\), the sum running over all nontrivial zeros \(\rho=\beta+i\gamma\) of \(\zeta\) in conjugate pairs. Every term is positive because \(\beta<1\) (the nonvanishing of \(\zeta\) on \(\operatorname{Re}s=1\)), so \(U_1\) is a positive measure with no hypothesis on the zeros.

Source: ch/basepoint.tex, label bp:prop:thorin.

Compensated sine squared formula at the basepoint one

Volume I, Proposition 12.1.3, page 227 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, with zeros \(\rho\), and let \(v_1(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(1+i\theta)\). Let \(\Lambda\) be the von Mangoldt function, \(\psi=\Gamma'/\Gamma\) the digamma function, \(\gamma_E\) Euler’s constant, and \(\nu_1=\sum_n\Lambda(n)n^{-1}\delta_{\log n}-\mathbf 1_{(0,\infty)}(x)\,dx\) the prime measure compensated by Lebesgue measure. For every real \(\theta\ne0\), [v_1()=-+(2)+E+{X}_0^{X}22(2)_1(dx),] that is, the limit is that of \(\sum_{n\le X}\Lambda(n)n^{-1}\,2\sin^2(\tfrac12\theta\log n)-\big(\log X-\sin(\theta\log X)/\theta\big)\). The limit exists by the prime number theorem, and at \(\theta=0\) the formula reduces to \(v_1(0)=\sum_\rho\rho^{-1}\) (the sum over the nontrivial zeros taken in conjugate pairs).

Earlier editions: Proposition 9.1.3, page 340 (edition 1).

Source: ch/basepoint.tex, label bp:prop:sin2.

De la Vallée Poussin’s zero free region from the Thorin density: the two frequency inequality 4Z(y)+Z(2y) at most 8 c_alpha

Volume I, Proposition 12.1.5, page 228 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(\alpha>1\) let \(c_\alpha=-\frac{\zeta'}\zeta(\alpha)\), \(Z(\alpha,y)=\sum_{n\ge2}\Lambda(n)n^{-\alpha}(1-\cos(y\log n))\) the prime part of the Thorin density \(g(\alpha,y)=\operatorname{Re}\frac{\xi'}\xi(\alpha+iy)\), so that \(Z=c_\alpha+g-\operatorname{Re}\mathcal{A}(\alpha+iy)\) with \(\mathcal{A}\) as in the quasi Riemann hypothesis theorem of the book. (i) \(0\le Z(\alpha,y)\le2c_\alpha\) and \(4Z(\alpha,y)+Z(\alpha,2y)\le8c_\alpha\). (ii) If \(\beta_0+i\gamma_0\) is a zero, then \(\frac4{\alpha-\beta_0}\le3c_\alpha+4\operatorname{Re}\mathcal{A}(\alpha+i\gamma_0)+\operatorname{Re}\mathcal{A}(\alpha+2i\gamma_0)\), and with \(\alpha=1+\frac1{4\log\gamma_0}\) this gives \(1-\beta_0\ge c/\log\gamma_0\) for \(\gamma_0\ge\gamma^*\), with absolute constants \(c,\gamma^*>0\) (the proof gives \(c=\frac3{116}-o(1)\) in the variable \(x=(1-\beta_0)\log\gamma_0\)). The proof is the inequality \(3+4\cos\theta+\cos2\theta=2(1+\cos\theta)^2\ge0\) termwise.

Earlier editions: Proposition 9.1.4, page 340 (edition 1).

Source: ch/basepoint.tex, label tx3:prop:dvp.

Thorin property of the reciprocal Xi function for basepoint at least 5.4962 from the sine squared lemma alone, with no information on the zeros

Volume I, Proposition 12.1.6, page 229 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+\frac s2)\zeta(s)\) be the completed Riemann zeta function. For \(\alpha>1\) let \(D(\alpha)=\frac{\xi'}{\xi}(\alpha)-\frac{1}{\alpha-1}=\frac12\psi(1+\frac\alpha2)-\frac12\log\pi+\frac{\zeta'}{\zeta}(\alpha)\), where \(\psi\) is the digamma function. Then \(D\) is increasing, \(D(\alpha)\to\infty\), and its root is \(\alpha_0=5.4962215670\ldots\). For every \(\alpha\ge\alpha_0\), \(u\mapsto\xi(\alpha)/\xi(\alpha+\sqrt u)\) is the Laplace transform of a generalised gamma convolution (GGC). The proof uses only the sine squared lemma (the identity expressing \(e^{-sx}-1+sx\) in the variable \(\sqrt u\) as a positive integral of \(\log(1+u/\theta^2)\) against a sine squared kernel) and the Euler product, with no information on the zeros.

Earlier editions: Proposition 9.1.5, page 341 (edition 1).

Source: ch/basepoint.tex, label tx:prop:alpha0.

Thorin condition with the prime part dropped: the sign of m(alpha) changes at alpha_1 = 3.5747764, computed in floating point and not proved on [alpha_1, alpha_0)

Volume I, Remark 12.1.7, page 229 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. In the variable \(y=\sqrt z\) the Thorin density of \(\xi(\alpha)/\xi(\alpha+\sqrt u)\) is \(\pi^{-1}g(\alpha,y)\) with \(g=P+D+A+Z\), where \(P=\frac{\alpha-1}{(\alpha-1)^2+y^2}\) comes from the pole, \(D=D(\alpha)\), \(A=\frac12(\operatorname{Re}\psi(1+\frac{\alpha+iy}2)-\psi(1+\frac\alpha2))\ge0\) from the Gamma factor and \(Z\ge0\) from the primes. Dropping \(Z\) leaves \(m(\alpha)=\min_y\{P+A\}+D(\alpha)\), equal to \(-0.0133\) at \(\alpha=3.5\), \(0.0638\) at \(4\), \(0.1650\) at \(5\) and \(0.2012\) at \(5.5\), changing sign at \(\alpha_1=3.5747764\ldots\) (minimum in \(y\) at \(3.4506\)). Where \(m(\alpha)\ge0\) the Thorin condition holds with only \(\Lambda(n)\ge0\) used; this is computed in floating point (20 digits) at sampled points and not proved on \([\alpha_1,\alpha_0)\), where \(m\) need not be monotone. No bound on the primes uniform in \(y\) goes below \(\alpha_1\), since \(\inf_yZ(\alpha,y)=0\) for every \(\alpha>1\) by Kronecker’s theorem. For every \(\alpha>1\) a march with a Lipschitz bound checks \(g>0\) from \(\zeta\) on \(\operatorname{Re}s=\alpha\) alone, which is not new (the book’s Poisson theorem gives \(\alpha\ge1\)); \(\alpha=1.3\) took \(21\,931\) evaluations.

Earlier editions: Remark 9.1.6, page 342 (edition 1).

Source: ch/basepoint.tex, label tx2:rem:alpha1.

Unconditional GGC property of the reciprocal Xi function in the fourth root variable, compensated by the first power sum, with explicit Thorin density from the heat trace

Volume I, Proposition 12.1.8, page 230 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+\frac s2)\zeta(s)\) be the completed Riemann zeta function, with the zeros \(\rho=\frac12+w_\rho\), and let \(W(t)=\sum_{\operatorname{Im}\rho>0}e^{(\rho-\frac12)^2t}\) be the heat trace of the zeros (one zero from each pair \(\{\rho,1-\rho\}\), with multiplicity). Put \(p_1=\int_0^\infty W(t)\,dt=\frac12(\log\xi)''(\frac12)=0.0231049931\ldots\), which equals \(\sum_{\gamma>0}\gamma^{-2}\) under RH. Unconditionally, for \(s\ge0\), \(\Psi(s):=\frac{\xi(\frac12+s^{1/4})}{\xi(\frac12)}e^{-p_1\sqrt s}=\exp\Big(-\int_0^\infty\log\Big(1+\frac sz\Big)U_*(dz)\Big)\) with \(U_*(dz)=\frac{dz}{\pi\sqrt z}\int_0^\infty\sin^2\Big(\frac{t\sqrt z}{2}\Big)W(t)\,dt\ge0\), so \(\Psi\) is the Laplace transform of a generalised gamma convolution (GGC) with an absolutely continuous Thorin measure. Under the Riemann Hypothesis, with \(\pm\gamma\) the ordinates of the zeros, \(\Psi(s)=\prod_{\gamma>0}\big(1+\sqrt s/\gamma^2\big)e^{-\sqrt s/\gamma^2}\).

Earlier editions: Proposition 9.1.7, page 342 (edition 1).

Source: ch/basepoint.tex, label tx:prop:W.

The square root is forced: xi(alpha)/xi(alpha+q) is not the Laplace transform of a positive measure in q for any alpha

Volume I, Remark 12.1.9, page 230 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For every real \(\alpha\), \(q\mapsto\xi(\alpha)/\xi(\alpha+q)\) is not \(\int_{[0,\infty)}e^{-qt}\mu(dt)\) for a positive measure \(\mu\ne0\) on any half plane. For real \(q\to\infty\), \(\xi(\alpha+q)\) grows like \(\exp(\frac q2\log q)\), so the function decays faster than \(e^{-qT}\) for every \(T\), while \(\int e^{-qt}\mu(dt)\ge\mu([0,T])e^{-qT}\); hence \(\mu([0,T])=0\) for all \(T\). Landau’s theorem gives it too: the function has no real singularity, since \(\xi\) has no real zeros, so the abscissa would be \(-\infty\) and the function entire, but it has poles. No law on \((0,1]\) has Mellin transform \(\xi(\alpha)/\xi(\alpha+q)\).

Earlier editions: Remark 9.1.8, page 343 (edition 1).

Source: ch/basepoint.tex, label tx3:rem:forced.

Barnes beta law of Ostrovsky with Lévy density sech squared over 4t: Mellin transform by Barnes G, GGC form with an explicit Thorin density, and zeros of the Mellin transform of that density

Volume I, Proposition 12.1.10, page 231 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Y_{\mathfrak b}\ge0\) be infinitely divisible with Lévy density \(\nu(t)=\operatorname{sech}^2(t/4)/(4t)\), so \(\mathbb{E}Y_{\mathfrak b}=1\), and \(\mathfrak b=e^{-Y_{\mathfrak b}}\), the Barnes beta law of Ostrovsky. (i) For \(\operatorname{Re}q>-\frac12\), \(\mathbb{E}\mathfrak b^q=\frac{\mathsf G(q+1)^2\mathsf G(\frac12)\mathsf G(\frac32)}{\mathsf G(q+\frac12)\mathsf G(q+\frac32)}=\frac{2^{2q^2}(2\pi)^{-q}\mathsf G(1+q)^4}{\mathsf G(1+2q)}\) with \(\mathsf G\) the Barnes function, \(\mathbb{E}\mathfrak b=2/\pi\), \(\mathbb{E}\mathfrak b^q\sim Cq^{-1/4}\), and since \(t\nu(t)=\sum_{k\ge1}(-1)^{k+1}ke^{-kt/2}\), \(Y_{\mathfrak b}\) is not a GGC. (ii) For \(s\ge0\), \(\mathbb{E}\mathfrak b^{\sqrt s}=e^{-\sqrt s}/\Psi_{\mathfrak b}(s)\) with \(\Psi_{\mathfrak b}(s)=\exp(-\int_0^\infty\log(1+\frac sz)\omega_{\mathfrak b}(z)dz)\) the Laplace transform of a GGC with Thorin density \(\omega_{\mathfrak b}(z)=\frac1{2\pi\sqrt z}-\frac1{\sinh(2\pi\sqrt z)}>0\). (iii) For \(-\frac12<\operatorname{Re}w<\frac12\), \(\int_0^\infty z^{w-1}\omega_{\mathfrak b}(z)dz=-4(2\pi)^{-2w}(1-2^{-2w})\Gamma(2w)\zeta(2w)\), whose zeros in that strip are the points \(\rho/2\) and \(\pi ik/\log2\), \(k\ne0\). The law is due to Ostrovsky; the statements (i) to (iii) are the book’s.

Earlier editions: Proposition 9.1.9, page 343 (edition 1).

Source: ch/basepoint.tex, label tx3:prop:barnes.

Lévy form of the completed zeta function at the basepoint one

Volume I, Proposition 12.1.11, page 232 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\psi=\Gamma'/\Gamma\) the digamma function, \(\gamma_E\) Euler’s constant and \(\Lambda\) the von Mangoldt function. For \(\operatorname{Re}s>0\), [=b_1s+_0^(e{-sx}-1+sx),1(dx),] with drift \(b_1=-\frac12\log\pi+\frac12\psi(\frac32)+\gamma_E=\sum_\rho\rho^{-1}=0.0230957089661\) (the sum over the nontrivial zeros \(\rho\) of \(\zeta\) in conjugate pairs) and the signed Lévy measure [1(dx)=,x-x+{n},{n}(dx),] the integral against the last two pieces (the pole term \(-dx/x\) and the prime atoms) being taken as \(\lim_{X\to\infty}\int_0^{\log X}\).

Earlier editions: Proposition 9.1.10, page 344 (edition 1).

Source: ch/basepoint.tex, label bp:prop:levy.

Minimum of the Thorin density at the origin and stable summand at the basepoint one

Volume I, Proposition 12.1.12, page 234 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho\), \(v_1(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(1+i\theta)\) and \(b_1=v_1(0)=\frac{\xi'}{\xi}(1)=\sum_\rho\rho^{-1}=0.0230957089661\). Let \(T_1\) be the generalized gamma convolution (GGC) with \(\mathbb{E}e^{-uT_1}=\xi(1)/\xi(1+\sqrt u)\) for \(u\ge0\), whose Thorin density in the variable \(\theta=\sqrt z\) is \(v_1(\theta)/\pi\). With no hypothesis on the zeros, \(v_1(\theta)>v_1(0)=b_1\) for every real \(\theta\ne0\); the proof uses a computation of the zeros below height \(2000\) in ball arithmetic and, above that height, explicit bounds for the argument of the zeta function on the critical line (Hasanalizade, Shen and Wong 2022, or the classical bounds of Backlund 1918). Consequently \(T_1\stackrel{d}{=}S+T_1^\circ\), where \(S\) and \(T_1^\circ\) are independent, \(S\) is positive stable of index one half with \(\mathbb{E}e^{-uS}=e^{-b_1\sqrt u}\), and \(T_1^\circ\) is a GGC with no drift whose Thorin density in the chart \(\theta=\sqrt z\) is \((v_1-b_1)/\pi\).

Earlier editions: Proposition 9.1.11, page 346 (edition 1).

Source: ch/basepoint.tex, label bp:prop:v1min.

Minimum at the origin and stable summand at every basepoint from one

Volume I, Proposition 12.1.14, page 237 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, and for real \(\alpha\) and \(\theta\) let \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\) and \(b_\alpha=v_\alpha(0)=\frac{\xi'}{\xi}(\alpha)\). With no hypothesis on the zeros, \(v_\alpha(\theta)>v_\alpha(0)=b_\alpha\) for every \(\alpha\ge1\) and every real \(\theta\ne0\); the proof uses the computation of the zeros below height \(2000\) in ball arithmetic of the case \(\alpha=1\), above that height explicit bounds for the argument of the zeta function on the critical line (Hasanalizade, Shen and Wong 2022), and a further computation in ball arithmetic. Consequently, for every \(\alpha\ge1\), the generalized gamma convolution (GGC) \(T_\alpha\) with \(\mathbb{E}e^{-uT_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt u)\) satisfies \(T_\alpha\stackrel{d}{=}S_\alpha+T_\alpha^\circ\), where \(S_\alpha\) and \(T_\alpha^\circ\) are independent, \(S_\alpha\) is positive stable of index one half with \(\mathbb{E}e^{-uS_\alpha}=e^{-b_\alpha\sqrt u}\), which is the law of the first passage time of Brownian motion to the level \(b_\alpha/\sqrt2\), and \(T_\alpha^\circ\) is a GGC with no drift, with Thorin density \((v_\alpha-b_\alpha)/\pi\) in the chart \(\theta=\sqrt z\) and mean \(\mathbb{E}T_\alpha^\circ=\frac12(\log\xi)''(\alpha)\).

Earlier editions: Proposition 9.1.12, page 349 (edition 1).

Source: ch/basepoint.tex, label bp:prop:valpha.

Regularly varying tail of the Thorin remainder at basepoints at least one

Volume I, Result 12.1.4, page 239 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha\ge1\), let \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}\xi(\alpha+i\theta)\) and \(b_\alpha=v_\alpha(0)\), let \(T_\alpha^\circ\) be the generalised gamma convolution without drift whose Thorin density in the chart \(\theta\) is \((v_\alpha-b_\alpha)/\pi\), and let \(c(\alpha)>0\) be the constant with \(v_\alpha(\theta)-b_\alpha=c(\alpha)\theta^2+O(\theta^4)\) near \(0\). Then the Lévy density of \(T_\alpha^\circ\) is asymptotic to \(c(\alpha)t^{-5/2}/(4\sqrt\pi)\) as \(t\to\infty\) and \(P(T_\alpha^\circ>t)\sim c(\alpha)t^{-3/2}/(6\sqrt\pi)\). The proof: \(t\) times the Lévy density is \(\pi^{-1}\int_0^\infty e^{-t\theta^2}(v_\alpha(\theta)-b_\alpha)d\theta\), and \(v_\alpha\) grows only logarithmically, so Watson’s lemma applies; the Lévy tail is regularly varying, hence subexponential, and for an infinitely divisible law with such a Lévy tail the tail of the law is asymptotic to it (Embrechts, Goldie and Veraverbeke 1979). A computation at \(\alpha=2\), where \(c(2)=3.25029\times10^{-4}\), gives the ratio of \(t\) times the Lévy density to \(c(2)/(4\sqrt\pi t^{3/2})\) as \(1.000372\), \(1.0000929\) and \(1.0000232\) at \(t=25\), \(100\) and \(400\); this check covers only these values. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Result 9.1.4, page 350 (edition 1).

Source: ch/basepoint.tex, label bp:res:tail.

The uniform factor is Thorin positive, so Kolmogorov dominates xi at every basepoint above one

Volume I, Theorem 12.1.15, page 240 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(U\) be the uniform factor of Kolmogorov’s Mellin transform, with \(\mathbb{E} U^{-s}=L\,\varphi_U(L(s-1))\), \(L=\log2\) and \(\varphi_U(s)=(1-e^{-s})/s\), and let \(\mu_U\) be its L'evy measure. For every \(\alpha>1\) the Thorin density \(\tau_\alpha^U\) that the sine squared lemma assigns to \(\mu_U\) is positive on \((0,\infty)\). Consequently \(\tau_\alpha^K=\tau_\alpha^\xi+\tau_\alpha^U>\tau_\alpha^\xi\): at every basepoint \(\alpha>1\) the Thorin density of Kolmogorov’s Mellin transform exceeds that of \(\xi\).

Source: ch/basepoint.tex, label bp:thm:uniformU.

Lévy form below the basepoint one and zero free half planes

Volume I, Proposition 12.1.17, page 241 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Lambda\) the von Mangoldt function, \(\psi=\Gamma'/\Gamma\) the digamma function and \(\Theta=\sup\{\operatorname{Re}\rho\}\) over the nontrivial zeros \(\rho\) of \(\zeta\). Let \(\frac12<\alpha<1\), \(b_\alpha=\frac{\xi'}{\xi}(\alpha)\) and \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\), and let \(\Pi_\alpha\) be the signed measure on \((0,\infty)\) with [x,(dx)=,dx+(n,{n}(dx)-e^{(1-)x},dx),] the integral against the part in parentheses being taken as \(\lim_{X\to\infty}\int_0^{\log X}\). (1) If \(\Theta<\alpha\), the Lévy form \(\log\frac{\xi(\alpha+s)}{\xi(\alpha)}=b_\alpha s+\int_0^\infty(e^{-sx}-1+sx)\,\Pi_\alpha(dx)\) holds for \(\operatorname{Re}s>0\), the compensated limit converging, and the identity \(e^{-sx}+sx-1=\frac{2x}\pi\int_0^\infty\log(1+s^2/\theta^2)\sin^2(\frac{x\theta}2)\,d\theta\) converts it into [v()=b_+A_()+_{X},] where \(A_\alpha(\theta)=\frac12\big[\operatorname{Re}\psi\big(\frac{\alpha+2+i\theta}2\big)-\psi\big(\frac{\alpha+2}2\big)\big]\); this value of \(v_\alpha(\theta)\) is positive. The error of the compensated sum with prime cutoff \(X\) (the bracket truncated at \(X\) minus its limit) is \(O(X^{\Theta-\alpha+\delta})\) for every \(\delta>0\). (2) Conversely, if the Lévy form holds for \(\operatorname{Re}s>0\), then \(\xi(\alpha+s)\) has no zeros with \(\operatorname{Re}s>0\), so \(\Theta\le\alpha\).

Earlier editions: Proposition 9.1.13, page 351 (edition 1).

Source: ch/basepoint.tex, label bp:prop:below.

Monotonicity of the ratio of a zero pair kernel in the squared width, constant 1 plus square root of 2 sharp

Volume I, Lemma 12.1.19, page 244 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0<\delta<\frac12\), let \(u\) be real, and for \(w>0\) put \(r_\delta(w,u)=2\operatorname{Re}\frac1{w+(u+i\delta)^2}\), the ratio of the kernel of a pair of zeros \(\frac12\pm\delta+i\gamma\) at height offset \(u\) to the width, as a function of \(w=(\alpha-\frac12)^2\) for the level \(\operatorname{Re}s=\alpha\). Then as a function of \(w>0\): (a) \(r_\delta(w,u)\) is nonincreasing on \((0,\infty)\) if \(|u|\ge(1+\sqrt2)\delta\); (b) it is nonincreasing on \([2\delta^2,\infty)\) for every \(u\), in particular on \(w\ge\frac12\), that is for \(\alpha\ge\frac12+\frac1{\sqrt2}\), whatever \(\delta\); (c) if \(0<|u|<(1+\sqrt2)\delta\) it increases exactly on the interval \(\max\big(0,2\delta^2-(|u|+\delta)^2\big)<w<2\delta^2-(|u|-\delta)^2\), which begins at \(0\) if and only if \(|u|\ge(\sqrt2-1)\delta\); at \(u=0\) it equals \(2/(w-\delta^2)\), which has a pole at \(w=\delta^2\). In particular the constant \(1+\sqrt2\) in (a) cannot be lowered.

Source: ch/basepoint.tex, label lem:dbeta:comp.

Comparison of the Thorin density across vertical lines

Volume I, Theorem 12.1.20, page 245 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be one of \(\xi\), \(\xi(\cdot,\chi_4)\) and \(\xi(\cdot,\chi_{12})\), where \(\xi\) is the completed Riemann zeta function and \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) and \(\xi(s,\chi_{12})=(12/\pi)^{s/2}\Gamma(\frac s2)L(s,\chi_{12})\) are the completed Dirichlet \(L\) functions of the nonprincipal character modulo \(4\) and of the real primitive character modulo \(12\); each \(f\) is entire of order one, real on the real axis, with \(f(s)=f(1-s)\) and no real zeros. Write \(v_\alpha(\theta)=\operatorname{Re}\frac{f'}{f}(\alpha+i\theta)\). Let \(\frac12<\alpha<\alpha'\) and let \(\theta\) be real. Suppose that every zero \(\frac12+\delta+i\gamma\) of \(f\) with \(\delta>(\alpha-\frac12)/\sqrt2\) satisfies \(|\theta-\gamma|\ge(1+\sqrt2)\delta\). Then [  .] The hypothesis holds at every \(\theta\) under the Riemann Hypothesis (RH) for \(\xi\), or the Generalized Riemann Hypothesis (GRH) for \(\chi_4\) or \(\chi_{12}\); at every \(\theta\) when \(\alpha\ge\frac12+\frac1{\sqrt2}\), with no hypothesis; and at every \(\theta\) with \(|\theta|\le H_0-\frac{1+\sqrt2}2\), where \(H_0\) is a height below which every zero of \(f\) is known to lie on the critical line: \(H_0=3\times10^{12}\) for \(\xi\) (Platt and Trudgian 2021), and \(H_0=10^8/q\) for \(\xi(\cdot,\chi_q)\), that is \(2.5\times10^7\) for \(\chi_4\) and \(10^8/12>8.33\times10^6\) for \(\chi_{12}\) (Platt 2016).

Earlier editions: Theorem 9.1.15, page 355 (edition 1).

Source: ch/basepoint.tex, label thm:dbeta:comp.

Explicit lower bound of the Thorin density by comparison with the Euler product

Volume I, Corollary 12.1.21, page 245 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be one of \(\xi\), \(\xi(\cdot,\chi_4)\) and \(\xi(\cdot,\chi_{12})\), where \(\xi\) is the completed Riemann zeta function and \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) and \(\xi(s,\chi_{12})=(12/\pi)^{s/2}\Gamma(\frac s2)L(s,\chi_{12})\) are the completed Dirichlet \(L\) functions of the nonprincipal character modulo \(4\) and of the real primitive character modulo \(12\); each \(f\) is entire of order one, real on the real axis, with \(f(s)=f(1-s)\) and no real zeros. Write \(v_\alpha(\theta)=\operatorname{Re}\frac{f'}{f}(\alpha+i\theta)\), let \(\psi=\Gamma'/\Gamma\) be the digamma function and \(\Lambda\) the von Mangoldt function, and let \(q=1\), \(4\), \(12\) for the three functions in the order above. For \(\sigma>1\) and real \(\theta\), \(v_\sigma(\theta)\ge A^\infty_f(\sigma,\theta)-S_f(\sigma)\), where [A^{(,)=-+(2),A^{4}(,)=+(2),A^{{12}}(,)=+(2),] and \(S_f(\sigma)=\sum_{(n,q)=1}\Lambda(n)n^{-\sigma}\), so that \(S_\xi=-\zeta'/\zeta\), \(S_{\chi_4}(\sigma)=S_\xi(\sigma)-\log2/(2^\sigma-1)\) and \(S_{\chi_{12}}(\sigma)=S_{\chi_4}(\sigma)-\log3/(3^\sigma-1)\); moreover \(A^\infty_f(\sigma,\theta)\) increases in \(|\theta|\). Hence, if \(\frac12<\alpha<\alpha'\) with \(\alpha'>1\), \(\theta\) is real, and every zero \(\frac12+\delta+i\gamma\) of \(f\) with \(\delta>(\alpha-\frac12)/\sqrt2\) satisfies \(|\theta-\gamma|\ge(1+\sqrt2)\delta\) (the comparison hypothesis, which holds at every \(\theta\) when all zeros of \(f\) lie on the line \(\operatorname{Re}s=\frac12\), that is under the Riemann Hypothesis for \(\xi\) or the Generalized Riemann Hypothesis for \(\chi_4\) or \(\chi_{12}\), and at every \(\theta\) when \(\alpha\ge\frac12+\frac1{\sqrt2}\) with no hypothesis), then [v()  (A}_f(‘,)-S_f(’)) = ||+O(1),] the \(O(1)\) being as \(|\theta|\to\infty\) with \(\alpha\), \(\alpha'\) and \(f\) fixed.

Earlier editions: Corollary 9.1.16, page 356 (edition 1).

Source: ch/basepoint.tex, label cor:dbeta:lower.

Minimum of the Thorin density at the origin and its equivalence with RH

Volume I, Theorem 12.1.23, page 246 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be one of \(\xi\), \(\xi(\cdot,\chi_4)\) and \(\xi(\cdot,\chi_{12})\), where \(\xi\) is the completed Riemann zeta function and \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) and \(\xi(s,\chi_{12})=(12/\pi)^{s/2}\Gamma(\frac s2)L(s,\chi_{12})\) are the completed Dirichlet \(L\) functions of the nonprincipal character modulo \(4\) and of the real primitive character modulo \(12\); each \(f\) is entire of order one, real on the real axis, with \(f(s)=f(1-s)\) and no real zeros. Write \(v_\alpha(\theta)=\operatorname{Re}\frac{f'}{f}(\alpha+i\theta)\), \(b_\alpha=v_\alpha(0)=\frac{f'}{f}(\alpha)\) and \(V_f=(\log f)''(\frac12)\), which is \(0.04620998623\), \(0.1560296499\) and \(0.3329236082\) for the three functions in the order above. Let \(H_0\) be a height below which every zero of \(f\) is known to lie on the critical line, \(H_0=3\times10^{12}\) (Platt and Trudgian 2021), \(2.5\times10^7\) and \(10^8/12\) (Platt 2016) for the three functions, and let \(\alpha_{\max}=7.35\), \(4.70\) and \(3.10\) for the three functions. Then: (i) for every \(\alpha>\frac12\), \(0<b_\alpha\le(\alpha-\frac12)V_f\), and the ratio \(b_\alpha/(\alpha-\frac12)\) decreases in \(\alpha\) and tends to \(V_f\) as \(\alpha\downarrow\frac12\); (ii) for \(\frac12<\alpha\le\alpha_{\max}\), \(v_\alpha(\theta)>b_\alpha\) for \(0<|\theta|\le H_0-\frac{1+\sqrt2}2\); (iii) for \(\frac12+\frac1{\sqrt2}\le\alpha\le\alpha_{\max}\), \(v_\alpha(\theta)>b_\alpha\) for every real \(\theta\ne0\); (iv) for \(f=\xi(\cdot,\chi_4)\) and \(f=\xi(\cdot,\chi_{12})\), \(v_\alpha(\theta)>b_\alpha\) for every \(\alpha\ge\frac12+\frac1{\sqrt2}\) and every real \(\theta\ne0\); (v) for \(f=\xi(\cdot,\chi_4)\) and \(f=\xi(\cdot,\chi_{12})\), \(v_\alpha(\theta)>b_\alpha\) for every \(\alpha\ge1\) and every real \(\theta\ne0\); (vi) the Riemann Hypothesis (RH) for \(\xi\), respectively the Generalized Riemann Hypothesis (GRH) for \(\chi_4\) or for \(\chi_{12}\) (all zeros of \(f\) on the line \(\operatorname{Re}s=\frac12\)), holds if and only if \(v_\alpha(\theta)>b_\alpha\) for every \(\alpha\in(\frac12,1)\) and every real \(\theta\ne0\).

Earlier editions: Theorem 9.1.18, page 357 (edition 1).

Source: ch/basepoint.tex, label thm:dbeta:minall.

Heat kernel form of the heat trace of the shifted clock with the Chebyshev error

Volume I, Proposition 12.1.29, page 252 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)\), let \(\psi\) be the Chebyshev function \(\psi(y)=\sum_{n\le y}\Lambda(n)\) with \(\Lambda\) the von Mangoldt function, and for \(\alpha\ge1\) let \(k_\alpha(t)=\pi^{-1}\int_0^\infty e^{-t\theta^2}v_\alpha(\theta)\,d\theta\) be the heat trace (the Laplace transform in \(z=\theta^2\) of the Thorin density of the shifted clock). Let \(b_\alpha=(\xi'/\xi)(\alpha)\), the constant in \(v_\alpha(\theta)=b_\alpha+\int_0^\infty2\sin^2(x\theta/2)\,x\Pi_\alpha(dx)\) for \(\alpha>1\), where \(x\Pi_\alpha(dx)=\big(\frac{e^{-(\alpha+2)x}}{1-e^{-2x}}-e^{-(\alpha-1)x}\big)dx+\sum_n\Lambda(n)n^{-\alpha}\delta_{\log n}(dx)\). Put \(g_t(x)=1-e^{-x^2/4t}\). Then for \(\alpha\ge1\) and \(t>0\), \(2\sqrt{\pi t}\,k_\alpha(t)=b_\alpha+\bar A_\alpha(t)+\int_{[1,\infty)}g_t(\log y)\,y^{-\alpha}\,d(\psi(y)-y)\), where the gamma piece \(\bar A_\alpha(t)=\int_0^\infty g_t(x)\frac{e^{-(\alpha+2)x}}{1-e^{-2x}}dx=\sum_{k\ge1}\big(\frac1{c_k}-\sqrt{\pi t}\,e^{c_k^2t}\operatorname{erfc}(c_k\sqrt t)\big)\) with \(c_k=\alpha+2k\) is positive, and the Stieltjes integral converges absolutely after one integration by parts. For \(\alpha>1\) the Stieltjes integral equals \(\sum_n\Lambda(n)n^{-\alpha}g_t(\log n)-L_\alpha(t)\) with \(L_\alpha(t)=\int_0^\infty g_t(x)e^{-(\alpha-1)x}dx=\frac1{\alpha-1}-\sqrt{\pi t}\,e^{(\alpha-1)^2t}\operatorname{erfc}((\alpha-1)\sqrt t)\).

Source: ch/basepoint.tex, label bp:prop:heat.

Certified positivity of the heat trace at the basepoint one

Volume I, Proposition 12.1.30, page 254 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(v_1(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(1+i\theta)\), and \(T_1\) the generalized gamma convolution with \(\mathbb{E}e^{-uT_1}=\xi(1)/\xi(1+\sqrt u)\). Let \(k_1(t)=\pi^{-1}\int_0^\infty e^{-t\theta^2}v_1(\theta)\,d\theta\) for \(t>0\) be the heat trace of \(T_1\), that is, the Laplace transform in the variable \(z=\theta^2\) of its Thorin measure. Then \(2\sqrt{\pi t}\,k_1(t)>0.021\) for every \(t>0\). The proof uses the primes up to \(10^7\), explicit bounds for \(\psi(y)-y\), where \(\psi(y)=\sum_{n\le y}\Lambda(n)\) is Chebyshev’s function and \(\Lambda\) the von Mangoldt function (Büthe 2018; Broadbent, Kadiri, Lumley, Ng and Wilk 2021; Johnston and Yang 2023), and a computation in ball arithmetic.

Earlier editions: Proposition 9.1.24, page 364 (edition 1).

Source: ch/basepoint.tex, label bp:prop:heatcert.

The heat trace identity at the centre alpha one half with a finite part

Volume I, Remark 12.1.31, page 256 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(W(t)=\sum_{\operatorname{Re}\gamma>0}e^{-t\gamma^2}\) over the zeros \(\gamma\) of \(\xi(\frac12+iz)\) with multiplicity, \(\psi\) the Chebyshev function, and \(\mathcal F(\frac12)=2-\frac{\zeta'}\zeta(\frac12)=2-\frac12\log\pi+\frac12\frac{\Gamma'}\Gamma(\frac14)=-0.686091709613\), the value at \(s=\frac12\) of the continuation of \(\int_1^\infty y^{-s}d(\psi(y)-y)=-\frac{\zeta'}\zeta(s)-\frac1{s-1}\). With no hypothesis on the zeros, for every \(t>0\), \(2\sqrt{\pi t}W(t)=\bar A_{1/2}(t)+\mathcal F(\frac12)-\int_{[1,\infty)}e^{-(\log y)^2/4t}y^{-1/2}d(\psi(y)-y)\), where \(\bar A_{1/2}(t)=\sqrt{t/\pi}\int_0^\infty e^{-tr^2}\big[\operatorname{Re}\frac{\Gamma'}\Gamma(\frac14+\frac{ir}2)-\frac{\Gamma'}\Gamma(\frac14)-4+\frac1{1/4+r^2}\big]dr\) is the gamma piece of the heat trace identity at \(\alpha=\frac12\) and the integral converges absolutely. With a sharp cutoff \(\int_{[1,X]}y^{-1/2}d(\psi(y)-y)\) has no limit as \(X\to\infty\), since a limit would give \(\psi(X)-X=o(\sqrt X)\), which the pole of \(\frac{\zeta'}\zeta\) at \(\frac12+i\gamma_1\) excludes; its Gaussian means converge, since as \(t\to\infty\) \(W(t)\to0\) exponentially and \(\bar A_{1/2}(t)=O(1/t)\), so the integral tends to \(\mathcal F(\frac12)\). As the left side is positive for every \(t\), the Gaussian smoothed Chebyshev error at the centre stays below \(\bar A_{1/2}(t)+\mathcal F(\frac12)\) at every time, unconditionally. A computation shows the two sides agree to \(3\times10^{-15}\) at \(t=0.01\), \(0.025\), \(0.125\), \(0.5\) and \(2\) (for \(t\ge0.5\) the left side is below \(10^{-40}\) and the residual is the rounding of the prime sum); this covers only these values. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Remark 9.1.25, page 365 (edition 1).

Source: ch/basepoint.tex, label bp:rem:centre.

No Brownian first passage to a random level at the basepoint one

Volume I, Proposition 12.2.1, page 258 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(T_1\) the generalized gamma convolution with \(\mathbb{E}e^{-uT_1}=\xi(1)/\xi(1+\sqrt u)\). Let \(B\) be a standard Brownian motion and, for \(c\in\mathbb{R}\) and \(a\ge0\), let \(H^{(c)}_a=\inf\{t:B_t+ct\ge a\}\) be the first passage time of \(B_t+ct\) to the level \(a\). For no drift \(c\ge0\) and no random level \(L\ge0\) independent of the Brownian motion is \(T_1\) equal in law to the first passage time \(H^{(c)}_L\). (For \(c<0\) the first passage time is infinite with positive probability, so equality is impossible.)

Earlier editions: Proposition 9.2.1, page 367 (edition 1).

Source: ch/bphit.tex, label bp:prop:nogo.

Subordination of the tilted centre clock at the basepoint one

Volume I, Theorem 12.2.2, page 258 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(T\) the centre clock, the positive random variable with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\) for \(u\ge0\), which exists with no hypothesis on the zeros (its existence uses the verification of the Riemann Hypothesis below height \(1000\)). Let \(\widetilde T\) have the tilted law \(P(\widetilde T\in dt)=e^{-t/4}P(T\in dt)/\mathbb{E}e^{-T/4}\), and let \((\sigma_\ell)_{\ell\ge0}\) be a subordinator independent of \(\widetilde T\) with Laplace exponent \(u+\sqrt u\), that is, \(\mathbb{E}e^{-u\sigma_\ell}=e^{-\ell(u+\sqrt u)}\). Then the shifted clock \(T_1\), with \(\mathbb{E}e^{-uT_1}=\xi(1)/\xi(1+\sqrt u)\), satisfies [T_1{T}T+H{T/},] where \(H_a\) is the first passage time of a standard Brownian motion to the level \(a\), independent of \(\widetilde T\).

Earlier editions: Theorem 9.2.2, page 368 (edition 1).

Source: ch/bphit.tex, label bp:thm:sub.

Kent string hitting time of the shifted clock at the basepoint one under RH

Volume I, Corollary 12.2.3, page 259 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis (every nontrivial zero of the Riemann zeta function has real part one half) and the simplicity of every nontrivial zero.

Statement. Let \(T\) be the centre clock, with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\), where \(\xi\) is the completed Riemann zeta function, let \(\widetilde T\) be the clock tilted by \(e^{-t/4}\), \(P(\widetilde T\in dt)=e^{-t/4}P(T\in dt)/\mathbb{E}e^{-T/4}\), and let \(T_1\) be the shifted clock with \(\mathbb{E}e^{-uT_1}=\xi(1)/\xi(1+\sqrt u)\), so that \(T_1\stackrel{d}{=}\sigma_{\widetilde T}\) for an independent subordinator \(\sigma\) with Laplace exponent \(u+\sqrt u\), the inverse local time at zero of a Brownian motion \(B\) that is sticky at zero. Assume RH and that all nontrivial zeros are simple. Then \(\widetilde T\) is Kent’s hitting time for a Krein string whose Dirichlet spectrum is \(\{\gamma^2+\frac14\}=\{|\rho|^2\}\) (\(\gamma>0\) running over the ordinates of the zeros \(\rho=\frac12+i\gamma\)), and \(T_1\) is the first hitting time of the level \(y_*\) by the process \(D(\ell_t)\), where \(D\) is that string diffusion started at its left end, \(y_*\) is the level whose hitting time by \(D\) is \(\widetilde T\), and \(\ell_t\) is the local time at zero of the independent sticky Brownian motion \(B\). Equivalently, \(T_1\) is the first hitting time of the line \(\{y=y_*\}\) by the two dimensional Markov process \((D(\ell_t),B_t)\).

Earlier editions: Corollary 9.2.3, page 369 (edition 1).

Source: ch/bphit.tex, label bp:cor:kent.

Subordination of the centre clock at every basepoint above the centre

Volume I, Theorem 12.2.5, page 264 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\beta+i\gamma\), and \(T\) the centre clock, with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\). For every \(\alpha>\frac12\), with no hypothesis on the zeros (the existence of the centre clock uses the verification of the Riemann Hypothesis below height \(1000\)), \(\varphi_\alpha(u)=\xi(\alpha)/\xi(\alpha+\sqrt u)\) is the Laplace transform of \(T_\alpha\stackrel{d}{=}\sigma^{(\alpha)}_{\widetilde T_\alpha}\), where \(\widetilde T_\alpha\) is the centre clock tilted by \(e^{-(\alpha-\frac12)^2T}\), that is, \(P(\widetilde T_\alpha\in dt)=e^{-(\alpha-\frac12)^2t}P(T\in dt)/\mathbb{E}e^{-(\alpha-\frac12)^2T}\), and \(\sigma^{(\alpha)}\) is an independent subordinator with exponent \(u+2(\alpha-\frac12)\sqrt u\), that is, \(T_\alpha\stackrel{d}{=}\widetilde T_\alpha+H_{\sqrt2(\alpha-\frac12)\widetilde T_\alpha}\) with \(H_a\) the first passage time of a standard Brownian motion to the level \(a\), independent of \(\widetilde T_\alpha\). The law has infinite mean and \(P(T_\alpha>x)\sim\frac{\xi'}{\xi}(\alpha)\,(\pi x)^{-1/2}\) as \(x\to\infty\). By the threshold theorem of Polson (2026), the variable \(T_\alpha\) is a generalized gamma convolution (GGC) if and only if no zero has \(\beta>\alpha\), in particular for every \(\alpha\ge1\), and when every zero has \(\beta<\alpha\) its Thorin density in the chart \(\theta=\sqrt z\) is \(\frac1\pi\sum_\rho\frac{\alpha-\beta}{(\alpha-\beta)^2+(\theta-\gamma)^2}\).

Earlier editions: Theorem 9.2.5, page 373 (edition 1).

Source: ch/bphit.tex, label bp:thm:alpha.

Lévy measure and self decomposability of the shifted clock

Volume I, Corollary 12.2.6, page 264 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho\), let \(G(u)=\xi(\frac12+\sqrt u)/\xi(\frac12)=\prod_j(1+u/a_j)\) with \(a_j=-(\rho_j-\frac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), and let \(W(x)=\sum_je^{-a_jx}\), so that the centre clock \(T\), with \(\mathbb{E}e^{-uT}=1/G(u)\), is infinitely divisible with Lévy measure \(W(x)\,dx/x\); with no hypothesis on the zeros (using the verification of the Riemann Hypothesis below height \(1000\)), \(W>0\) and \(W'<0\) on \((0,\infty)\). Let \(\alpha>\frac12\), \(a=\alpha-\frac12\), and let \(T_\alpha\) be the shifted clock with \(\mathbb{E}e^{-uT_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt u)\). Then \(T_\alpha\) is infinitely divisible, with no drift and Lévy measure \(W_\alpha(x)\,dx/x\), where [W_(x)=_0^x(x-s){-3/2}e^{-a2xs/(x-s)},W(s),ds=,] with \(V=a^2/(2Z^2)\) and \(Z\) a standard normal variable. Hence \(W_\alpha>0\) and \(W_\alpha'<0\) on \((0,\infty)\), so that \(T_\alpha\) is self decomposable; and on every interval \((0,x_0]\) on which \(W\) is completely monotone, so is \(W_\alpha\).

Earlier editions: Corollary 9.2.6, page 374 (edition 1).

Source: ch/bphit.tex, label bp:cor:levyalpha.

Volume I, Chapter 13. The critical strip, Riesz’s criterion and the squarefree series

Reciprocal Xi function on the real axis as a normal scale mixture

Volume I, Proposition 13.1.1, page 268 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(b_\alpha=\xi'(\alpha)/\xi(\alpha)\). For \(\alpha\ge\frac12\) let \(T_\alpha\) be the positive random variable with Laplace transform \(\mathbb{E}\,e^{-\lambda T_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt\lambda)\), \(\lambda\ge0\) (it exists with no hypothesis on the zeros of \(\zeta\); \(T_{1/2}\) is the centre clock), let \(Z\) be a standard normal variable independent of \(T_\alpha\), and put \(Y_\alpha=\sqrt{2T_\alpha}\,Z\). Then, for every \(\alpha\ge\frac12\), the function \(u\mapsto\xi(\alpha)/\xi(\alpha+|u|)\) on \(\mathbb{R}\) is the characteristic function of \(Y_\alpha\), and the law of \(Y_\alpha\) is symmetric, unimodal and infinitely divisible, with no hypothesis on the zeros. At the centre its variance is \(2\,\mathbb{E}\,T_{1/2}=2\sum_k a_k^{-1}=0.046210\), where \(a_k=-(\rho_k-\frac12)^2\) with one zero \(\rho_k\) taken from each pair \(\{\rho,1-\rho\}\) of nontrivial zeros of \(\zeta\). For \(\alpha>\frac12\), \(\xi(\alpha)/\xi(\alpha+|u|)=1-b_\alpha|u|+O(u^2)\) as \(u\to0\) (a corner at the origin), so \(Y_\alpha\) lies in the domain of attraction of the Cauchy law, and \(P(|Y_\alpha|>y)\sim 2b_\alpha/(\pi y)\) as \(y\to\infty\).

Earlier editions: Proposition 10.1.1, page 378 (edition 1).

Source: ch/strip.tex, label sm:prop:cf.

Thorin form of the reciprocal Xi function and the EGGC criterion

Volume I, Theorem 13.1.2, page 270 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, let \(\rho=\beta+i\gamma\) run over the nontrivial zeros of \(\zeta\), and for \(\alpha\ge\frac12\) let \(T_\alpha\) be the positive random variable with Laplace transform \(\mathbb{E}\,e^{-\lambda T_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt\lambda)\), \(\lambda\ge0\), let \(Z\) be a standard normal variable independent of \(T_\alpha\), \(Y_\alpha=\sqrt{2T_\alpha}\,Z\), and \(v_\alpha(\theta)=\operatorname{Re}\,(\xi'/\xi)(\alpha+i\theta)\) for real \(\theta\). Recall that a symmetric law is an extended generalised gamma convolution (EGGC) if and only if it is the law of \(Z\sqrt{2T'}\) with \(T'\) a generalised gamma convolution (GGC) independent of \(Z\), where a GGC is a law on \([0,\infty)\) with Laplace transform \(\exp\{-a\lambda-\int_{(0,\infty)}\log(1+\lambda/z)\,U(dz)\}\) for a drift \(a\ge0\) and a Thorin measure \(U\). Let \(\alpha\ge1\), or more generally let \(\alpha>\frac12\) be such that \(\operatorname{Re}\rho<\alpha\) for every zero \(\rho\). Then \(v_\alpha>0\) and \(\xi(\alpha)/\xi(\alpha+|u|)=\exp\{-\frac1\pi\int_0^\infty\log(1+u^2/\theta^2)\,v_\alpha(\theta)\,d\theta\}\) for real \(u\), so that \(T_\alpha\) is a GGC without drift whose Thorin measure is \((v_\alpha(\theta)/\pi)\,d\theta\) under the substitution \(z=\theta^2\), and \(Y_\alpha\) is a symmetric EGGC. With no hypothesis on the zeros, for \(\frac12<\alpha<1\) the law of \(Y_\alpha\) is an EGGC if and only if \(\zeta\) has no zeros with real part greater than \(\alpha\), and \(Y_{1/2}\) is an EGGC if and only if the Riemann Hypothesis holds (that centre statement is the earlier rigidity theorem of Polson, 2026, for the centre clock \(T_{1/2}\), read through \(Y_{1/2}=\sqrt{2T_{1/2}}\,Z\)). Under the Riemann Hypothesis, \(\xi(\frac12)/\xi(\frac12+|u|)=\prod_{\gamma>0}(1+u^2/\gamma^2)^{-1}\), the product over the ordinates of the zeros \(\frac12+i\gamma\) with \(\gamma>0\), counted with multiplicity, and \(Y_{1/2}\) has the law of \(\sum_kL_k/\gamma_k\) with independent standard Laplace variables \(L_k\) (density \(\frac12e^{-|y|}\)).

Earlier editions: Theorem 10.1.2, page 380 (edition 1).

Source: ch/strip.tex, label sm:thm:thorin.

EGGC property of the reciprocal Xi law across the critical strip

Volume I, Theorem 13.1.3, page 272 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, let \(\rho=\beta+i\gamma\) run over the nontrivial zeros of \(\zeta\) (with multiplicity), and for \(\alpha\ge\frac12\) let \(T_\alpha\) be the positive random variable with \(\mathbb{E}\,e^{-\lambda T_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt\lambda)\), \(\lambda\ge0\), \(Z\) a standard normal variable independent of \(T_\alpha\), \(Y_\alpha=\sqrt{2T_\alpha}\,Z\) and \(v_\alpha(\theta)=\operatorname{Re}\,(\xi'/\xi)(\alpha+i\theta)\). Recall that a symmetric law is an extended generalised gamma convolution (EGGC) if and only if it is the law of \(Z\sqrt{2T'}\) with \(T'\) a generalised gamma convolution independent of \(Z\). Let \(\frac12<\alpha<1\) and \(\varepsilon=\alpha-\frac12\). (1) With no hypothesis on the zeros, \(\xi(\alpha)/\xi(\alpha+|u|)\) is the characteristic function of the infinitely divisible, symmetric, unimodal law \(Y_\alpha\). If every zero has real part less than \(\alpha\), then \(\xi(\alpha)/\xi(\alpha+|u|)=\exp\{-\frac1\pi\int_0^\infty\log(1+u^2/\theta^2)\,v_\alpha(\theta)\,d\theta\}\) holds with the positive density \(v_\alpha(\theta)/\pi=\frac1\pi\sum_\rho\frac{\alpha-\beta}{(\alpha-\beta)^2+(\theta-\gamma)^2}\). (2) \(Y_\alpha\) is an EGGC if and only if no zero has \(\beta>\alpha\), that is, if and only if \(\zeta\) has no zeros in the half plane \(\operatorname{Re}s>\alpha\). (3) Let \(T_0\ge2\) be a height such that all zeros \(\rho\) with \(0<\gamma\le T_0\) lie on the critical line. Then \(\xi(\alpha)/\xi(\alpha+|u|)=\prod_{0<\gamma\le T_0}\frac{\varepsilon^2+\gamma^2}{(\varepsilon+|u|)^2+\gamma^2}\times\mathcal{E}_{\alpha,T_0}(u)\), where the finite product, over the ordinates \(\gamma\) of the zeros \(\frac12+i\gamma\) with \(0<\gamma\le T_0\) (with multiplicity), is the characteristic function of an EGGC with the positive Thorin density \(\sum_{0<\gamma\le T_0}[P_\varepsilon(\theta-\gamma)+P_\varepsilon(\theta+\gamma)]\), \(P_\varepsilon(x)=\varepsilon/(\pi(\varepsilon^2+x^2))\), and the remainder satisfies \(|\log\mathcal{E}_{\alpha,T_0}(u)|\le C(|u|+u^2)\frac{\log T_0}{T_0}\) for an absolute constant \(C\) (the proof gives \(C=4\)).

Earlier editions: Theorem 10.1.3, page 382 (edition 1).

Source: ch/strip.tex, label sm:thm:strip.

Complete monotonicity of the Poisson trace is equivalent to RH

Volume I, Corollary 13.1.4, page 275 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, let \(\rho_k=\beta_k+i\gamma_k\) (\(\gamma_k>0\)) be the nontrivial zeros of \(\zeta\) in the upper half plane, counted with multiplicity, and put \(a_k=-(\rho_k-\frac12)^2\) and \(\sqrt{a_k}=\gamma_k+i(\frac12-\beta_k)\) (the root with positive real part). Define the heat trace \(W(t)=\sum_ke^{-a_kt}\) and the Poisson trace \(K_{1/2}(y)=\sum_ke^{-y\sqrt{a_k}}\) for \(t,y>0\). For \(\alpha\ge1\) put \(v_\alpha(\theta)=\operatorname{Re}\,(\xi'/\xi)(\alpha+i\theta)\), \(b_\alpha=\xi'(\alpha)/\xi(\alpha)\) and \(K_\alpha(y)=\frac1\pi\int_0^\infty e^{-\theta y}v_\alpha(\theta)\,d\theta\). Then the Riemann Hypothesis holds if and only if \(K_{1/2}\) is completely monotone on \((0,\infty)\), and if and only if \(W\) is completely monotone on \((0,\infty)\) (the equivalence with \(W\) is the earlier result of Polson, 2026, which the corollary recalls). For every \(\alpha\ge1\), with no hypothesis, \(K_\alpha\) is completely monotone and \(K_\alpha(y)\sim b_\alpha/(\pi y)\) as \(y\to\infty\); consequently the Lévy density \(K_\alpha(|y|)/|y|\) of the law with characteristic function \(\xi(\alpha)/\xi(\alpha+|u|)\) is asymptotic to \(b_\alpha/(\pi y^2)\), the Lévy density of the Cauchy component.

Earlier editions: Corollary 10.1.4, page 385 (edition 1).

Source: ch/strip.tex, label sm:cor:poisson.

Positive half of the reciprocal Xi law as a centred infinitely divisible law with Poisson trace Lévy density

Volume I, Proposition 13.1.5, page 276 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, \(K_{1/2}(x)=\sum_ke^{-x\sqrt{a_k}}\) the Poisson trace over the nontrivial zeros \(\rho_k=\beta_k+i\gamma_k\) (\(\gamma_k>0\)) with \(a_k=-(\rho_k-\frac12)^2\) and \(\sqrt{a_k}=\gamma_k-i(\beta_k-\frac12)\), which is positive on \((0,\infty)\), and let \(T\) be the positive random variable with \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\), so that \(Y_{1/2}\) is the symmetric infinitely divisible law with characteristic function \(\xi(\frac12)/\xi(\frac12+|u|)\) and Lévy density \(K_{1/2}(|y|)/|y|\). Let \(A\) be the centred infinitely divisible law without Gaussian part whose Lévy measure is \(K_{1/2}(x)\,dx/x\) on \((0,\infty)\), and \(A'\) an independent copy. Then \(Y_{1/2}\overset d=A-A'\), and the cumulants are \(\kappa_m(A)=\int_0^\infty x^{m-1}K_{1/2}(x)\,dx\) for \(m\ge2\); in particular \(\operatorname{Var}A=\mathbb{E}T\). (i) Under the Riemann Hypothesis, with \(\gamma_k\) the positive ordinates counted with multiplicity and \(E_k\) independent standard exponential variables, \(A\) has the law of \(\sum_k(E_k-1)/\gamma_k\), a series that converges almost surely and in \(L^2\); \(\mathbb{E}e^{-sA}=1/\mathcal{P}(s)\) for \(s>-\gamma_1\), where \(\mathcal{P}(s)=\prod_k(1+s/\gamma_k)e^{-s/\gamma_k}\) and \(\mathcal{P}(s)\mathcal{P}(-s)=\xi(\frac12+is)/\xi(\frac12)\); and \(\kappa_m(A)=(m-1)!\sum_k\gamma_k^{-m}\). (ii) If \(A\) has the law of \(\sum_k(E_k-1)/c_k\) for positive numbers \(c_k\) with \(\sum_kc_k^{-2}<\infty\), then the Riemann Hypothesis holds. (iii) For positive \(\gamma_k\) with \(\sum_k\gamma_k^{-2}<\infty\), the function \(u\mapsto\mathcal{P}(\sqrt u)\) is the Laplace transform of a generalized gamma convolution whose Thorin density in the variable \(\theta=\sqrt z\) is \(\frac1\pi\sum_k\theta^2/\bigl(\gamma_k(\gamma_k^2+\theta^2)\bigr)\). For the heat flow function \(\xi_t\) (\(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\)) and \(t\ge\Lambda\), unconditionally for \(t\ge0.2\), where \(\Lambda\le0.2\) is the de Bruijn Newman constant (Platt and Trudgian 2021) and the zeros of \(\xi_t\) are \(\frac12\pm i\theta_k(t)\), the series \(A_t=\sum_k(E_k-1)/\theta_k(t)\) satisfies \(B_{2T_t}\overset d=A_t-A_t'\), with \(A_t'\) an independent copy of \(A_t\), for a Brownian motion \(B\) independent of \(T_t\), the positive variable with \(\mathbb{E}e^{-uT_t}=\xi_t(\frac12)/\xi_t(\frac12+\sqrt u)\), and \(\mathbb{E}e^{-sA_t}=1/\mathcal{P}_t(s)\) with \(\mathcal{P}_t(s)\mathcal{P}_t(-s)=\xi_t(\frac12+is)/\xi_t(\frac12)\), \(\mathcal{P}_t\) being the product of (i) over the \(\theta_k(t)\).

Earlier editions: Proposition 10.1.5, page 386 (edition 1).

Source: ch/strip.tex, label sm:prop:half.

Primes as poles of the Poisson trace: the residues at plus or minus i times the logarithms of prime powers

Volume I, Proposition 13.1.6, page 277 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho=\beta+i\gamma\) run over the nontrivial zeros of the Riemann zeta function with \(\gamma>0\), counted with multiplicity, let \(V(z)=\sum_{\gamma>0}e^{\rho z}\) be Cramér’s function, and let \(K_{1/2}(x)=\sum_{\gamma>0}e^{-x(\gamma-i(\beta-1/2))}\), which is the Poisson trace of the zeros. Then \(K_{1/2}(x)=e^{-ix/2}V(ix)\), and the series converges for \(\operatorname{Re}x>0\). The function \(K_{1/2}\) continues to a meromorphic function on \(\mathbb{C}\setminus(-\infty,0]\) (the point \(x=0\) is a logarithmic branch point), whose poles are simple and lie at \(x=\pm i\log n\) for the prime powers \(n\), with residue \(-\Lambda(n)/(2\pi\sqrt n)\), where \(\Lambda\) is von Mangoldt’s function. The continuation of \(V\) with its simple poles is due to Cramér (1919); the value of the residue is proved in the book by Landau’s formula \(\sum_{0<\gamma\le T}n^{\rho}=-\frac T{2\pi}\Lambda(n)+O(\log T)\) and Abel summation.

Earlier editions: Proposition 10.1.6, page 387 (edition 1).

Source: ch/strip.tex, label sm:prop:cramer.

Unconditional range of the complete monotonicity conditions of the Poisson trace

Volume I, Theorem 13.1.8, page 279 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_k=\beta_k+i\gamma_k\) (\(\gamma_k>0\)) be the nontrivial zeros of \(\zeta\) in the upper half plane, counted with multiplicity, \(a_k=-(\rho_k-\frac12)^2\), \(\sqrt{a_k}=\gamma_k+i\delta_k\) with \(\delta_k=\frac12-\beta_k\), \(\varphi_k=\arctan(\delta_k/\gamma_k)\), and let \(K_{1/2}(y)=\sum_ke^{-y\sqrt{a_k}}\) be the Poisson trace of the zeros. For \(n=0,1,2,\dots\) and \(y>0\), \((-1)^nK_{1/2}^{(n)}(y)=\sum_k|\sqrt{a_k}|^n\,e^{-\gamma_ky}\cos(n\varphi_k-\delta_ky)\). Suppose that every zero with \(0<\gamma_k\le H\) lies on the critical line. Then for \(0<y<\pi\) and every integer \(n\) with \(0\le n<H(\pi-y)\), every term of this series is positive, and \((-1)^nK_{1/2}^{(n)}(y)>0\). With \(H=T_0=3\times10^{12}\), the height up to which Platt and Trudgian verified that all zeros lie on the critical line, this covers all \(n<9.12\times10^{12}\) at \(y=0.1\), all \(n<6.42\times10^{12}\) at \(y=1\) and all \(n<4.24\times10^{11}\) at \(y=3\).

Earlier editions: Theorem 10.1.8, page 389 (edition 1).

Source: ch/strip.tex, label thm:biane:cmrange.

Quasi Riemann hypothesis version of the range of the Poisson trace conditions: y below 4 pi over 3 and n below 4 pi T_0 over 3 under (Q_{7/8})

Volume I, Remark 13.1.9, page 279 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The quasi Riemann hypothesis \((\mathrm Q_{7/8})\) of the book, which rests on the unrefereed half plane of OpenAI (2026); the variant with Liu’s claimed half plane \(\operatorname{Re}s>0.874957\) rests on an unrefereed claim as well.

Statement. In the rectangle theorem for the Poisson trace conditions, assume \((\mathrm Q_{7/8})\), which rests on the unrefereed half plane of OpenAI (2026). Then \(\Delta=\frac38\) and the theorem holds for \(0<y<\frac{4\pi}3=4.1888\) and \(n<\frac{4\pi}3T_0=1.26\times10^{13}\), \(T_0=3\times10^{12}\). Liu’s claimed half plane \(\operatorname{Re}s>0.874957\), also unrefereed, would give \(y<4.1893\). The half plane at \(\frac78\) changes the constants by the factor \(\frac43\) and nothing else. The rectangle uses that \(\varphi_k\) and \(\delta_k\) have the same sign, which the sum bound of the termwise proposition discards; the observation is from the first author’s note on Ostrovsky’s laws and the quasi Riemann hypothesis.

Earlier editions: Remark 10.1.9, page 389 (edition 1).

Source: ch/strip.tex, label tx3:rem:qrhrange.

Positive definite and bounded forms of the Poisson trace criterion for RH

Volume I, Proposition 13.1.10, page 280 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_k=\beta_k+i\gamma_k\) (\(\gamma_k>0\)) be the nontrivial zeros of \(\zeta\) in the upper half plane, counted with multiplicity, and put \(a_k=-(\rho_k-\frac12)^2\) and \(\sqrt{a_k}=\gamma_k+i(\frac12-\beta_k)\). For complex \(z\) with \(\operatorname{Re}z>0\) let \(K_{1/2}(z)=\sum_ke^{-z\sqrt{a_k}}\) (the Poisson trace of the zeros, absolutely and locally uniformly convergent, hence analytic). The following are equivalent: (i) the Riemann Hypothesis; (ii) \(K_{1/2}\) is completely monotone on \((0,\infty)\); (iii) for every \(x>0\) the function \(y\mapsto K_{1/2}(x+iy)\) is positive definite on \(\mathbb{R}\); (iv) for some \(x>0\) the function \(y\mapsto K_{1/2}(x+iy)\) is bounded on \([0,\infty)\). Under these conditions \(|K_{1/2}(x+iy)|\le K_{1/2}(x)\) for \(x>0\) and real \(y\).

Earlier editions: Proposition 10.1.10, page 390 (edition 1).

Source: ch/strip.tex, label prop:biane:pd.

Small y asymptotics of the Poisson trace and Pólya’s trace

Volume I, Proposition 13.1.11, page 282 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_k=\beta_k+i\gamma_k\) (\(\gamma_k>0\)) be the nontrivial zeros of \(\zeta\) in the upper half plane, counted with multiplicity, \(a_k=-(\rho_k-\frac12)^2\), \(\sqrt{a_k}=\gamma_k+i(\frac12-\beta_k)\), and \(K_{1/2}(y)=\sum_ke^{-y\sqrt{a_k}}\) (the Poisson trace of the zeros). Let \(P(\theta)=K_{i\theta/2}(2\pi)\), with \(K_\nu\) the modified Bessel function of the second kind (Pólya’s function), let \(\theta_1\le\theta_2\le\cdots\) be its positive zeros, counted with multiplicity, and \(K^P(y)=\sum_ne^{-\theta_ny}\). Let \(\gamma_E\) be Euler’s constant, \(M(y)=\frac1{2\pi y}\big(\log\frac1{2\pi y}-\gamma_E\big)\) and \(c=2\pi-\frac1{6\pi}-\frac1{48\pi}=6.2235022035\). As \(y\to0+\): (i) \(K_{1/2}(y)=M(y)+\frac78+O(y\log\frac1y)\), and under the Riemann Hypothesis \(K_{1/2}(y)=M(y)+\frac78+\frac1{48\pi}y\log\frac1y+o(y\log\frac1y)\); (ii) \(K^P(y)=M(y)-\frac14+\big(2\pi-\frac1{6\pi}\big)y\log\frac1y+O(y)\); (iii) \(D(y)=K_{1/2}(y)-K^P(y)=\frac98+O(y\log\frac1y)\), and under the Riemann Hypothesis \(D(y)=\frac98-c\,y\log\frac1y+o(y\log\frac1y)\).

Earlier editions: Proposition 10.1.11, page 392 (edition 1).

Source: ch/strip.tex, label prop:biane:traces.

Explicit obstruction and approximation cost for translates of the tilted Pólya kernel

Volume I, Proposition 13.1.12, page 284 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and let \(\Phi(u)=2\sum_{n\ge1}\big(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\big)\exp(-\pi n^2e^{2u})\) be Pólya’s kernel, an even, positive function that decays doubly exponentially and satisfies \(\int_{\mathbb{R}}\Phi(u)e^{su}\,du=\xi(\frac12+s)\) for every complex \(s\). For \(\alpha\ge\frac12\) put \(\Phi_\alpha(u)=e^{(\alpha-\frac12)u}\Phi(u)\), so that \(\int_{\mathbb{R}}\Phi_\alpha(u)e^{i\theta u}\,du=\xi(\alpha+i\theta)\), and for \(g\in L^1(\mathbb{R})\) put \(\hat g(\theta)=\int_{\mathbb{R}}g(u)e^{i\theta u}\,du\). (1) The translates \(\Phi_\alpha(\cdot-x)\), \(x\in\mathbb{R}\), span a dense subspace of \(L^1(\mathbb{R})\) if and only if \(\zeta\) has no zeros on the line \(\operatorname{Re}s=\alpha\). (2) At the centre (\(\alpha=\frac12\)) they are not dense: for every finite combination \(h=\sum_kc_k\Phi(\cdot-x_k)\) and every \(g\in L^1(\mathbb{R})\), \(\|g-h\|_1\ge\sup_\gamma|\hat g(\gamma)|\), the supremum over the ordinates \(\gamma\) of the zeros of \(\zeta\) on the critical line. (3) At the basepoint one they are dense, with no hypothesis on the zeros. (4) For \(\alpha_0\in[\frac12,1)\), \(\zeta\) has no zeros in the half plane \(\operatorname{Re}s>\alpha_0\) if and only if the translates of \(\Phi_\alpha\) are dense in \(L^1(\mathbb{R})\) for every \(\alpha>\alpha_0\); in particular the Riemann Hypothesis holds if and only if they are dense for every \(\alpha>\frac12\). (5) If the translates of \(\Phi_\alpha\) are dense, \(h=\sum_kc_k\Phi_\alpha(\cdot-x_k)\), \(g\in L^1(\mathbb{R})\) and \(\|g-h\|_1\le\delta\), then \(\sum_k|c_k|\ge(|\hat g(\gamma)|-\delta)/|\xi(\alpha+i\gamma)|\) for every ordinate \(\gamma\) of a zero of \(\zeta\). Parts (1), (3) and (4) restate Wiener’s closure theorem (Wiener 1932) for \(\Phi_\alpha\); parts (2) and (5) are what the completeness form adds.

Earlier editions: Proposition 10.1.12, page 394 (edition 1).

Source: ch/strip.tex, label wr:prop:wiener.

Pólya kernel analogue of Salem’s criterion

Volume I, Corollary 13.1.13, page 286 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(\Phi\) be Pólya’s kernel, the even positive function with \(\int_{\mathbb{R}}\Phi(u)e^{su}\,du=\xi(\frac12+s)\) for every complex \(s\), and put \(\Phi_\alpha(u)=e^{(\alpha-\frac12)u}\Phi(u)\) and \(\Xi(\gamma)=\xi(\frac12+i\gamma)\). The Riemann Hypothesis holds if and only if, for every \(\frac12<\alpha<1\), the equation \(\int_{\mathbb{R}}\Phi_\alpha(x-y)f(y)\,dy=0\) for all \(x\in\mathbb{R}\) has no bounded solution \(f\ne0\). At the centre (\(\alpha=\frac12\)) the equation has the explicit bounded solutions \(f(y)=\cos(\gamma y)\), one for each ordinate \(\gamma\) of a zero of \(\zeta\) on the critical line, because \(\Xi(\gamma)=0\) there; at the basepoint one (\(\alpha=1\)) it has none. Salem (1953) proved the corresponding statement with the Fermi Dirac kernel in place of \(\Phi_\alpha\).

Earlier editions: Corollary 10.1.13, page 396 (edition 1).

Source: ch/strip.tex, label wr:cor:salem.

Twisted Riesz function and the basepoint form of Riesz’s criterion

Volume I, Proposition 13.2.1, page 289 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\) be the Möbius function and, for real \(h\) and \(x>0\), let \(R_h(x)=x\sum_{n\ge1}\frac{\mu(n)}{n^2}\big(\frac x{n^2}\big)^{ih}e^{-x/n^2}\); for \(h=0\) this is Riesz’s function \(R(x)=x\sum_{n\ge1}\mu(n)n^{-2}e^{-x/n^2}=-\sum_{k\ge1}\frac{(-x)^k}{(k-1)!\,\zeta(2k)}\). For every real \(h\) and \(\frac12<\operatorname{Re}s<1\), \(\int_0^\infty R_h(x)\,x^{-s-1}\,dx=\Gamma(1-s+ih)/\zeta(2s)\). For \(\frac12\le\alpha<1\), \(\zeta\) has no zeros in the half plane \(\operatorname{Re}s>\alpha\) if and only if \(R_h(x)=O(x^{\alpha/2+\delta})\) as \(x\to\infty\) for every \(\delta>0\). In particular the Riemann Hypothesis holds if and only if \(R_h(x)=O(x^{1/4+\delta})\) for every \(\delta>0\). For \(h=0\) and \(\alpha=\frac12\) this is Riesz’s criterion (Riesz 1916).

Earlier editions: Proposition 10.2.1, page 400 (edition 1).

Source: ch/riesz.tex, label ri:prop:twisted.

The Riesz function as an average over the centre clock

Volume I, Proposition 13.2.4, page 292 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(R(x)=x\sum_{n\ge1}\mu(n)n^{-2}e^{-x/n^2}=-\sum_{k\ge1}\frac{(-x)^k}{(k-1)!\,\zeta(2k)}\) be Riesz’s function (\(\mu\) the Möbius function), \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) the completed Riemann zeta function, and \(T\) the centre clock, the positive random variable with \(\mathbb{E}\,e^{-\lambda T}=\xi(\frac12)/\xi(\frac12+\sqrt\lambda)\), \(\lambda\ge0\). For every \(x>0\), with no hypothesis on the zeros of \(\zeta\), \(R(x)=-\frac1{\xi(1/2)}\,\mathbb{E}\Big[\sum_{k\ge1}k(2k-1)\Big(-\frac x\pi\Big)^ke^{-(2k-\frac12)^2T}\Big]\), the expectation of the series being the series of the expectations.

Earlier editions: Proposition 10.2.4, page 402 (edition 1).

Source: ch/riesz.tex, label ri:prop:avg.

Riesz coefficients as values of GGC Laplace transforms

Volume I, Proposition 13.2.5, page 293 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, and for \(\alpha\ge\frac12\) let \(T_\alpha\) be the positive random variable with Laplace transform \(\mathbb{E}\,e^{-\lambda T_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt\lambda)\), \(\lambda\ge0\) (it exists with no hypothesis on the zeros). Put \(\kappa_k=2k(2k-1)(k-1)!/\pi^k\) and \(\rho_{1/2}(\theta)=\frac1\pi\operatorname{Re}\,(\xi'/\xi)(1+i\theta)\). For every \(\alpha\in[\frac12,1]\) and every integer \(k\ge1\), \(\frac1{\zeta(2k)}=\frac{\kappa_k}{2\xi(\alpha)}\,\mathbb{E}\,e^{-(2k-\alpha)^2T_\alpha}\). In particular, for the centre clock \(T_{1/2}\) and the basepoint one clock \(T_1\), where \(\xi(1)=\frac12\), \(\frac1{\zeta(2k)}=\frac{\kappa_k}{2\xi(\frac12)}\,\mathbb{E}\,e^{-(2k-\frac12)^2T_{1/2}}=\kappa_k\,\mathbb{E}\,e^{-(2k-1)^2T_1}\). At \(\alpha=1\), with no hypothesis on the zeros, \(T_1\) is a generalised gamma convolution without drift whose Thorin measure is \(\rho_{1/2}(\theta)\,d\theta\) under the substitution \(z=\theta^2\), and \(\frac1{\zeta(2k)}=\kappa_k\exp\Big\{-\int_0^\infty\log\Big(1+\frac{(2k-1)^2}{\theta^2}\Big)\rho_{1/2}(\theta)\,d\theta\Big\}\).

Earlier editions: Proposition 10.2.5, page 403 (edition 1).

Source: ch/riesz.tex, label ri:prop:ggccoef.

GGC form of Riesz’s criterion

Volume I, Theorem 13.2.7, page 295 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, \(R(x)=x\sum_{n\ge1}\mu(n)n^{-2}e^{-x/n^2}=-\sum_{k\ge1}\frac{(-x)^k}{(k-1)!\,\zeta(2k)}\) Riesz’s function (\(\mu\) the Möbius function), \(\Lambda\) the von Mangoldt function and \(v_\alpha(\theta)=\operatorname{Re}\,(\xi'/\xi)(\alpha+i\theta)\). For \(\alpha>\frac12\) let \(T_\alpha\) be the positive random variable with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt\lambda)\), \(\lambda\ge0\) (it exists with no hypothesis on the zeros); a generalised gamma convolution (GGC) is a law on \([0,\infty)\) with Laplace transform \(\exp\{-a\lambda-\int_{(0,\infty)}\log(1+\lambda/z)\,U(dz)\}\) for a drift \(a\ge0\) and a Thorin measure \(U\). For \(\frac12\le\alpha<1\) the following are equivalent. (1) \(R(x)=O(x^{\alpha/2+\delta})\) as \(x\to\infty\) for every \(\delta>0\). (2) \(\zeta\) has no zeros with \(\operatorname{Re}s>\alpha\). (3) \(T_{\alpha'}\) is a GGC for every \(\alpha'>\alpha\). (4) For every \(\alpha'>\alpha\), \(v_{\alpha'}(\theta)\ge0\) at every real \(\theta\) for which \(\alpha'+i\theta\) is not a zero of \(\xi\). (5) For every \(\alpha'>\alpha\) and every real \(\theta\) the limit \(\lim_{L\to\infty}\Big[\sum_{\log n\le L}\frac{\Lambda(n)}{n^{\alpha'}}\,2\sin^2\big(\tfrac12\theta\log n\big)-\int_0^L2\sin^2\big(\tfrac12\theta x\big)\,e^{(1-\alpha')x}\,dx\Big]\) exists. In particular (\(\alpha=\frac12\)) the Riemann Hypothesis holds if and only if \(T_\alpha\) is a GGC for every \(\alpha>\frac12\). For \(\alpha=\frac12\) the equivalence of (1) and (2) is Riesz’s criterion (Riesz 1916), and that of (2) and (4) is Lagarias’s positivity criterion (Lagarias 1999), whose argument gives the equivalence of (2) and (4) for every \(\alpha\).

Earlier editions: Theorem 10.2.7, page 405 (edition 1).

Source: ch/riesz.tex, label ri:thm:ggcriesz.

Closed form constant of the squarefree reciprocal sum over number fields

Volume I, Proposition 13.3.1, page 303 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K\) be a number field of degree \(n\ge2\) with Dedekind zeta function \(\zeta_K\), let \(c_K=\lim_{s\to1}(s-1)\zeta_K(s)\), let \(\gamma_K\) be its Euler Kronecker constant, defined by \(\zeta_K(s)=c_K/(s-1)+c_K\gamma_K+O(s-1)\), and put \(C_K=\gamma_K-2\zeta_K'(2)/\zeta_K(2)\). Then, as \(x\to\infty\), \(\sum_{N(\mathfrak a)\le x}\frac{\mu(\mathfrak a)^2}{N(\mathfrak a)}=\frac{c_K}{\zeta_K(2)}\big(\log x+C_K\big)+O\big(x^{-1/n}\log x\big)\), the sum running over the integral ideals \(\mathfrak a\) of \(K\) (\(\mu\) the Möbius function of ideals, \(N\) the norm).

Earlier editions: Proposition 10.3.1, page 413 (edition 1).

Source: ch/sqfree.tex, label prop:sqfree:alpha1.

Squarefree Dirichlet series as a gamma factor times a ratio of Pólya moment generating functions

Volume I, Proposition 13.3.3, page 304 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(\Phi\) be Pólya’s kernel, the even positive function with \(\int_{\mathbb{R}}\Phi(u)e^{su}\,du=\xi(\frac12+s)\) for every complex \(s\), and let \(X\) be the symmetric random variable with density \(\Phi/\xi(\frac12)\), so that \(\mathbb{E}\,e^{sX}=\xi(\frac12+s)/\xi(\frac12)\) for all complex \(s\). Put \(\Gamma_{\mathrm{sq}}(s)=\frac{2s-1}{s-1}\,2^s\pi^{-(s+1)/2}\,\Gamma\big(\frac{s+1}2\big)\). For every \(s\) with \(\xi(2s)\ne0\), \(\sum_{n\ge1}\frac{\mu(n)^2}{n^s}=\Gamma_{\mathrm{sq}}(s)\,\frac{\mathbb{E}\,e^{(s-\frac12)X}}{\mathbb{E}\,e^{(2s-\frac12)X}}\), where \(\mu\) is the Möbius function and the left side is read as the continuation \(\zeta(s)/\zeta(2s)\) where the series diverges.

Earlier editions: Proposition 10.3.3, page 415 (edition 1).

Source: ch/sqfree.tex, label prop:sqfree:polya.

Squarefree constant as a combination of two tilted Pólya means

Volume I, Theorem 13.3.5, page 305 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, \(b_\alpha=\xi'(\alpha)/\xi(\alpha)\) (the mean of the Pólya variable \(V_\alpha\) tilted to the basepoint \(\alpha\), with \(\mathbb{E}\,e^{sV_\alpha}=\xi(\alpha+s)/\xi(\alpha)\)), \(a_k=-(\rho_k-\frac12)^2\) with one zero \(\rho_k\) taken from each pair \(\{\rho,1-\rho\}\) of nontrivial zeros of \(\zeta\), \(G(u)=\xi(\frac12+\sqrt u)/\xi(\frac12)\), \(W(t)=\sum_ke^{-a_kt}\), and let \(\gamma_E\) be Euler’s constant. The constant \(C=\gamma_E-2\zeta'(2)/\zeta(2)\) of \(\sum_{n\le x}\mu(n)^2/n=\frac6{\pi^2}(\log x+C)+O(x^{-1/2})\) equals \(2+\log2-\frac12\log\pi-\frac12\gamma_E+b_1-2b_2\). With no hypothesis on the zeros, \(b_1-2b_2=\frac{G'}G\big(\frac14\big)-6\,\frac{G'}G\big(\frac94\big)=\sum_k\Big(\frac1{\frac14+a_k}-\frac6{\frac94+a_k}\Big)=\int_0^\infty\big(e^{-t/4}-6e^{-9t/4}\big)\,W(t)\,dt\), the sum running over the pairs \(\{\rho,1-\rho\}\) and converging absolutely. Under the Riemann Hypothesis, \(a_k=\gamma_k^2\) and \(b_1-2b_2=\int\big((z+\frac14)^{-1}-6(z+\frac94)^{-1}\big)\,U_{\mathrm{RH}}(dz)\), where \(U_{\mathrm{RH}}=\sum_{\gamma>0}\delta_{\gamma^2}\) (over the ordinates \(\gamma\) of the zeros \(\frac12+i\gamma\), \(\gamma>0\), with multiplicity) is the Thorin measure of the clock \(1/G\).

Earlier editions: Theorem 10.3.5, page 416 (edition 1).

Source: ch/sqfree.tex, label thm:sqfree:const.

Dedekind zeta clock is a GGC if and only if RH holds for the field

Volume I, Proposition 13.3.6, page 306 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K\) be a number field with \(r_1\) real and \(r_2\) pairs of complex embeddings and discriminant \(d_K\), let \(\zeta_K\) be its Dedekind zeta function, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\), and \(\xi_K(s)=s(s-1)\,|d_K|^{s/2}\,\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\,\zeta_K(s)\), an entire function of order one with \(\xi_K(s)=\xi_K(1-s)\) whose zeros are the nontrivial zeros of \(\zeta_K\). Suppose \(\zeta_K(\frac12)\ne0\) and let \(G_K(u)=\xi_K(\frac12+\sqrt u)/\xi_K(\frac12)\). Then \(1/G_K\) is the Laplace transform of a generalised gamma convolution if and only if every zero of \(\zeta_K\) in the critical strip lies on the line \(\operatorname{Re}s=\frac12\) (the Riemann Hypothesis for \(\zeta_K\)). Its drift is then \(0\) and its Thorin measure is \(\sum_{\gamma>0}\delta_{\gamma^2}\), over the ordinates \(\gamma\) of the zeros counted with multiplicity. This is the number field form of the rigidity theorem of Polson, 2026, with the same proof. A real zero \(\beta\in(\frac12,1)\) of \(\zeta_K\) would be a pole of \(1/G_K\) at the point \((\beta-\frac12)^2\) of the positive axis.

Earlier editions: Proposition 10.3.6, page 417 (edition 1).

Source: ch/sqfree.tex, label prop:sqfree:thorinK.

Squarefree constant of a number field through the zeros of its Dedekind zeta function

Volume I, Theorem 13.3.7, page 306 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K\) be a number field with \(r_1\) real and \(r_2\) pairs of complex embeddings and discriminant \(d_K\), with Dedekind zeta function \(\zeta_K\), residue \(c_K=\lim_{s\to1}(s-1)\zeta_K(s)\) and Euler Kronecker constant \(\gamma_K\) (defined by \(\zeta_K(s)=c_K/(s-1)+c_K\gamma_K+O(s-1)\)), and put \(C_K=\gamma_K-2\zeta_K'(2)/\zeta_K(2)\). Let \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\) and \(\xi_K(s)=s(s-1)\,|d_K|^{s/2}\,\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\,\zeta_K(s)\), and let \(\gamma_E\) be Euler’s constant and \(A_K=2+\frac12\log|d_K|+r_1\big(\log2-\frac12\log\pi-\frac12\gamma_E\big)+r_2\big(2-\gamma_E-\log2\pi\big)\). Then \(C_K=A_K+\frac{\xi_K'}{\xi_K}(1)-2\,\frac{\xi_K'}{\xi_K}(2)=A_K+\sum_k\Big(\frac1{\frac14+a_{K,k}}-\frac6{\frac94+a_{K,k}}\Big)\), the sum running over the pairs \(\{\rho,1-\rho\}\) of zeros of \(\xi_K\), with multiplicity, where \(a_{K,k}=-(\rho_k-\frac12)^2\) and a zero of order \(2m\) at \(s=\frac12\) counts as \(m\) pairs with \(a_{K,k}=0\). Under the Riemann Hypothesis for \(\zeta_K\) and \(\zeta_K(\frac12)\ne0\), the zero part is \(\frac{G_K'}{G_K}\big(\frac14\big)-6\,\frac{G_K'}{G_K}\big(\frac94\big)\) with \(G_K(u)=\xi_K(\frac12+\sqrt u)/\xi_K(\frac12)\), the same combination of values of the Stieltjes transform of the Thorin measure \(\sum_{\gamma>0}\delta_{\gamma^2}\) of \(1/G_K\) as over \(\mathbb{Q}\).

Earlier editions: Theorem 10.3.7, page 417 (edition 1).

Source: ch/sqfree.tex, label thm:sqfree:constK.

Exponentially and Gamma(2) smoothed squarefree counts are equivalent to RH

Volume I, Theorem 13.3.9, page 310 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\) be the Möbius function, and for \(x>0\) let \(Q_1(x)=\sum_{n\ge1}\mu(n)^2e^{-n/x}\) and \(Q_2(x)=\sum_{n\ge1}\mu(n)^2\big(1+\frac{2n}x\big)e^{-2n/x}\), the number of squarefree integers up to a random point of mean \(x\) with the exponential law and with the Gamma(2) law respectively. (i) The Riemann Hypothesis holds if and only if \(Q_1(x)=\frac6{\pi^2}x+O(x^{1/4+\varepsilon})\) as \(x\to\infty\) for every \(\varepsilon>0\), and if and only if \(Q_2(x)=\frac6{\pi^2}x+O(x^{1/4+\varepsilon})\) for every \(\varepsilon>0\). (ii) Under the Riemann Hypothesis, if the zeros are simple, \(Q_1(x)=\frac6{\pi^2}x+1+\sum_\rho A_\rho x^{\rho/2}+\frac{c_1\log x+c_2}x+O(x^{-2})\) and \(Q_2(x)=\frac6{\pi^2}x+1+\sum_\rho B_\rho x^{\rho/2}+\frac{2c_1}x+O(x^{-2})\), where \(A_\rho=\frac{\Gamma(\rho/2)\,\zeta(\rho/2)}{2\zeta'(\rho)}\), \(B_\rho=\big(1+\frac\rho2\big)\frac{2^{-\rho/2}\,\Gamma(\rho/2)\,\zeta(\rho/2)}{2\zeta'(\rho)}\), the sums over the nontrivial zeros \(\rho\) are taken over \(|\operatorname{Im}\rho|<2T_j\) along a sequence \(T_j\to\infty\), \(c_1=-\frac{\zeta(-1)}{2\zeta'(-2)}=-1.36843277762021\) and \(c_2=c_1\big(1-\gamma_E+\frac{\zeta'(-1)}{\zeta(-1)}-\frac{\zeta''(-2)}{\zeta'(-2)}\big)=-0.339381226129599\), with \(\gamma_E\) Euler’s constant.

Earlier editions: Theorem 10.3.9, page 420 (edition 1).

Source: ch/sqfree.tex, label thm:sqfree:smooth.

Lower bound for the squarefree error of the order of the fourth root of x

Volume I, Corollary 13.3.11, page 312 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Q(x)=\sum_{n\le x}\mu(n)^2\) be the number of squarefree integers up to \(x\) and \(\Delta(x)=Q(x)-\frac6{\pi^2}x\). Let \(\rho_1\) be the zero of \(\zeta\) with least positive imaginary part, and \(R_{\rho_1}=\frac{\zeta(\rho_1/2)}{\rho_1\zeta'(\rho_1)}\), so that \(|R_{\rho_1}|=0.100404\). With no hypothesis, \(\limsup_{x\to\infty}\Delta(x)/x^{1/4}\ge|R_{\rho_1}|\) and \(\liminf_{x\to\infty}\Delta(x)/x^{1/4}\le-|R_{\rho_1}|\). The same holds for \(Q_2(x)-\frac6{\pi^2}x-1\), where \(Q_2(x)=\sum_{n\ge1}\mu(n)^2\big(1+\frac{2n}x\big)e^{-2n/x}\) is the squarefree count smoothed by a Gamma(2) law of mean \(x\), with \(|B_{\rho_1}|=9.944\times10^{-5}\) in place of \(|R_{\rho_1}|\), where \(B_\rho=\big(1+\frac\rho2\big)\frac{2^{-\rho/2}\,\Gamma(\rho/2)\,\zeta(\rho/2)}{2\zeta'(\rho)}\).

Earlier editions: Corollary 10.3.11, page 423 (edition 1).

Source: ch/sqfree.tex, label cor:sqfree:omega.

Squarefree count at a Gamma(n) random point under RH

Volume I, Proposition 13.3.12, page 313 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis.

Statement. Let \(Q(x)=\sum_{m\le x}\mu(m)^2\) be the number of squarefree integers up to \(x\), and for \(n\ge1\) let \(G_{n,x}\) have the Gamma(\(n\)) law with mean \(x\) (shape \(n\), rate \(n/x\)). Assume the Riemann Hypothesis. Then, for every \(\varepsilon>0\) and uniformly in \(n\ge1\), \(\mathbb{E}\,Q(G_{n,x})=\frac6{\pi^2}\,x+O_\varepsilon\big(x^{1/4+\varepsilon}\,n^{1/8+\varepsilon}\big)\) as \(x\to\infty\).

Earlier editions: Proposition 10.3.12, page 424 (edition 1).

Source: ch/sqfree.tex, label prop:sqfree:gamman.

Gamma(2) smoothed squarefree error as a positive average of the Mertens function

Volume I, Theorem 13.3.15, page 315 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\) be the Möbius function, \(M(t)=\sum_{d\le t}\mu(d)\) the Mertens function, and \(Q_2(x)=\sum_{n\ge1}\mu(n)^2\big(1+\frac{2n}x\big)e^{-2n/x}\), the squarefree count smoothed by a Gamma(2) law of mean \(x\). For \(y>0\) let \(g_2(y)=\frac2y-\frac1{e^y-1}-\frac y{4\sinh^2(y/2)}\) and \(f_2(y)=-2g_2'(y)=\frac4{y^2}-\frac{y\cosh(y/2)}{2\sinh^3(y/2)}\). Then \(f_2>0\) (this is Lazarević’s inequality \((\sinh z/z)^3>\cosh z\) with \(z=y/2\)), \(f_2(y)=y^2/60+O(y^4)\) as \(y\to0\) and \(f_2(y)\sim4/y^2\) as \(y\to\infty\), and \(f_2\) is the density of a probability law \(\mathcal{V}_2\) on \((0,\infty)\) with \(P(\mathcal{V}_2\ge y)=2g_2(y)\) and \(\mathbb{E}\,\mathcal{V}_2^s=-2\,\Gamma(s+2)\,\zeta(s)\) for \(-3<\operatorname{Re}s<1\). For every \(x>0\), \(Q_2(x)=\frac6{\pi^2}\,x-\frac12\,\mathbb{E}\,M\Big(\sqrt{x\mathcal{V}_2/2}\Big)\). Moreover \(f_2\) is twice the difference between the continuous and the integer mixtures of the Gamma(2) densities \(\gamma_{2,\lambda}(y)=\lambda^2ye^{-\lambda y}\) over their rate: \(f_2(y)=2\Big(\int_0^\infty\gamma_{2,\lambda}(y)\,d\lambda-\sum_{m\ge1}\gamma_{2,m}(y)\Big)\).

Earlier editions: Theorem 10.3.15, page 426 (edition 1).

Source: ch/sqfree.tex, label thm:sqfree:mertens.

Gamma smoothing threshold, shape two

Volume I, Theorem 13.3.18, page 317 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\) be the Möbius function, \(Q(x)=\sum_{n\le x}\mu(n)^2\) the number of squarefree integers up to \(x\) and \(M(t)=\sum_{d\le t}\mu(d)\) the Mertens function. For a shape \(\nu>0\) let \(R_\nu(y)=\frac{y^{\nu+1}}{\Gamma(\nu+1)}\sum_{m\ge1}m^\nu e^{-my}\), the right Riemann sum with step \(y\) of the Gamma(\(\nu+1\)) density, and \(f_\nu(y)=\frac{2\nu\,(1-R_\nu(y))}{y^2}\) for \(y>0\). The function \(f_\nu\) is a signed density of total mass \(1\), and for \(Y\) with the Gamma(\(\nu\)) law of rate \(\lambda\), \(\mathbb{E}\,Q(Y)=\frac6{\pi^2}\,\mathbb{E}\,Y-\frac12\int_0^\infty M\big(\sqrt{y/\lambda}\big)f_\nu(y)\,dy\). Then \(\{\nu>0:\ f_\nu(y)\ge0\text{ for all }y>0\}=(0,2]\). Thus two is the largest shape for which the gamma smoothed squarefree error is a positive average of the Mertens function, and the obstruction is the trivial zero of \(\zeta\) at \(-2\).

Earlier editions: Theorem 10.3.18, page 428 (edition 1).

Source: ch/sqfree.tex, label thm:sqfree:threshold.

Pointwise bound for the squarefree error under RH (open)

Volume I, Question 13.3.19, page 319 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(Q(x)=\sum_{n\le x}\mu(n)^2\) be the number of squarefree integers up to \(x\) and \(\Delta(x)=Q(x)-\frac6{\pi^2}x\). Open question: prove, under the Riemann Hypothesis, that \(\Delta(x)=O(x^{1/4+\varepsilon})\) for every \(\varepsilon>0\) (by the book, the order \(x^{1/4}\) cannot be lowered, and under the Riemann Hypothesis the best bound proved there is \(O(x^{2/5+\varepsilon})\), weaker than known conditional exponents such as \(7/22\) (Baker and Pintz 1985)). The book gives these forms: remove the loss \(n^{1/8}\) from the bound \(\mathbb{E}\,Q(G_{n,x})=\frac6{\pi^2}x+O(x^{1/4+\varepsilon}n^{1/8+\varepsilon})\) valid under the Riemann Hypothesis for the count at a Gamma(\(n\)) random point \(G_{n,x}\) of mean \(x\) (the conjecture implies the bound with no loss, and a loss \(n^\delta\) in place of \(n^{1/8}\) would give \(\Delta(x)=O(x^{1-3/(8\delta+4)+\varepsilon})\), whose exponent tends to \(\frac14\) as \(\delta\to0\)); or show that for large \(X\) no \(x\in[X,2X]\) has \(|\Delta(x)|>CX^{1/4+\varepsilon}\), for some constant \(C\) (under the Riemann Hypothesis the book shows that for \(C=1\) the set of such \(x\) has measure \(O(X^{1-\varepsilon})\)), which would follow from a bound \(O(X^{1/4+\varepsilon})\) for the oscillation of \(\Delta\) on the subintervals of \([X,2X]\) of length \(X^{1-\varepsilon/2}\).

Earlier editions: Question 10.3.19, page 430 (edition 1).

Source: ch/sqfree.tex, label q:sqfree:pointwise.

Sharpening the lower bound constant for the squarefree error (open)

Volume I, Question 13.3.20, page 319 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(Q(x)=\sum_{n\le x}\mu(n)^2\) be the number of squarefree integers up to \(x\), \(\Delta(x)=Q(x)-\frac6{\pi^2}x\), \(\rho_1\) the zero of \(\zeta\) with least positive imaginary part and \(R_{\rho_1}=\frac{\zeta(\rho_1/2)}{\rho_1\zeta'(\rho_1)}\), so that \(|R_{\rho_1}|=0.1004\) (to four digits) and, by the book, \(\limsup\Delta(x)/x^{1/4}\ge|R_{\rho_1}|\) and \(\liminf\Delta(x)/x^{1/4}\le-|R_{\rho_1}|\) with no hypothesis. Open questions: sharpen the constant \(|R_{\rho_1}|\) by using many zeros at once with a Fejér kernel, as Ingham did for the Mertens function, and compare it with the observed \(\max|\Delta(x)|/x^{1/4}\), about \(1\) for \(x\) up to \(2\times10^8\); and decide whether \(\limsup|\Delta(x)|/x^{1/4}=\infty\), as the linear independence of the ordinates of the zeros would suggest.

Earlier editions: Question 10.3.20, page 430 (edition 1).

Source: ch/sqfree.tex, label q:sqfree:fejer.

Volume I, Chapter 14. Below the basepoint one: a map of processes, free probability, almost GGCs and the Lambert law

Background driving subordinator of the centre clock

Volume I, Proposition 14.1.1, page 322 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function and \(G(u)=\xi(\frac{1}{2}+\sqrt{u})/\xi(\frac{1}{2})\). The centre clock \(T\) is the positive random variable with Laplace transform \(\mathbb{E}e^{-uT}=1/G(u)\), \(u\ge0\). Let \(W(t)=\sum_k e^{-a_kt}\), \(t>0\), be the heat trace of the zeros, where \(a_k=-(\rho_k-\frac{1}{2})^2\) and \(\rho_k\) runs over the zeros of \(\xi\) with positive imaginary part, counted with multiplicity; under the Riemann Hypothesis (RH), \(W(t)=\sum_{\gamma>0}e^{-\gamma^2t}\) with \(\gamma\) the ordinates of the zeros. With no hypothesis (the proof rests on the inequalities \(W>0\) and \(W'<0\), whose proof uses only the verification of RH below height \(1000\)), \(T\overset{d}{=}\int_0^\infty e^{-t}\,dZ_t\) for a subordinator \(Z\) with Lévy density \(h(x)=-W'(x)\), which is \(\sum_{\gamma>0}\gamma^2e^{-\gamma^2x}\) under RH and is given for every \(x>0\) by the derivative of Weil’s explicit formula for \(W\) with the test function \(e^{-tr^2}\). The condition \(\mathbb{E}\log(1+Z_1)<\infty\) holds because \(h\) decays exponentially. Moreover RH holds if and only if \(h\) is completely monotone.

Earlier editions: Proposition 11.1.1, page 432 (edition 1).

Source: ch/sato.tex, label sb:prop:bdlp.

Unimodality of the centre clock and the shifted clocks

Volume I, Corollary 14.1.2, page 323 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function. The centre clock \(T\) is the positive random variable with Laplace transform \(\mathbb{E}e^{-uT}=\xi(\frac{1}{2})/\xi(\frac{1}{2}+\sqrt{u})\), and for \(\alpha>\frac{1}{2}\) the shifted clock \(T_\alpha\) is the positive random variable with Laplace transform \(\mathbb{E}e^{-uT_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt{u})\). With no hypothesis, the law of \(T\) is unimodal, and so is the law of \(T_\alpha\) for every \(\alpha>\frac{1}{2}\). Here a law is unimodal if its distribution function is convex to the left of some point, a mode, and concave to the right of it.

Earlier editions: Corollary 11.1.2, page 433 (edition 1).

Source: ch/sato.tex, label sb:cor:unimodal.

The tilted Pólya variable at the basepoint one is not quasi infinitely divisible

Volume I, Proposition 14.1.3, page 325 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function and let \(V_1\) be the tilted Pólya variable at the basepoint one, the random variable with moment generating function \(\mathbb{E}e^{sV_1}=\xi(1+s)/\xi(1)\) (the Esscher tilt by \(e^{x/2}\) of the Pólya variable \(X\), which has \(\mathbb{E}e^{sX}=\xi(\frac{1}{2}+s)/\xi(\frac{1}{2})\)). A law is quasi infinitely divisible if its characteristic function is the quotient of two infinitely divisible characteristic functions, equivalently if it has a Lévy Khintchine form with a signed Lévy measure \(\nu^+-\nu^-\) in which \(\nu^+\) and \(\nu^-\) are both Lévy measures. Then \(V_1\) is not quasi infinitely divisible, and so, an infinitely divisible law being quasi infinitely divisible, it is not infinitely divisible. The reflection \(x\mapsto-x\) of the measure \(\Pi_1(dx)=\big(e^{-3x}(1-e^{-2x})^{-1}-1\big)\,dx/x+\sum_{n\ge2}\Lambda(n)(n\log n)^{-1}\delta_{\log n}(dx)\), with \(\Lambda\) the von Mangoldt function, is the candidate signed Lévy measure of \(V_1\); its pole term \(-dx/x\) and its prime atoms at \(\log n\) each have infinite mass on \((1,\infty)\), so its positive part and its negative part both have infinite mass there, and its Lévy form converges only as a compensated limit, which exists by the prime number theorem.

Earlier editions: Proposition 11.1.3, page 436 (edition 1).

Source: ch/sato.tex, label sb:prop:qid.

Thorin Bernstein form of the shifted clock criterion

Volume I, Proposition 14.1.4, page 328 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, \(G(u)=\xi(\frac{1}{2}+\sqrt{u})/\xi(\frac{1}{2})\) and, for \(\alpha>\frac{1}{2}\), \(f_\alpha(u)=u+2(\alpha-\frac{1}{2})\sqrt{u}\). A Thorin Bernstein function is the Laplace exponent of a generalised gamma convolution (GGC), equivalently a Bernstein function whose derivative is a Stieltjes function. For every \(\alpha\ge1\) the composition \(\log G\big((\alpha-\frac{1}{2})^2+f_\alpha(u)\big)-\log G\big((\alpha-\frac{1}{2})^2\big)\), which is the Laplace exponent of the shifted clock \(T_\alpha\) (the positive random variable with \(\mathbb{E}e^{-uT_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt{u})\)), is a Thorin Bernstein function of \(u\) with no hypothesis, while \(\log G\) itself, the Laplace exponent of the centre clock, is a Thorin Bernstein function if and only if the Riemann Hypothesis holds. For \(\frac{1}{2}<\alpha<1\) the composition is a Thorin Bernstein function if and only if \(\zeta\) has no zeros with real part greater than \(\alpha\).

Earlier editions: Proposition 11.1.4, page 439 (edition 1).

Source: ch/sato.tex, label sb:prop:tbf.

Upsilon image and free counterpart of the centre clock

Volume I, Theorem 14.2.3, page 338 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, \(G(u)=\xi(\frac{1}{2}+\sqrt{u})/\xi(\frac{1}{2})=\prod_k(1+u/a_k)\) with \(a_k=-(\rho_k-\frac{1}{2})^2\), where \(\rho_k\) runs over the zeros of \(\xi\) with positive imaginary part, counted with multiplicity, and let \(T\) be the centre clock, the positive random variable with \(\mathbb{E}e^{-uT}=1/G(u)\). Put \(W(t)=\sum_ke^{-a_kt}\), \(h=-W'\), \(m_n=\frac{1}{n!}\int_0^\infty t^nW(t)\,dt=\sum_ka_k^{-n-1}\), let \(\gamma_1=14.1347\ldots\) be the least ordinate of a zero and \(N(\theta)\) the number of zeros with ordinate in \((0,\theta]\). With no hypothesis \(T\) is infinitely divisible with no drift and Lévy measure \(W(t)\,dt/t\), and self decomposable (because \(W>0\) and \(W'<0\) on \((0,\infty)\)). For a law \(\mu\) on \([0,\infty)\) write \(\Psi_\mu(s)=-\log\int e^{-sx}\mu(dx)\) for its Laplace exponent. A generalised gamma convolution (GGC) is a law on \([0,\infty)\) with \(\Psi(s)=ds+\int\log(1+s/t)\,U(dt)\) for a drift \(d\ge0\) and a Thorin measure \(U\), and BO is the class of infinitely divisible laws on \([0,\infty)\) with a completely monotone Lévy density. The Upsilon transform \(\Upsilon\) maps an infinitely divisible law \(\mu\), the law of \(X_1\) for a Lévy process \(X\), to the law of \(\int_0^1\log(1/t)\,dX_t\). The Bercovici Pata bijection \(\Lambda_{\mathrm{BP}}\) maps an infinitely divisible law to the freely infinitely divisible law with the same Lévy Khintchine triplet; \(G_\mu(z)=\int\mu(dx)/(z-x)\) is the Cauchy transform of \(\mu\) and \(\mathcal{C}_\mu(z)=z\varphi_\mu(1/z)\) its free cumulant transform, with \(\varphi_\mu\) the Voiculescu transform. A freely infinitely divisible law \(\mu\) is free regular if its free convolution powers \(\mu^{\boxplus t}\) are carried by \([0,\infty)\) for every \(t>0\) (these are the images under \(\Lambda_{\mathrm{BP}}\) of the infinitely divisible laws on \([0,\infty)\)). (a) With no hypothesis, \(\Upsilon(T)\) is a GGC with no drift and Thorin measure \(\widetilde{U}(du)=u^{-2}h(1/u)\,du\), so that \(\widetilde{U}((0,u])=W(1/u)>0\) for every \(u>0\). Its Laplace exponent is \(\Psi_{\Upsilon(T)}(s)=\int_0^\infty\log G(sx)\,e^{-x}\,dx=\sum_ke^{a_k/s}E_1(a_k/s)\), \(s>0\), with the principal branch of the exponential integral \(E_1\); its Lévy density is the completely monotone function \(\ell(z)=z^{-1}\int_0^\infty W(z/w)\,e^{-w}\,dw\); its cumulants are \(\kappa_n(\Upsilon(T))=n!\,(n-1)!\,m_{n-1}\); and \(\int u^{-n}\,\widetilde{U}(du)=n!\,m_{n-1}\) for \(n\ge1\). Moreover \(\mathbb{E}e^{\lambda\Upsilon(T)}=\infty\) for every \(\lambda>0\), while \(\mathbb{E}e^{\lambda T}<\infty\) exactly when \(\lambda<\gamma_1^2\). (b) With no hypothesis, \(\Lambda_{\mathrm{BP}}(T)\) is free regular, freely self decomposable (for every \(c\in(0,1)\) it is the free convolution of its dilation by \(c\) with some law) and unimodal. Its free cumulant transform is \(\mathcal{C}_{\Lambda_{\mathrm{BP}}(T)}(z)=\int_0^\infty zW(x)/(1-zx)\,dx\), \(z\in\mathbb{C}^-\) (the lower half plane), where the right side continues it analytically to \(\mathbb{C}\setminus(0,\infty)\), with \(\mathcal{C}_{\Lambda_{\mathrm{BP}}(T)}(-s)=-\Psi_{\Upsilon(T)}(s)\) for \(s>0\); its free cumulants are \((n-1)!\,m_{n-1}\), the cumulants of \(T\). Its Cauchy transform \(G_{\Lambda_{\mathrm{BP}}(T)}\) satisfies \(\Psi_{\Upsilon(T)}\big(-G_{\Lambda_{\mathrm{BP}}(T)}(z)\big)=1-z\,G_{\Lambda_{\mathrm{BP}}(T)}(z)\) for \(z\in\mathbb{C}^+\) (the upper half plane), where \(\Psi_{\Upsilon(T)}\) is continued analytically to \(\mathbb{C}\setminus(-\infty,0)\) by \(\Psi_{\Upsilon(T)}(s)=\int_0^\infty sW(y)/(1+sy)\,dy\), and its support lies in \([c,\infty)\), where \(c=\max_{s>0}(\Psi_{\Upsilon(T)}(s)-1)/s>0\). (c) RH holds if and only if \(T\) is the Upsilon image of an infinitely divisible law on \([0,\infty)\). In that case \(T=\Upsilon(\mu_0)\), where \(\mu_0\) is the self decomposable law with no drift and Lévy measure \(N(x^{-1/2})\,dx/x\) (canonical function \(N(v^{-1/2})\)); \(\mu_0\) is neither a GGC nor in BO. With no hypothesis, \(T\) is not the Upsilon image of a law in BO.

Earlier editions: Theorem 11.2.3, page 450 (edition 1).

Source: ch/wigfree.tex, label thm:wigfree:clock.

RH holds if and only if a freely infinitely divisible law nu_xi with free cumulant transform log xi(1/2)/xi(1/2+sqrt(-z)) exists

Volume I, Proposition 14.2.4, page 340 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mathcal{C}_\xi(z)=\log\{\xi(\frac12)/\xi(\frac12+\sqrt{-z})\}=\sum_{n\ge1}m_{n-1}z^n/n\) for \(|z|<\gamma_1^2\). RH holds if and only if there is a freely infinitely divisible probability measure \(\nu_\xi\) whose free cumulant transform is \(\mathcal{C}_\xi\). Under RH, \(\nu_\xi=\Lambda_{\rm BP}(\mu_0)\), where \(\mu_0\) is the self decomposable law with Lévy measure \(N(x^{-1/2})dx/x\); its free cumulants are \(m_{n-1}/n\), its free Lévy measure is \(N(y^{-1/2})dy/y\), its support is compact and it is freely self decomposable. For the converse, compact support and the free cumulants \(m_{n-1}/n\) put the third and higher cumulants as moments of a free Lévy measure \(\rho\) with \(t^2\rho\) on \([-\gamma_1^{-2},\gamma_1^{-2}]\), so the series \(\sum_{n\ge3}m_{n-1}z^n/n\) is analytic off a subset of the real axis, while it has a logarithmic singularity at every \(a_k\); so every \(a_k\) is real and RH holds.

Earlier editions: Proposition 11.2.4, page 452 (edition 1).

Source: ch/wigfree.tex, label tx2:prop:free.

Dickman reading of the preimage mu_0 of the clock under the Upsilon transform: a sum of independent Dickman variables over the zeros

Volume I, Result 14.2.3, page 340 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(D\) be the Dickman law, \(\mathbb{E}e^{-sD}=\exp(-\operatorname{Ein}(s))\), \(\operatorname{Ein}(s)=\int_0^s(1-e^{-v})dv/v\). Since \(\int_0^\infty\operatorname{Ein}(ax)e^{-x}dx=\log(1+a)\), \(\Upsilon(D/\gamma^2)\) is the exponential law of rate \(\gamma^2\), and under RH \(\mu_0\), the preimage of the clock \(T\) under \(\Upsilon\), is the law of \(\sum_{\gamma>0}D_\gamma/\gamma^2\) with \(D_\gamma\) independent Dickman variables, with Lévy measure \(\sum_\gamma\mathbf 1_{(0,\gamma^{-2})}(y)dy/y=N(y^{-1/2})dy/y\) and all jumps at most \(\gamma_1^{-2}\). With no hypothesis \(\psi_0(s)=\sum_k\operatorname{Ein}(s/a_k)\) is entire and \(\int_0^\infty\psi_0(sx)e^{-x}dx=\log G(s)\) for \(s>0\), termwise integration being justified by \(|\operatorname{Ein}(z)|\le|z|\) for \(\operatorname{Re}z\ge0\). Under RH the density of \(\mu_0\) has mass \(0.99993\) on \([0.008,0.045]\) and mean \(0.023104\), against \(p_1=0.0231050\) (numerical).

Earlier editions: Result 11.2.3, page 452 (edition 1).

Source: ch/wigfree.tex, label tx3:res:dickman.

Peeling a gamma layer from the Pólya generating function keeps zeros real under RH

Volume I, Proposition 14.2.9, page 348 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function and \(G(u)=\xi(\frac{1}{2}+\sqrt{u})/\xi(\frac{1}{2})=\sum_{k\ge0}b_ku^k\), so that the Pólya variable \(X\), with \(\mathbb{E}e^{sX}=G(s^2)\), has \(b_k=\mathbb{E}X^{2k}/(2k)!\). For \(a>0\) let \((a)_k=a(a+1)\cdots(a+k-1)\) and \(G_a(u)=\sum_{k\ge0}b_ku^k/(a)_k\). If the Riemann Hypothesis holds, then for every \(a>0\) the function \(G_a\) is in the Laguerre Pólya class (the real entire functions that are locally uniform limits of real polynomials with only real zeros) with only real negative zeros, and all its Jensen polynomials \(\sum_{i=0}^{d}\binom{d}{i}(n+i)!\,b_{n+i}x^i/(a)_{n+i}\), \(d,n\ge0\), have only real negative zeros. Multiplying the coefficients by \((a)_k\) instead does not preserve real zeros: it turns \((1+u)^2=1+2u+u^2\) into \(1+2au+a(a+1)u^2\), whose discriminant is \(-4a<0\).

Earlier editions: Proposition 11.2.9, page 460 (edition 1).

Source: ch/wigfree.tex, label prop:wigfree:peel.

No gamma scale mixture for the Pólya variable and Williams’ variable

Volume I, Theorem 14.2.10, page 349 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function. The Pólya variable \(X\) is the real random variable with \(\mathbb{E}e^{sX}=\xi(\frac{1}{2}+s)/\xi(\frac{1}{2})\), \(s\in\mathbb{C}\), and Williams’ variable \(S_2\) is the positive random variable with \(\mathbb{E}e^{-\lambda S_2}=\big(\sqrt{2\lambda}/\sinh\sqrt{2\lambda}\big)^2\), \(\lambda\ge0\) (the sum of two independent hitting times of the level one by a three dimensional Bessel process), for which \(\mathbb{E}(\frac{\pi}{2}S_2)^{s/2}=2\xi(s)\). A law is a scale mixture of Gamma(\(a\)) laws if it is the law of \(\Theta\,\Gamma\) with \(\Gamma\) gamma distributed with shape \(a\) and unit scale and independent of a random variable \(\Theta\ge0\). Then for no \(a>0\) is \(|X|\), \(X^2\) or \(S_2\) a scale mixture of Gamma(\(a\)) laws.

Earlier editions: Theorem 11.2.10, page 461 (edition 1).

Source: ch/wigfree.tex, label thm:wigfree:nomix.

Almost GGC theorem for the shifted clocks

Volume I, Theorem 14.3.2, page 355 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, let \(\rho=\beta+i\gamma\) run over its zeros (the nontrivial zeros of \(\zeta\)), counted with multiplicity, let \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)\) for \(\alpha>\frac{1}{2}\) and let \(P_w(x)=w/(\pi(w^2+x^2))\) be the Cauchy kernel of width \(w>0\), with \(P_0\) the unit point mass at \(0\). Write \(v_\alpha=v_\alpha^+-v_\alpha^-\) with \(v_\alpha^+(\theta)/\pi=\sum_{\beta\le\alpha,\,\gamma>0}[P_{\alpha-\beta}(\theta-\gamma)+P_{\alpha-\beta}(\theta+\gamma)]\) and \(v_\alpha^-(\theta)/\pi=\sum_{\beta>\alpha,\,\gamma>0}[P_{\beta-\alpha}(\theta-\gamma)+P_{\beta-\alpha}(\theta+\gamma)]\) (a zero on the line \(\operatorname{Re}s=\alpha\) contributes a unit atom at its ordinate to the positive part), and put \(m_\alpha^\pm(T)=\pi^{-1}\int_0^Tv_\alpha^\pm(\theta)\,d\theta\). Recall that the shifted clock \(T_\alpha\), with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt{u})\), is a generalised gamma convolution (GGC) exactly when no zero has \(\beta>\alpha\), that is when \(v_\alpha^-\equiv0\). For every \(\frac{1}{2}<\alpha<1\), with no hypothesis, \(m_\alpha^-(T)/m_\alpha^+(T)\le T^{e(\alpha)-1+o(1)}\) as \(T\to\infty\), where \(e(\alpha)=3(1-\alpha)/(2-\alpha)\) for \(\frac{1}{2}<\alpha<\frac{7}{10}\) and \(e(\alpha)=15(1-\alpha)/(3+5\alpha)\) for \(\frac{7}{10}\le\alpha<1\), and \(e(\alpha)<1\) throughout; so the negative part \(v_\alpha^-/\pi\), the Cauchy kernels of the zeros to the right of \(\alpha\), carries on \([0,T]\) a vanishing fraction of the mass of the positive part, with a power saving. For \(\alpha\ge1\) no zero lies to the right of \(\alpha\) and \(m_\alpha^-=0\). Under the density hypothesis (\(N(\sigma,T)\le T^{2(1-\sigma)+o(1)}\), where \(N(\sigma,T)\) counts the zeros with \(\beta\ge\sigma\) and \(0<\gamma\le T\)) the same bound holds with \(e(\alpha)=2(1-\alpha)\). RH holds if and only if \(v_\alpha^-\) vanishes identically for every \(\alpha>\frac{1}{2}\). The proof combines the zero density estimates of Ingham, of Guth and Maynard and of Huxley.

Earlier editions: Theorem 11.4.2, page 472 (edition 1).

Source: ch/almost.tex, label ag:thm:almost.

Mass bookkeeping of the positive and negative parts of the Thorin density below basepoint one

Volume I, Lemma 14.3.1, page 355 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) with zeros \(\rho=\beta+i\gamma\) and \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)=v_\alpha^+-v_\alpha^-\) with \(\frac{v_\alpha^+(\theta)}\pi=\sum_{\beta\le\alpha,\gamma>0}[P_{\alpha-\beta}(\theta-\gamma)+P_{\alpha-\beta}(\theta+\gamma)]\) and \(\frac{v_\alpha^-(\theta)}\pi=\sum_{\beta>\alpha,\gamma>0}[P_{\beta-\alpha}(\theta-\gamma)+P_{\beta-\alpha}(\theta+\gamma)]\), where \(P_w(x)=w/(\pi(w^2+x^2))\) is the Cauchy kernel. Put \(m_\alpha^\pm(T)=\pi^{-1}\int_0^Tv_\alpha^\pm(\theta)\,d\theta\) and let \(N(\alpha,t)\) be the number of zeros with \(0<\gamma\le t\) and \(\beta\ge\alpha\). Let \(\alpha>\frac12\) and suppose \(N(\alpha,t)\le t^{e+o(1)}\) as \(t\to\infty\) for some \(e<1\). Then \(m_\alpha^-(T)\le N(\alpha,2T)+O(T^{e-1+o(1)})\) and \(m_\alpha^+(T)=\frac T{2\pi}\log\frac T{2\pi}+O(T)\).

Source: ch/almost.tex, label ag:lem:mass.

Canonical factorisation of the shifted clock transform into a GGC and a defect

Volume I, Proposition 14.3.4, page 357 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, let \(\rho=\beta+i\gamma\) run over its zeros (the nontrivial zeros of \(\zeta\)), counted with multiplicity, let \(P_w(x)=w/(\pi(w^2+x^2))\) be the Cauchy kernel of width \(w>0\), with \(P_0\) the unit point mass at \(0\), and for \(\alpha>\frac{1}{2}\) let \(\varphi_\alpha(u)=\xi(\alpha)/\xi(\alpha+\sqrt{u})\) be the Laplace transform of the shifted clock \(T_\alpha\) and \(v_\alpha^+(\theta)/\pi=\sum_{\beta\le\alpha,\,\gamma>0}[P_{\alpha-\beta}(\theta-\gamma)+P_{\alpha-\beta}(\theta+\gamma)]\) (a zero on the line \(\operatorname{Re}s=\alpha\) contributes a unit atom at its ordinate). For every \(\alpha>\frac{1}{2}\) and \(u\ge0\), \(\varphi_\alpha(u)=\varphi_\alpha^+(u)\,D_\alpha(u)\), where \(\varphi_\alpha^+(u)=\exp\big\{-\int_0^\infty\log\big(1+u/\theta^2\big)\,\frac{v_\alpha^+(\theta)}{\pi}\,d\theta\big\}\) and \(D_\alpha(u)=\prod_{\beta>\alpha,\,\gamma>0}\frac{(\beta-\alpha)^2+\gamma^2}{(\sqrt{u}-\beta+\alpha)^2+\gamma^2}\). Here \(\varphi_\alpha^+\) is the Laplace transform of a generalised gamma convolution (GGC) \(T_\alpha^+\) with Thorin density \(v_\alpha^+/\pi\) in the chart \(\theta=\sqrt{z}\), and the product defining the defect \(D_\alpha\) runs over the zeros to the right of the line \(\operatorname{Re}s=\alpha\) and converges absolutely. The defect is identically one if and only if \(\zeta\) has no zeros with \(\operatorname{Re}s>\alpha\), and then \(T_\alpha\overset{d}{=}T_\alpha^+\). Otherwise \(D_\alpha(u)\to0\) as \(u\to\infty\), and \(T_\alpha^+\) is not the sum of \(T_\alpha\) and an independent nonnegative variable.

Earlier editions: Proposition 11.4.3, page 473 (edition 1).

Source: ch/almost.tex, label ag:prop:factor.

Signed Thorin representation of the reciprocal Xi function at every basepoint alpha at least one half, with damped sinusoid factors for zeros to the right

Volume I, Theorem 14.3.5, page 358 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+\frac s2)\zeta(s)\) be the completed Riemann zeta function, with the nontrivial zeros \(\rho=\beta+i\gamma\) counted with multiplicity. For \(\alpha\ge\frac12\) and \(u\ge0\) put \(\varphi_\alpha(u)=\xi(\alpha)/\xi(\alpha+\sqrt u)\) and \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)\). Let \(U_\alpha^\circ\) be the signed measure on \(\theta>0\) with density \(v_\alpha(\theta)/\pi\) plus a unit point mass at \(\gamma\) for every zero \(\beta+i\gamma\) with \(\gamma>0\) and \(\beta=\alpha\) (counted with multiplicity). Then for every \(\alpha\ge\frac12\) and \(u\ge0\), with no hypothesis on the zeros, \(\varphi_\alpha(u)=\exp\Big\{-\int_0^\infty\log\Big(1+\frac{u}{\theta^2}\Big)U_\alpha^\circ(d\theta)\Big\}\prod_{\beta>\alpha,\ \gamma>0}\Big|1-\frac{u}{(\rho-\alpha)^2}\Big|^{-2}\), the integral and the product converging absolutely. For a zero \(\rho\) to the right of \(\alpha\), write \(p=(\rho-\alpha)^2=-c+id\) with \(c=\gamma^2-(\beta-\alpha)^2>0\) and \(d=2\gamma(\beta-\alpha)>0\); then \(\Big|1-\frac up\Big|^{-2}=\frac{|p|^2}{(u+c)^2+d^2}=|p|^2\int_0^\infty e^{-ut}\frac{e^{-ct}\sin dt}{d}\,dt\), the Laplace transform of a damped sinusoid, which changes sign, so this factor is not completely monotone.

Earlier editions: Theorem 11.4.4, page 475 (edition 1).

Source: ch/almost.tex, label tx:thm:signed.

The critical line basepoint: the signed Thorin measure at alpha one half is the sum of point masses at the ordinates of the zeros on the line

Volume I, Corollary 14.3.6, page 358 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+\frac s2)\zeta(s)\) be the completed Riemann zeta function, with the nontrivial zeros \(\rho=\beta+i\gamma\) counted with multiplicity, and for \(u\ge0\) let \(\varphi_{1/2}(u)=\xi(\frac12)/\xi(\frac12+\sqrt u)\). Let \(U_{1/2}^\circ\) be the signed measure on \(\theta>0\) with density \(v_{1/2}(\theta)/\pi\), \(v_{1/2}(\theta)=\operatorname{Re}(\xi'/\xi)(\frac12+i\theta)\), plus a unit point mass at \(\gamma\) for every zero \(\frac12+i\gamma\) with \(\gamma>0\) (with multiplicity). Unconditionally, \(U_{1/2}^\circ=\sum_{\beta=1/2,\ \gamma>0}\delta_\gamma\ge0\) (the density vanishes away from these atoms) and \(\varphi_{1/2}(u)=\prod_{\beta=1/2,\ \gamma>0}\frac{\gamma^2}{\gamma^2+u}\ \prod_{\beta>1/2,\ \gamma>0}\Big|1-\frac{u}{(\rho-\frac12)^2}\Big|^{-2}\).

Earlier editions: Corollary 11.4.5, page 475 (edition 1).

Source: ch/almost.tex, label tx:cor:half.

Explicit bounds on the defect from the verification of RH

Volume I, Theorem 14.3.7, page 359 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, let \(\rho=\beta+i\gamma\) run over its zeros (the nontrivial zeros of \(\zeta\)), counted with multiplicity, let \(P_w(x)=w/(\pi(w^2+x^2))\) be the Cauchy kernel of width \(w>0\), with \(P_0\) the unit point mass at \(0\), and for \(\alpha>\frac{1}{2}\) let \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)=v_\alpha^+(\theta)-v_\alpha^-(\theta)\) with \(v_\alpha^+(\theta)/\pi=\sum_{\beta\le\alpha,\,\gamma>0}[P_{\alpha-\beta}(\theta-\gamma)+P_{\alpha-\beta}(\theta+\gamma)]\) and \(v_\alpha^-(\theta)/\pi=\sum_{\beta>\alpha,\,\gamma>0}[P_{\beta-\alpha}(\theta-\gamma)+P_{\beta-\alpha}(\theta+\gamma)]\). The shifted clock \(T_\alpha\) has Laplace transform \(\mathbb{E}e^{-uT_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt{u})=\varphi_\alpha^+(u)D_\alpha(u)\), where \(\varphi_\alpha^+(u)=\exp\{-\int_0^\infty\log(1+u/\theta^2)\,v_\alpha^+(\theta)\,\pi^{-1}d\theta\}\) is the Laplace transform of a generalised gamma convolution with Thorin density \(v_\alpha^+/\pi\) and \(D_\alpha(u)=\prod_{\beta>\alpha,\,\gamma>0}\frac{(\beta-\alpha)^2+\gamma^2}{(\sqrt{u}-\beta+\alpha)^2+\gamma^2}\) is the defect. Let \(T_0=3\times10^{12}\), below which all zeros lie on the critical line (the verification of Platt and Trudgian), and let \(\Sigma_0=\sum_{\gamma>T_0}\gamma^{-2}\), the sum over all zeros above \(T_0\) counted with multiplicity. By the Riemann von Mangoldt formula with the explicit error term of Hasanalizade, Shen and Wong, \(\Sigma_0\) differs from \(\frac{\log(T_0/2\pi)+1}{2\pi T_0}=1.4797\times10^{-12}\) by less than \(2\times10^{-24}\), so \(\Sigma_0<1.48\times10^{-12}\). Then for every \(\alpha>\frac{1}{2}\) and every \(u\ge0\): (1) \(|\log D_\alpha(u)|\le(u+\sqrt{u})\,\Sigma_0\), so that \(e^{-1.5\times10^{-12}(u+\sqrt{u})}\varphi_\alpha^+(u)\le\mathbb{E}e^{-uT_\alpha}\le e^{1.5\times10^{-12}(u+\sqrt{u})}\varphi_\alpha^+(u)\); (2) the negative part of the density satisfies \(v_\alpha^-(\theta)/\pi\le2\Sigma_0/\pi<9.43\times10^{-13}\) for \(0\le\theta\le T_0/2\); (3) consequently \(v_\alpha(\theta)>0\) for all \(0\le\theta\le T_0/2\) whenever \(\alpha-\frac{1}{2}>6\times10^{-10}\). Part of the proof is certified in ball arithmetic.

Earlier editions: Theorem 11.4.6, page 476 (edition 1).

Source: ch/almost.tex, label ag:thm:explicit.

Positivity of the Thorin density on the verified window

Volume I, Corollary 14.3.8, page 360 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, let \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)\) for \(\alpha>\frac{1}{2}\) (the function whose quotient by \(\pi\) is the Thorin density of the shifted clock \(T_\alpha\), the positive random variable with \(\mathbb{E}e^{-uT_\alpha}=\xi(\alpha)/\xi(\alpha+\sqrt{u})\), in the chart \(\theta=\sqrt{z}\), when every zero lies to the left of the line \(\operatorname{Re}s=\alpha\)), let \(T_0=3\times10^{12}\) (every zero off the critical line has \(|\gamma|>T_0\), by the verification of Platt and Trudgian) and let \(V_\xi=(\log\xi)''(\frac{1}{2})=0.04620998623\). (1) For every \(\alpha>\frac{1}{2}\), \(v_\alpha(\theta)>0\) for \(|\theta|\le T_0-\frac{1}{2}\). (2) For every \(\alpha\in(\frac{1}{2},7.35]\), \(v_\alpha(\theta)>v_\alpha(0)\) for \(0<|\theta|\le T_0-\frac{1+\sqrt{2}}{2}\), and \(0<v_\alpha(0)\le(\alpha-\frac{1}{2})V_\xi\).

Earlier editions: Corollary 11.4.7, page 477 (edition 1).

Source: ch/almost.tex, label cor:dbeta:window.

Cancellation lemma for a zero to the right of the basepoint

Volume I, Lemma 14.3.10, page 361 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(v_\alpha(\theta)=\operatorname{Re}(\xi'/\xi)(\alpha+i\theta)\) and \(P_w(x)=w/(\pi(w^2+x^2))\) for \(w>0\). Let \(\rho=\beta+i\gamma\) be a zero with \(\beta>\alpha\), and let \(m>0\) be a lower bound on an interval \(I\) for the rest of the density, \(v_\alpha/\pi+P_{\beta-\alpha}(\cdot-\gamma)\) (that is, the density with the negative kernel of \(\rho\) removed). If \(\beta-\alpha\ge1/(\pi m)\), the kernel of \(\rho\) cannot make \(v_\alpha\) negative on \(I\). The proof is that on \(I\), \(v_\alpha/\pi\ge m-\max_xP_{\beta-\alpha}(x)=m-1/(\pi(\beta-\alpha))\ge0\).

Source: ch/almost.tex, label ag:lem:cancel.

Size biased clock and the exponential mixture form of RH

Volume I, Proposition 14.4.1, page 371 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), an entire function of \(u\) whose zeros are the numbers \(-a_k\), \(a_k=-(\rho_k-\tfrac12)^2\), one for each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\) counted with multiplicity (so \(a_k=\gamma_k^2>0\) for a zero \(\rho_k=\tfrac12+i\gamma_k\) on the critical line). With no hypothesis \(1/G\) is the Laplace transform of a probability law on \([0,\infty)\), the law of the clock \(T\), so that \(\mathbb{E}e^{-uT}=1/G(u)\) for \(u\ge0\). Let \(W(t)=\sum_ke^{-a_kt}\), \(t>0\), be the heat trace, let \(m_0=\mathbb{E}T=\sum_ka_k^{-1}=0.0231049931\ldots\), and let \(\widehat T\) be the size biased clock, the variable with law \(t\,\mathbb{P}(T\in dt)/m_0\). With no hypothesis, \(\widehat T\overset{d}{=}T+A\), where \(A\) is independent of \(T\) and has density \(W(t)/m_0\) on \((0,\infty)\). The Riemann Hypothesis (RH) holds if and only if \(A\) is a mixture of exponential laws, that is, \(A\overset{d}{=}E/V\) with \(E\) a standard exponential variable independent of a variable \(V>0\); and then \(V\) has the law \(m_0^{-1}\sum_{\gamma>0}\gamma^{-2}\delta_{\gamma^2}\), the ordinates \(\gamma\) of the zeros \(\tfrac12+i\gamma\) counted with multiplicity.

Earlier editions: Proposition 11.5.1, page 489 (edition 1).

Source: ch/pakes.tex, label pk:prop:sizebias.

Centre clock lies in Urbanik class L2 and not in L3

Volume I, Theorem 14.4.8, page 377 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), an entire function of \(u\) whose zeros are the numbers \(-a_k\), \(a_k=-(\rho_k-\tfrac12)^2\), one for each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\) counted with multiplicity. With no hypothesis \(1/G\) is the Laplace transform of a probability law on \([0,\infty)\), the law of the clock \(T\) (the centre clock), and \(W(t)=\sum_ke^{-a_kt}\), \(t>0\), is the heat trace. The ladder functions are \(W^{[0]}=W\) and \(W^{[j+1]}(t)=-t\,\frac{d}{dt}W^{[j]}(t)\). With no hypothesis the clock is self decomposable: \(T\overset{d}{=}\int_0^\infty e^{-t}\,dZ_t\) for a subordinator \(Z=Z^{[1]}\), its driving process, with Laplace exponent \(u(\log G)'(u)\) and Lévy density \(-W'(x)\); and \(Z^{[j+1]}\) denotes the driving process of the law of \(Z^{[j]}_1\) whenever that law is self decomposable. A law \(\mu\) on \(\mathbb{R}\) is self decomposable if for every \(c\in(0,1)\) there is a law \(\mu_c\) with \(\widehat\mu(z)=\widehat\mu(cz)\widehat{\mu_c}(z)\) for the characteristic functions; Urbanik’s class \(L_0\) consists of the self decomposable laws and, for \(n\ge1\), \(L_n\) consists of the laws in \(L_{n-1}\) all of whose components \(\mu_c\) lie in \(L_{n-1}\). Unconditionally, that is, using only the verification of the Riemann Hypothesis below height \(2197.3\) (Platt and Trudgian): \(W^{[3]}(t)>0\) for every \(t>0\), and \(W^{[4]}(e^{-4.15})<0\) (in ball arithmetic \(W^{[4]}(e^{-4.15})=-0.11089729616255\pm10^{-23}\)). Consequently \(W\), \(W^{[1]}\) and \(W^{[2]}\) are positive on \((0,\infty)\); the centre clock lies in \(L_2\) and not in \(L_3\); its driving process \(Z=Z^{[1]}\) and the driving process \(Z^{[2]}\) of \(Z\) are self decomposable at every time (the laws of \(Z_s\) and of \(Z^{[2]}_s\) are self decomposable for every \(s>0\)); and the driving process \(Z^{[3]}\) of \(Z^{[2]}\) is self decomposable at no time.

Earlier editions: Theorem 11.5.8, page 496 (edition 1).

Source: ch/pakes2.tex, label pk:thm:ladder.

First two driving processes of the clock are self decomposable and unimodal

Volume I, Corollary 14.4.9, page 379 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), and \(T\) the clock, the random variable with \(\mathbb{E}e^{-uT}=1/G(u)\) for \(u\ge0\) (it exists with no hypothesis). With no hypothesis \(T\) is self decomposable: \(T\overset{d}{=}\int_0^\infty e^{-t}\,dZ_t\) for a subordinator \(Z\), its driving process, with Laplace exponent \(u(\log G)'(u)\); and \(Z^{[2]}\) denotes the driving process of the self decomposable law of \(Z_1\). A law \(\mu\) on \(\mathbb{R}\) is self decomposable if for every \(c\in(0,1)\) there is a law \(\mu_c\) with \(\widehat\mu(z)=\widehat\mu(cz)\widehat{\mu_c}(z)\) for the characteristic functions; Urbanik’s class \(L_1\) consists of the self decomposable laws all of whose components \(\mu_c\) are self decomposable. With no hypothesis, for every \(s>0\) the laws of \(Z_s\) and of \(Z^{[2]}_s\) are self decomposable, hence unimodal, and the law of \(Z_s\) lies in \(L_1\).

Earlier editions: Corollary 11.5.9, page 498 (edition 1).

Source: ch/pakes2.tex, label pk:cor:unimodal.

The clock is not the martingale limit of a supercritical continuous state branching process

Volume I, Theorem 14.4.11, page 383 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), and \(T\) the clock, \(\mathbb{E}e^{-uT}=1/G(u)\) for \(u\ge0\). With no hypothesis \(T\overset{d}{=}\int_0^\infty e^{-t}\,dZ_t\) for a subordinator \(Z\) (its driving process) with Laplace exponent \(\Psi_Z(u)=u(\log G)'(u)\). Let \(H\) be the inverse function of \(\log G\) near \(0\) and \(g=\Psi_Z\circ H=H/H'=\sum_{k\ge1}g_kw^k\), so that \(g_1=1\). A supercritical continuous state branching process with finite Malthusian parameter is obtained, by Lamperti’s time change, from a spectrally positive Lévy process \(Y\) with Laplace exponent \(\Psi_Y(u)=-\log\mathbb{E}e^{-uY_1}\) and \(\Psi_Y'(0)\in(0,\infty)\); started at \(z>0\) it is a process \((\mathsf{B}_t)_{t\ge0}\) with \(\mathbb{E}\mathsf{B}_t=ze^{\Psi_Y'(0)t}\), and the nonnegative martingale \(e^{-\Psi_Y'(0)t}\mathsf{B}_t\) converges almost surely to a limit \(\mathsf{M}\). With no hypothesis, \((-1)^{15}g_{16}<0\): in ball arithmetic \(g_{16}=1.49524474945112\times10^{-15}\pm2\times10^{-30}\). Consequently, for no \(\lambda>0\), no such \(Y\) and no initial state \(z>0\) does \(\lambda T\) have the law of \(\mathsf{M}\): no positive multiple of the centre clock is the martingale limit of a supercritical continuous state branching process with finite Malthusian parameter, from any initial state.

Earlier editions: Theorem 11.5.11, page 502 (edition 1).

Source: ch/pakes2.tex, label pk:thm:nobranch.

Criticality filter for properties of the Pólya kernel

Volume I, Proposition 14.4.12, page 384 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(\Phi(v)=2\sum_{n\ge1}(2\pi^2n^4e^{9v/2}-3\pi n^2e^{5v/2})\exp(-\pi n^2e^{2v})\) the Pólya kernel, an even positive function with \(\xi(\tfrac12+i\theta)=\int_{\mathbb{R}}\Phi(v)\cos(\theta v)\,dv\); for \(t\le0\) put \(\Phi_t(v)=e^{tv^2/4}\Phi(v)\). For an even positive integrable function \(g\) on \(\mathbb{R}\) put \(\widehat g(\theta)=\int_{\mathbb{R}}g(v)\cos(\theta v)\,dv\) and \(R_g(s)=\widehat g(0)/\widehat g(i\sqrt s)=\int_{\mathbb{R}}g(v)\,dv\big/\int_{\mathbb{R}}g(v)\cosh(\sqrt s\,v)\,dv\) for the \(s\ge0\) at which the last integral converges; for \(g=\Phi\) this is \(R_\Phi(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\), the Laplace transform of the clock. A GGC (generalised, or generalized, gamma convolution) is a random variable \(H\ge0\) with \(\mathbb{E}e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), where the drift \(d\ge0\) and the Thorin measure \(U\), a positive measure on \((0,\infty)\), satisfy \(\int_{(0,1)}|\log z|\,U(dz)+\int_{(1,\infty)}z^{-1}U(dz)<\infty\). Let \(\mathcal{P}\) be a property of even positive integrable functions on \(\mathbb{R}\) that \(\Phi_t\) has for some \(t<0\). Then \(\mathcal{P}\) implies neither that \(\widehat g\) has only real zeros nor that \(R_g\) is the Laplace transform of a GGC, for the functions \(g\) with \(\mathcal{P}\): the function \(g=\Phi_t\) has \(\mathcal{P}\), while \(\widehat g\) has nonreal zeros and \(R_g\) is not the Laplace transform of a GGC. At every \(t\le0\) the kernel \(\Phi_t\) (i) is positive and even, and strictly decreasing on \((0,\infty)\); (ii) is log concave; (iii) satisfies \(\log\Phi_t(v)=-\pi e^{2|v|}+O(v^2)\) as \(|v|\to\infty\); (iv) extends analytically to the strip \(|\operatorname{Im}v|<\pi/4\). So no combination of (i) to (iv) implies that \(\widehat g\) has only real zeros, or that \(R_g\) is the Laplace transform of a GGC. The proof uses the theorem of Rodgers and Tao that the de Bruijn Newman constant is nonnegative, which gives a nonreal zero of \(\widehat{\Phi_t}\) for every \(t<0\).

Earlier editions: Proposition 11.5.12, page 504 (edition 1).

Source: ch/pakes3.tex, label pk:prop:filter.

Damped Pólya law as a mixture of arcsine laws

Volume I, Proposition 14.4.13, page 386 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(\Phi(v)=2\sum_{n\ge1}(2\pi^2n^4e^{9v/2}-3\pi n^2e^{5v/2})\exp(-\pi n^2e^{2v})\) the Pólya kernel, an even positive function with \(\int_{\mathbb{R}}\Phi(v)\cos(\theta v)\,dv=\xi(\tfrac12+i\theta)\). For \(t\le0\) let \(\Phi_t(v)=e^{tv^2/4}\Phi(v)\), let \(p_t=\Phi_t/\int_{\mathbb{R}}\Phi_t\) be the corresponding probability density, which is even and decreasing on \((0,\infty)\), and let \(X_t\) be a variable with density \(p_t\) (so \(X_0\) is the Pólya variable). For every \(t\le0\) the function \(h_t(\rho)=-2\rho\int_\rho^\infty p_t'(r)\,dr/\sqrt{r^2-\rho^2}\), \(\rho>0\), is a positive probability density, and \(p_t(v)=\frac1\pi\int_{|v|}^\infty h_t(\rho)\,d\rho/\sqrt{\rho^2-v^2}\) for \(v\ne0\). Equivalently, \(X_t\overset{d}{=}\varrho_t\cos\Theta\) with \(\Theta\) uniform on \([0,2\pi)\) and \(\varrho_t\) of density \(h_t\) independent of \(\Theta\): \(X_t\) is a mixture of arcsine laws, the first coordinate of the rotation invariant vector \(\varrho_t(\cos\Theta,\sin\Theta)\) in \(\mathbb{R}^2\) and also of a rotation invariant vector in \(\mathbb{R}^3\).

Earlier editions: Proposition 11.5.13, page 505 (edition 1).

Source: ch/pakes3.tex, label pk:prop:arcsine.

Tilted Biane Pitman Yor variable is not a GGC near t = 0 and at ten certified values of t

Volume I, Proposition 14.4.15, page 388 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(f(b)=\sum_{m\ge1}(4\pi^2m^4b-6\pi m^2)e^{-\pi m^2b}\), \(b>0\), the density of the variable \(\mathcal{B}\) of Biane, Pitman and Yor, with \(\mathbb{E}e^{-\lambda\mathcal{B}}=(\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda})^2\) and \(\mathbb{E}\mathcal{B}^{s/2}=2\xi(s)\); \(\mathcal{B}\) is a GGC with Thorin measure \(2\sum_{m\ge1}\delta_{\pi m^2}\). For real \(t\) let \(\mathcal{B}_t\) be the variable with density proportional to \(e^{t(\log b)^2/16}f(b)\) on \((0,\infty)\), so that \(\mathcal{B}_0=\mathcal{B}\). A GGC (generalised, or generalized, gamma convolution) is a random variable \(H\ge0\) with \(\mathbb{E}e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), with drift \(d\ge0\) and Thorin measure \(U\), a positive measure on \((0,\infty)\) with \(\int_{(0,1)}|\log z|\,U(dz)+\int_{(1,\infty)}z^{-1}U(dz)<\infty\). (i) \(\mathcal{B}_t\) is not a GGC for \(t=-20\), \(-5\), \(-1\), \(-\tfrac12\), \(-\tfrac1{10}\), \(-\tfrac1{100}\), \(\tfrac1{100}\), \(\tfrac15\), \(1\) and \(5\). (ii) There is \(t_0>0\) such that \(\mathcal{B}_t\) is not a GGC for \(0<|t|<t_0\).

Earlier editions: Proposition 11.5.15, page 508 (edition 1).

Source: ch/pakes3.tex, label pk:prop:bpytilt.

Theta variable of the odd real character modulo 8 and its reciprocal are not GGCs

Volume I, Proposition 14.4.18, page 391 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: counterexample, proved.

Statement. Let \(\chi_{-8}\) be the real primitive Dirichlet character modulo \(8\) with \(\chi_{-8}(-1)=-1\) (so \(\chi_{-8}(n)=1\) for \(n\equiv1,3\), \(\chi_{-8}(n)=-1\) for \(n\equiv5,7 \pmod{8}\), and \(\chi_{-8}(n)=0\) for even \(n\)), let \(\vartheta(x)=\sum_{n\ge1}n\chi_{-8}(n)e^{-\pi n^2x/8}\) for \(x>0\), a positive function with \(\vartheta(1/x)=x^{3/2}\vartheta(x)\), and \(\xi(s,\chi_{-8})=\int_0^\infty\vartheta(x)x^{(s+1)/2}\,dx/x=(8/\pi)^{(s+1)/2}\Gamma(\tfrac{s+1}2)L(s,\chi_{-8})\). Let \(\mathcal{B}_{\chi_{-8}}\) be the variable with density \(b^{-1/2}\vartheta(b)/\xi(0,\chi_{-8})\) on \((0,\infty)\), so that \(\mathbb{E}\mathcal{B}_{\chi_{-8}}^{s/2}=\xi(s,\chi_{-8})/\xi(0,\chi_{-8})\); its reciprocal \(1/\mathcal{B}_{\chi_{-8}}\) has density \(\vartheta(x)/\xi(0,\chi_{-8})\). A GGC (generalised, or generalized, gamma convolution) is a random variable \(H\ge0\) with \(\mathbb{E}e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), with drift \(d\ge0\) and Thorin measure \(U\), a positive measure on \((0,\infty)\) with \(\int_{(0,1)}|\log z|\,U(dz)+\int_{(1,\infty)}z^{-1}U(dz)<\infty\). Neither \(\mathcal{B}_{\chi_{-8}}\) nor \(1/\mathcal{B}_{\chi_{-8}}\) is a GGC. (The proof is a certificate in ball arithmetic: the cumulants \(\kappa_2,\kappa_3,\kappa_4\) of a GGC with a finite fourth moment satisfy \(2\kappa_2\kappa_4\ge3\kappa_3^2\), and \(3\kappa_3^2-2\kappa_2\kappa_4\) equals \(7.55465911093\pm4\times10^{-12}\) for \(\mathcal{B}_{\chi_{-8}}\) and \(78.8854168657\pm2\times10^{-11}\) for \(1/\mathcal{B}_{\chi_{-8}}\), both positive.)

Earlier editions: Proposition 11.5.18, page 511 (edition 1).

Source: ch/pakes3.tex, label pk:prop:chi8.

Reciprocal theta variable of odd characters is a GGC for discriminants -3 and -4, not for -8, for d congruent to 1 mod 8 and for the function of Davenport and Heilbronn

Volume I, Proposition 14.4.19, page 392 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(q\ge3\) and let \(a=(a_n)_{n\ge1}\) be a real sequence of period \(q\) with \(a_1=1\) and \(a_{q - n}=-a_n\), whose theta series \(\vartheta_a(x)=\sum_{n\ge1}na_ne^{-\pi n^2x/q}\), \(x>0\), satisfies \(\vartheta_a(1/x)=x^{3/2}\vartheta_a(x)\) and is positive. Put \(\xi(s,a)=\int_0^\infty\vartheta_a(x)x^{(s+1)/2}\,dx/x\), let \(\mathcal{B}_a\) be the variable with density \(b^{-1/2}\vartheta_a(b)/\xi(0,a)\) on \((0,\infty)\), so that \(1/\mathcal{B}_a\) has density \(\vartheta_a(x)/\xi(0,a)\), and for \(\lambda>0\) put \(w=\sqrt{\pi q\lambda}\) and \(\Sigma_a(w)=\sum_{1\le n<q/2}a_n\sinh\frac{(q - 2n)w}{q}\). A GGC (generalised, or generalized, gamma convolution) is a random variable \(H\ge0\) with \(\mathbb{E}e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), with drift \(d\ge0\) and Thorin measure \(U\) (a positive measure on \((0,\infty)\) with \(\int_{(0,1)}|\log z|\,U(dz)+\int_{(1,\infty)}z^{-1}U(dz)<\infty\)); the Laplace transform of a GGC extends holomorphically and without zeros to \(\mathbb{C}\setminus(-\infty,0]\) and cannot vanish on \((-\infty,0)\) where it extends meromorphically. Then \(\int_0^\infty e^{-\lambda x}\vartheta_a(x)\,dx=\sqrt q\,\Sigma_a(w)/\sinh w\) and \(\mathbb{E}e^{-\lambda/\mathcal{B}_a}=\sqrt q\,\Sigma_a(w)/(\xi(0,a)\sinh w)\), and \(\xi(0,a)=q^{-1/2}\sum_{1\le n<q/2}(q - 2n)a_n\); the first formula is formula (96) of Biane, Pitman and Yor in this normalisation. In particular: (i) for the character of discriminant \(-4\) (\(q=4\), \(a=(1,0,-1,0)\)), \(\mathbb{E}e^{-\lambda/\mathcal{B}_a}=\operatorname{sech}\sqrt{\pi\lambda}\), and \(1/\mathcal{B}_a\), which is \(\tfrac\pi2\) times the exit time of Brownian motion from \((-1,1)\), is a GGC; (ii) for the character of discriminant \(-3\) (\(q=3\), \(a=(1,-1,0)\)), \(\mathbb{E}e^{-\lambda/\mathcal{B}_a}=3\sinh(w/3)/\sinh w=\prod_{3\nmid m}(1+3\lambda/(\pi m^2))^{-1}\), so that \(1/\mathcal{B}_a\) is a GGC with Thorin measure \(\sum_{3\nmid m}\delta_{\pi m^2/3}\), and it is \(\tfrac{3\pi}2\) times the hitting time of \(1\) by a three dimensional Bessel process started at \(\tfrac13\); (iii) for the character of discriminant \(-8\) (\(q=8\), \(a=(1,0,1,0,-1,0,-1,0)\)), \(\mathbb{E}e^{-\lambda/\mathcal{B}_a}=\cosh(w/4)/\cosh(w/2)=\prod_{k\ge1}(1+\lambda/(4\alpha_k))/(1+\lambda/\alpha_k)\) with \(\alpha_k=\tfrac\pi8(2k-1)^2\), so that \(1/\mathcal{B}_a\) is infinitely divisible with Lévy density \(\sum_k(e^{-\alpha_kx}-e^{-4\alpha_kx})/x>0\), and it is \(\pi\) times the hitting time of \(1\) by a reflected Brownian motion started at \(\tfrac12\) (the identifications of (ii) and (iii) with hitting times are those of Biane, Pitman and Yor); it is not a GGC, because its Laplace transform, continued meromorphically, vanishes at \(\lambda=-\pi/2\); (iv) if \(a=\chi_d\) for a real primitive character of fundamental discriminant \(d<0\) with \(\chi_d(2)=1\), that is \(d\equiv1 \pmod{8}\), and \(q=|d|\), the Laplace transform of \(1/\mathcal{B}_a\), continued meromorphically by the closed form, vanishes at a point of \((-4\pi/q,-\pi/q)\), so \(1/\mathcal{B}_a\) is not a GGC; for \(\chi_{-7}\) the point is \(\lambda=-1.033610573\). The same holds for the function of Davenport and Heilbronn, with \(q=5\), \(a=(1,\varkappa,-\varkappa,-1,0)\) and \(\varkappa=(\sqrt{10-2\sqrt5}-2)/(\sqrt5-1)\), at \(\lambda=-2.046575671\).

Earlier editions: Proposition 11.5.19, page 512 (edition 1).

Source: ch/pakes3.tex, label pk:prop:reflected.

Open problem on theta type functions whose GGC property forces zeros onto the line

Volume I, Problem 14.7, page 395 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(q\ge3\), \(p\in\{0,1\}\) and let \(a=(a_n)_{n\ge1}\) be a real sequence of period \(q\) with \(a_1=1\) and \(a_{q - n}=(-1)^pa_n\), whose theta series \(\vartheta_a(x)=\sum_{n\ge1}n^pa_ne^{-\pi n^2x/q}\), \(x>0\), satisfies \(\vartheta_a(1/x)=x^{p+1/2}\vartheta_a(x)\); put \(\xi(s,a)=\int_0^\infty\vartheta_a(x)x^{(s+p)/2}\,dx/x=(q/\pi)^{(s+p)/2}\Gamma(\tfrac{s+p}2)\sum_{n\ge1}a_nn^{-s}\), an entire function with \(\xi(s,a)=\xi(1-s,a)\). When \(\vartheta_a>0\) (a positive kernel) let \(\mathcal{B}_a\) be the variable with density \(b^{p/2-1}\vartheta_a(b)/\xi(0,a)\) on \((0,\infty)\), so that \(\mathbb{E}\mathcal{B}_a^{s/2}=\xi(s,a)/\xi(0,a)\). For the completed Riemann zeta function \(\xi\) the corresponding variable is \(\mathcal{B}\) with \(\mathbb{E}e^{-\lambda\mathcal{B}}=(\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda})^2\) and \(\mathbb{E}\mathcal{B}^{s/2}=2\xi(s)\), which is a GGC (generalised, or generalized, gamma convolution: a variable \(H\ge0\) with \(\mathbb{E}e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\), \(u>0\), for a drift \(d\ge0\) and a positive measure \(U\) on \((0,\infty)\), the Thorin measure). The test by Hankel matrices of cumulants: for a positive variable \(\mathcal{V}\) with \(\ell(\lambda)=\mathbb{E}e^{-\lambda\mathcal{V}}\) put \(\kappa_k(\lambda)=(-1)^k\frac{d^k}{d\lambda^k}\log\ell(\lambda)\) and \(c_k(\lambda)=\kappa_{k+1}(\lambda)/k!\); if \(\mathcal{V}\) is a GGC, the Hankel matrices \(\mathcal{K}_n(\lambda)=[c_{i+j}(\lambda)]_{i,j=0}^{n-1}\) and \(\mathcal{K}'_n(\lambda)=[c_{i+j+1}(\lambda)]_{i,j=0}^{n-1}\) are positive semidefinite for all \(n\) and \(\lambda\ge0\), so a variable for which one of them has a negative eigenvalue is not a GGC, and is said to fail the test (applied numerically for \(n\le12\) at \(\lambda=0\), \(\tfrac12\), \(2\), \(10\), \(20\), \(50\) and \(200\)). Question. Find a natural class of functions \(\xi(s,a)\) with positive kernels, containing \(\xi\), in which the GGC property of \(\mathcal{B}_a\) implies that every zero of \(\xi(s,a)\) lies on the critical line \(\operatorname{Re}s=\tfrac12\), or find a function in such a class whose variable \(\mathcal{B}_a\) is a GGC and which has a zero off the line. Since \(\mathcal{B}\) is a GGC, the first answer for a class containing \(\xi\) proves the Riemann Hypothesis. No candidate for the second is known: the function of Davenport and Heilbronn (coefficients of period \(5\), \((1,\varkappa,-\varkappa,-1,0)\) with \(\varkappa=(\sqrt{10-2\sqrt5}-2)/(\sqrt5-1)\)), the members with a positive kernel, among the \(17\) tested, of the real one parameter family \(f_r\) of combinations of \(L(s,\chi_{-7})\), \(L(s,\chi)\) and \(L(s,\bar\chi)\), with \(\chi\) and \(\bar\chi\) the complex odd characters modulo \(7\), that have real coefficients, \(a_1=1\) and the functional equation of the class, and the tilts \(\mathcal{B}_t\) of \(\mathcal{B}\) with density proportional to \(e^{t(\log b)^2/16}f(b)\) at the eleven values of \(t<0\) tested (\(f\) the density of \(\mathcal{B}\)) all fail the test by Hankel matrices of cumulants.

Earlier editions: Problem 11.1, page 514 (edition 1).

Source: ch/pakes3.tex, label pk:prob:ggc.

Open problem, is the tilted Biane Pitman Yor variable a GGC for no t other than 0

Volume I, Problem 14.8, page 395 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\mathcal{B}\) be the variable of Biane, Pitman and Yor, a GGC (generalised, or generalized, gamma convolution) with density \(f(b)=\sum_{m\ge1}(4\pi^2m^4b-6\pi m^2)e^{-\pi m^2b}\), \(b>0\), \(\mathbb{E}e^{-\lambda\mathcal{B}}=(\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda})^2\) and \(\mathbb{E}\mathcal{B}^{s/2}=2\xi(s)\) for the completed Riemann zeta function \(\xi\), and for real \(t\) let \(\mathcal{B}_t\) be the variable with density proportional to \(e^{t(\log b)^2/16}f(b)\) on \((0,\infty)\), so that \(\mathcal{B}_0=\mathcal{B}\). Let \(\Phi(v)=2\sum_{n\ge1}(2\pi^2n^4e^{9v/2}-3\pi n^2e^{5v/2})\exp(-\pi n^2e^{2v})\) be the Pólya kernel, an even positive function with \(\xi(\tfrac12+i\theta)=\int_{\mathbb{R}}\Phi(v)\cos(\theta v)\,dv\), so that \(f(b)=b^{-5/4}\Phi(\tfrac12\log b)\), and for real \(t\) let \(\Phi_t(v)=e^{tv^2/4}\Phi(v)\) be the damped kernel; then \(\mathcal{B}_t\) has density proportional to \(b^{-5/4}\Phi_t(\tfrac12\log b)\). Question. Is \(\mathcal{B}_t\) a GGC for no \(t\ne0\)? For \(t<0\) a positive answer would mean that the GGC property of the variable \(\mathcal{B}_t\) built from the damped Pólya kernel \(\Phi_t(v)=e^{tv^2/4}\Phi(v)\) is lost at every \(t<0\), although the Pólya kernel \(\Phi\) itself has it (\(\mathcal{B}_0=\mathcal{B}\) is a GGC). Known: \(\mathcal{B}_t\) is not a GGC for \(0<|t|<t_0\) for some \(t_0>0\), and for \(t=-20\), \(-5\), \(-1\), \(-\tfrac12\), \(-\tfrac1{10}\), \(-\tfrac1{100}\), \(\tfrac1{100}\), \(\tfrac15\), \(1\) and \(5\).

Earlier editions: Problem 11.2, page 515 (edition 1).

Source: ch/pakes3.tex, label pk:prob:tilt.

Open problem on the GGC property of theta variables of characters

Volume I, Problem 14.9, page 395 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a real primitive Dirichlet character \(\chi_d\) of fundamental discriminant \(d\) (\(p=1\) for \(d<0\), \(p=0\) for \(d>0\)) with positive theta series \(\vartheta_{\chi_d}(x)=\sum_{n\ge1}n^p\chi_d(n)e^{-\pi n^2x/|d|}\), let \(\mathcal{B}_{\chi_d}\) be the variable with density proportional to \(b^{p/2-1}\vartheta_{\chi_d}(b)\) on \((0,\infty)\). A GGC (generalised, or generalized, gamma convolution) is a random variable \(H\ge0\) with \(\mathbb{E}e^{-uH}=\exp\{-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), with drift \(d\ge0\) and a positive Thorin measure \(U\) on \((0,\infty)\). For a positive variable \(\mathcal{V}\) with \(\ell(\lambda)=\mathbb{E}e^{-\lambda\mathcal{V}}\) put \(\kappa_k(\lambda)=(-1)^k\frac{d^k}{d\lambda^k}\log\ell(\lambda)\) and \(c_k(\lambda)=\kappa_{k+1}(\lambda)/k!\); if \(\mathcal{V}\) is a GGC, the Hankel matrices \(\mathcal{K}_n(\lambda)=[c_{i+j}(\lambda)]_{i,j=0}^{n-1}\) and \(\mathcal{K}'_n(\lambda)=[c_{i+j+1}(\lambda)]_{i,j=0}^{n-1}\) are positive semidefinite for all \(n\) and \(\lambda\ge0\), so a negative eigenvalue shows that \(\mathcal{V}\) is not a GGC. A variable passes the test to order \(12\) if no \(\mathcal{K}_n(\lambda)\) or \(\mathcal{K}'_n(\lambda)\) with \(n\le12\) has a negative eigenvalue at \(\lambda=0\), \(\tfrac12\), \(2\), \(10\), \(20\), \(50\) and \(200\) (a numerical test). Question. Which of the variables \(\mathcal{B}_{\chi_d}\) for \(d=-4\), \(-3\), \(5\), \(8\) and \(12\), which pass the test to order \(12\), are GGCs? (For \(d=24\) the test fails at order \(11\).) On the reciprocal side, \(1/\mathcal{B}_{\chi_d}\) is a GGC for \(d=-3\) and \(d=-4\); for \(d=-4\) it is \(\tfrac\pi2\) times the exit time of Brownian motion from \((-1,1)\), whose density is not hyperbolically completely monotone (HCM), so the question for \(\mathcal{B}_{\chi_{-4}}\) cannot be settled through HCM. For even characters the Laplace transform of the reciprocal variable is the series \(\frac2{\xi(0,a)}\sum_na_nK_0\big(2n\sqrt{\pi\lambda/q}\big)\) of Bessel functions \(K_0\) (with \(q=|d|\), \(a_n=\chi_d(n)\) and \(\xi(0,a)\) the normalising constant of the density of \(\mathcal{B}_{\chi_d}\)), and no product formula for it is known to the authors.

Earlier editions: Problem 11.3, page 515 (edition 1).

Source: ch/pakes3.tex, label pk:prob:side.

Open problem on unconditional positivity of the Li coefficients

Volume I, Problem 14.10, page 395 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), with zeros \(-a_k\), \(a_k=-(\rho_k-\tfrac12)^2\), one for each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\) counted with multiplicity. Let \(W(t)=\sum_ke^{-a_kt}\) be the heat trace, \(m_0=\sum_ka_k^{-1}\), and \(A\) a variable with density \(W(t)/m_0\) on \((0,\infty)\) (the size biased clock is the clock plus an independent copy of \(A\)). The Li coefficients are \(\lambda_n=\sum_\rho\big(1-(1-1/\rho)^n\big)\), \(n\ge1\), the sum over the zeros \(\rho\) of \(\xi\) grouped in pairs \(\{\rho,1-\rho\}\); they satisfy \(\lambda_n=\sum_k\varphi_n(a_k)\) with \(\varphi_n(z)=2-\big(\frac{2\sqrt z+i}{2\sqrt z-i}\big)^n-\big(\frac{2\sqrt z-i}{2\sqrt z+i}\big)^n\), a rational function of \(z\) with a pole of order \(n\) at \(z=-\tfrac14\) and no constant term in \((z+\tfrac14)^{-1}\) (for example \(\varphi_1(z)=(z+\tfrac14)^{-1}\) and \(\varphi_2(z)=4z(z+\tfrac14)^{-2}\)). Let \(k_n\) be the function with \(\int_0^\infty e^{-zt}k_n(t)\,dt=\varphi_n(z)\); it has the form \(k_n(t)=e^{-t/4}p_n(t)\) with \(p_n\) a polynomial of degree \(n-1\) (for example \(k_1(t)=e^{-t/4}\) and \(k_2(t)=(4-t)e^{-t/4}\)). With no hypothesis, \(\lambda_n=\int_0^\infty k_n(t)W(t)\,dt=m_0\,\mathbb{E}k_n(A)\). Question. Which unconditional properties of \(W\), or of \(A\), force \(\mathbb{E}k_n(A)\ge0\) for every \(n\), which is the Riemann Hypothesis by Li’s criterion? Positivity of \(W\) gives \(n=1\); complete monotonicity of \(W\), which is equivalent to RH, gives every \(n\), since \(\varphi_n\ge0\) on \((0,\infty)\).

Earlier editions: Problem 11.4, page 515 (edition 1).

Source: ch/pakes3.tex, label pk:prob:li.

Open problem on the ladder index along the de Bruijn Newman flow

Volume I, Problem 14.11, page 395 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi\) be the Pólya kernel, with \(\xi(\tfrac12+i\theta)=\int_{\mathbb{R}}\Phi(v)\cos(\theta v)\,dv\) for the completed Riemann zeta function \(\xi\), let \(\Phi_t(v)=e^{tv^2/4}\Phi(v)\), \(\Xi_t(\theta)=\int_{\mathbb{R}}\Phi_t(v)\cos(\theta v)\,dv\) and \(R_t(s)=\Xi_t(0)/\Xi_t(i\sqrt s)\), so that \(R_0(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) is the Laplace transform of the clock. The zeros of \(\Xi_t\) are all real exactly when \(t\ge\Lambda\), where \(\Lambda\) is the de Bruijn Newman constant, \(0\le\Lambda\le0.2\) (Rodgers and Tao; Platt and Trudgian). For \(t\ge\Lambda\), in particular unconditionally for every \(t\ge0.2\), let \(\pm\theta_k(t)\), \(0<\theta_1(t)<\theta_2(t)<\cdots\), be the zeros of \(\Xi_t\) and \(a_k(t)=\theta_k(t)^2\); then \(R_t(s)=\prod_k(1+s/a_k(t))^{-1}\) is the Laplace transform of a GGC (generalised, or generalized, gamma convolution) with Thorin measure \(\sum_k\delta_{a_k(t)}\). At \(t=0\) put \(a_k(0)=a_k=-(\rho_k-\tfrac12)^2\), one for each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\) (real when RH holds), the numbers of the clock. Put \(W(x;t)=\sum_ke^{-a_k(t)x}\) and \(W^{[j]}(x;t)=(-x\,\partial_x)^jW(x;t)\). A law with this Laplace transform lies in Urbanik’s class \(L_n\) exactly when \(W^{[n+1]}(\cdot\,;t)\ge0\) on \((0,\infty)\) (the classes \(L_n\): \(L_0\) is the class of self decomposable laws and \(L_n\) the class of laws in \(L_{n-1}\) all of whose components lie in \(L_{n-1}\)); the index on the window \(0\le-\log x\le8\) is the largest \(n\) with \(W^{[1]},\dots,W^{[n+1]}\ge0\) there. Question. Describe the ladder index of \(R_t\) along the de Bruijn Newman flow. A numerical computation (not certified for \(t>0\)) finds on the window the index two of the clock at the tested values of \(t\) up to \(27.8\) and the index one at the tested values \(t=27.9\), \(30\), \(40\) and \(50\). Is the index nonincreasing in \(t\), and is it one for all large \(t\)?

Earlier editions: Problem 11.5, page 515 (edition 1).

Source: ch/pakes3.tex, label pk:prob:ladder.

Volume I, Chapter 15. The Dedekind eta function, L(s, chi_12) and Ramanujan’s Delta

Ramanujan Delta as the distribution of a maximum of independent exponential variables

Volume I, Lemma 15.1.1, page 404 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(E_{n,j}\), \(n\ge1\), \(1\le j\le24\), be independent, with \(E_{n,j}\) exponential of rate \(2\pi n\), and put \(M^*=\sup_{n,j}E_{n,j}\) and \(M^{(j)}=\sup_nE_{n,j}\). Almost surely these suprema are attained and finite, \(\Pr(M^*\le y)=F(y)=\prod_{n\ge1}(1-e^{-2\pi ny})^{24}\) for \(y>0\) (so that \(\Delta(iy)=e^{-2\pi y}F(y)\) for Ramanujan’s \(\Delta\)), and each \(M^{(j)}\) has the distribution function \(F_1(y)=\prod_{n\ge1}(1-e^{-2\pi ny})\). So \(M^*=\max_{j\le24}M^{(j)}\) is the maximum of \(24\) independent variables with distribution function \(F_1\).

Source: ch/ch16.tex, label lem:maxexp:max.

Max representation of the L function of Ramanujan’s Delta

Volume I, Theorem 15.1.2, page 405 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(E_{n,j}\), \(n\ge1\), \(1\le j\le24\), be independent exponential random variables, \(E_{n,j}\) of rate \(2\pi n\), and let \(M^*=\sup_{n,j}E_{n,j}\) (almost surely attained and finite, with \(\Pr(M^*\le y)=\prod_{n\ge1}(1-e^{-2\pi ny})^{24}\) for \(y>0\)). Let \(\Delta(z)=\eta(z)^{24}=\sum_{n\ge1}\tau(n)e^{2\pi inz}\) be Ramanujan’s cusp form, \(\eta\) Dedekind’s eta function, \(L(s,\Delta)=\sum_{n\ge1}\tau(n)n^{-s-11/2}\) its L function, normalised so that the critical line is \(\operatorname{Re}s=\frac12\), and \(\xi(s,\Delta)=(2\pi)^{-s-11/2}\Gamma(s+\frac{11}2)L(s,\Delta)\) its completed function. With \(\Gamma(a,x)=\int_x^\infty e^{-v}v^{a-1}\,dv\) and \(Q(a,x)=\Gamma(a,x)/\Gamma(a)\), for every complex \(s\), \(\xi(s,\Delta)=(2\pi)^{-s-11/2}\,\mathbb{E}\,\Gamma(s+\frac{11}2,2\pi M^*)\) and \(L(s,\Delta)=\mathbb{E}\,Q(s+\frac{11}2,2\pi M^*)\), and both expectations converge absolutely. In particular \(L(s,\Delta)=0\) at the points \(s=-\frac{11}2-k\), \(k=0,1,2,\dots\), where \(Q(-k,x)=0\).

Earlier editions: Theorem 8.5.2, page 269 (edition 1).

Source: ch/ch16.tex, label thm:maxexp:max.

Gamma mixture and mixed Poisson forms of L values of Delta

Volume I, Corollary 15.1.3, page 405 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M^*\) be the maximum of independent exponential random variables, \(24\) of each rate \(2\pi n\), \(n\ge1\), with distribution function \(F(y)=\prod_{n\ge1}(1-e^{-2\pi ny})^{24}\), and let \(L(s,\Delta)\) be the L function of Ramanujan’s cusp form \(\Delta\), normalised so that the critical line is \(\operatorname{Re}s=\frac12\). Let \(G_a\) be a gamma variable of shape \(a>0\) and scale one and \(N\) a Poisson process of unit rate, both independent of \(M^*\). For real \(s>-\frac{11}2\), with \(a=s+\frac{11}2\), \(L(s,\Delta)=\Pr(G_a>2\pi M^*)=\mathbb{E}\,F(G_a/2\pi)=\mathbb{E}\prod_{n\ge1}(1-e^{-nG_a})^{24}\), and for every integer \(k\ge1\), \(L(k-\frac{11}2,\Delta)=\Pr\big(N(2\pi M^*)\le k-1\big)=\mathbb{E}\Big[e^{-2\pi M^*}\sum_{j<k}\frac{(2\pi M^*)^j}{j!}\Big]\).

Earlier editions: Corollary 8.5.3, page 269 (edition 1).

Source: ch/ch16.tex, label cor:maxexp:forms.

L values of Delta and chi12 as infinitely divisible probabilities

Volume I, Proposition 15.1.7, page 410 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M^*\) be the maximum of independent exponential random variables, \(24\) of each rate \(2\pi n\), \(n\ge1\), and for real \(a>0\) let \(G_a\) be a gamma variable of shape \(a\) and scale one independent of \(M^*\) and \(V_a=G_a/(2\pi M^*)\). Let \(L(s,\Delta)\) be the L function of Ramanujan’s cusp form \(\Delta\), normalised so that the critical line is \(\operatorname{Re}s=\frac12\). Then \(V_a\) is a scale mixture of Gamma(\(a\)) laws, the mixing law being that of \(1/(2\pi M^*)\), and \(L(a-\frac{11}2,\Delta)=\Pr(V_a>1)\). For \(0<a\le2\), that is \(-\frac{11}2<s\le-\frac72\), \(V_a\) is infinitely divisible. In the same way, with \(M_1\) the maximum of independent exponential random variables, one of each rate \(2\pi n\), \(n\ge1\), \(G_{s/2}\) a gamma variable of shape \(s/2\) and scale one independent of \(M_1\) and \(\chi_{12}\) the real primitive character modulo \(12\) (equal to \(1\) at \(\pm1\) and to \(-1\) at \(\pm5\) modulo \(12\)), \(L(s,\chi_{12})=\Pr\big(12G_{s/2}/(\pi M_1)>1\big)\) for real \(s>0\), where \(12G_{s/2}/(\pi M_1)\) is infinitely divisible for \(s\le4\), and is a mixture of exponential laws for \(s\le2\), in particular at the centre \(s=\frac12\).

Earlier editions: Proposition 8.5.7, page 274 (edition 1).

Source: ch/ch16.tex, label prop:maxexp:krist.

RH is equivalent to a Gamma(k) scale mixture form of the clock canonical function, and the Kolmogorov law

Volume I, Remark 15.1.8, page 410 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(W(t)=\sum_{\operatorname{Re}\gamma>0}e^{-t\gamma^2}\) be the canonical function of the clock \(T\), whose Lévy density is \(W(t)/t\), the sum running over the zeros \(\gamma\) of \(\xi(\frac12+iz)\) with multiplicity. For every integer \(k\ge1\), the Riemann hypothesis holds if and only if \(t^{k-1}W(t)\) is proportional to a scale mixture of Gamma\((k)\) densities. The proof is that RH is equivalent to the complete monotonicity of \(W\), that is \(W(t)=\int e^{-tz}U(dz)\) with \(U\ge0\) (Bernstein’s theorem), and that a Gamma\((k)\) scale mixture divided by \(t^{k-1}\) is completely monotone. Under RH the law with density proportional to \(t^{k-1}W(t)\) is that of \(G_k/\gamma_J^2\) with \(G_k\) a Gamma\((k)\) variable independent of \(J\) and \(\Pr(J=\gamma)\) proportional to \(\gamma^{-2k}\). For \(k=1\) it is a mixture of exponential laws, infinitely divisible by the theorem of Goldie and Steutel; for \(k=2\) it is the Kolmogorov law of \(T\) with density \(tW(t)/\operatorname{Var}T\), infinitely divisible by Kristiansen’s theorem. Neither is a GGC. Kristiansen’s theorem applies only once the Gamma(2) mixture is known, which is RH itself. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Remark 8.5.8, page 275 (edition 1).

Source: ch/ch16.tex, label rem:maxexp:kolmogorov.

Laplace transform of the maximum M1 of exponentials in closed form

Volume I, Proposition 15.1.9, page 411 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M_1\) be the maximum of independent exponential random variables, one of each rate \(2\pi n\), \(n\ge1\), so that \(\Pr(M_1\le y)=\prod_{n\ge1}(1-e^{-2\pi ny})\) for \(y>0\), and put \(b_n=\pi(n^2-1)/12\). For \(\theta>0\) let \(w=\sqrt{1-12\theta/\pi}\), positive or purely imaginary. Then \(\mathbb{E}e^{-\theta M_1}=\frac{\pi(1-w^2)\sin(\pi w/3)}{2\sqrt3\,w\cos(\pi w/2)}=\prod_{k\ge1}\frac{1+\theta/b_{6k}}{(1+\theta/b_{6k-1})(1+\theta/b_{6k+1})}\).

Earlier editions: Proposition 8.5.9, page 276 (edition 1).

Source: ch/ch16.tex, label prop:ch16:M1.

A discrete mixture of Gamma(k) laws with two rates is not a GGC

Volume I, Result 15.1.8, page 411 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(k\) be a positive integer and \(\varphi(z)=\sum_nw_n(1+z/\lambda_n)^{-k}\) the Laplace transform of a mixture of Gamma\((k)\) laws with rates \(\lambda_1<\lambda_2<\cdots\) (finitely many, or tending to infinity), at least two of them, and weights \(w_n>0\) summing to one. Then the law is not a generalized gamma convolution. The proof: \(\varphi\) is meromorphic with a pole of order \(k\) at each \(-\lambda_n\); if it were a GGC transform, \(-\varphi'/\varphi=a+\int U(dt)/(z+t)\) with \(a\ge0\) and \(U\ge0\) would be holomorphic off \((-\infty,0]\) with poles of positive residue, so \(\varphi\) would have no zeros, \(U=k\sum_n\delta_{\lambda_n}\) and \(\varphi(z)=e^{-az}\prod_n(1+z/\lambda_n)^{-k}\le(1+z/\lambda_1)^{-k}(1+z/\lambda_2)^{-k}\) for \(z>0\), contradicting \(\varphi(z)\ge w_1(1+z/\lambda_1)^{-k}\) as \(z\to\infty\). Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Result 8.5.8, page 275 (edition 1).

Source: ch/ch16.tex, label res:maxexp:nonggc.

The completed zeta function as the Mellin transform of an infinitely divisible Gamma(2) mixture, not a GGC

Volume I, Result 15.1.8, page 411 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(N\) have the zeta law \(\Pr(N=n)=n^{-4}/\zeta(4)\), let \(G_2\) be a Gamma(2) variable independent of \(N\), and put \(Q=G_2/(\pi N^2)\). Then \(Q\) has density \(\frac{\pi^2}{\zeta(4)}x\omega(x)\) with \(\omega(x)=\sum_{n\ge1}e^{-\pi n^2x}\), \(\mathbb{E}Q^q=\Gamma(q+2)\pi^{-q}\zeta(2q+4)/\zeta(4)\), and for \(\operatorname{Re}s>1\), \(\pi^{-s/2}\Gamma(\frac s2)\zeta(s)=\frac{\pi^2}{90}\mathbb{E}[Q^{s/2-2}]\) and \(\xi(s)=\frac{\pi^2}{180}s(s-1)\mathbb{E}[Q^{s/2-2}]\), a normalisation of Riemann’s integral \(\int_0^\infty x^{s/2-1}\omega(x)dx\); beyond \(\operatorname{Re}s>1\) the identities hold after continuation, not as expectations. So \(Q\) is a scale mixture of Gamma(2) laws and infinitely divisible by Kristiansen’s theorem, and, being a discrete mixture of Gamma(2) laws with the rates \(\pi n^2\), not a GGC. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Result 8.5.8, page 275 (edition 1).

Source: ch/ch16.tex, label res:maxexp:zetaq.

Rényi type decomposition of the maximum M1 of exponentials

Volume I, Corollary 15.1.10, page 412 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M_1\) be the maximum of independent exponential random variables, one of each rate \(2\pi n\), \(n\ge1\), and \(b_n=\pi(n^2-1)/12\). Then \(M_1\overset{d}{=}\sum_{k\ge1}Y_k\) with independent \(Y_k\), where \(Y_k\) is exponential of rate \(b_{6k-1}\) with probability \(p_k\) and of rate \(b_{6k+1}\) with probability \(1-p_k\), and \(p_k=\big(1-\frac{b_{6k-1}}{b_{6k}}\big)\frac{b_{6k+1}}{b_{6k+1}-b_{6k-1}}=\frac{(12k-1)(3k+1)}{2(36k^2-1)}\), so that \(p_1=\frac{22}{35}\), \(p_2=\frac{161}{286}\), and \(p_k\downarrow\frac12\).

Earlier editions: Corollary 8.5.10, page 276 (edition 1).

Source: ch/ch16.tex, label cor:maxexp:renyi.

Lévy density and Bondesson function of the maximum M1

Volume I, Proposition 15.1.12, page 413 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M_1\) be the maximum of independent exponential random variables, one of each rate \(2\pi n\), \(n\ge1\), and \(b_n=\pi(n^2-1)/12\). Put \(a(n)=1\) for \(n\equiv\pm1\pmod6\), \(a(n)=-1\) for \(6\mid n\) and \(a(n)=0\) otherwise, \(k_1(t)=\sum_{n\ge2}a(n)e^{-b_nt}\), so that \(-\log\mathbb{E}e^{-\theta M_1}=\int_0^\infty(1-e^{-\theta t})\,\nu_1(t)\,dt\) with the Lévy density \(\nu_1(t)=k_1(t)/t\) (the canonical function of \(M_1\) is \(k_1\)), and let \(U_1=\sum_{n\ge2}a(n)\delta_{b_n}\) be the signed Thorin measure of \(M_1\) and \(N_1(z)=U_1((0,z])\) its Bondesson function. Then \(k_1(t)=\frac12e^{\pi t/12}\Big(1+\frac{\vartheta_0-3\vartheta_{1/2}-4\vartheta_{1/3}}{\sqrt{3t}}\Big)-1\) with \(\vartheta_\alpha=\sum_{m\in\mathbb{Z}}e^{-12\pi(m+\alpha)^2/t}\), so that \(\nu_1(t)=(2\sqrt3)^{-1}t^{-3/2}-(2t)^{-1}+O(t^{-1/2})\) as \(t\downarrow0\) and \(\nu_1(t)\sim e^{-2\pi t}/t\) as \(t\to\infty\). Moreover \(N_1(z)\ge0\) for all \(z>0\), and \(N_1(z)\sim\frac13\sqrt{3z/\pi}\) as \(z\to\infty\).

Earlier editions: Proposition 8.5.12, page 278 (edition 1).

Source: ch/ch16.tex, label prop:maxexp:levy.

Dictionary between a signed Thorin measure, the Levy density and the modulus of the characteristic function

Volume I, Lemma 15.1.11, page 413 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\ge0\) have Laplace exponent \(-\log\mathbb{E}e^{-\theta X}=\int_{(0,\infty)}\log(1+\theta/z)\,U(dz)\) for a signed measure \(U\) with \(\int\log(1+1/z)\,|U|(dz)<\infty\), and put \(N(z)=U((0,z])\), with \(N(z)\log(1+u^2/z^2)\to0\) as \(z\to\infty\) for real \(u\). Then \(-\log\mathbb{E}e^{-\theta X}=\int_0^\infty(1-e^{-\theta t})\nu(t)\,dt\) for \(\theta\ge0\) with the signed function \(\nu(t)=t^{-1}\int e^{-zt}U(dz)=\int_0^\infty e^{-zt}N(z)\,dz\), and for real \(u\), \(-\log|\mathbb{E}e^{iuX}|^2=\int_0^\infty4\sin^2(ut/2)\nu(t)\,dt=\int\log(1+u^2/z^2)\,U(dz)=\int_0^\infty N(z)\frac{2u^2}{z(z^2+u^2)}dz\). The law of \(X\) is a generalised gamma convolution if and only if \(U\ge0\), and it lies in the Bondesson class BO (it is infinitely divisible with a completely monotone Levy density) if and only if \(N\ge0\).

Source: ch/ch16.tex, label lem:maxexp:dict.

Thorin measure of M1 as a difference of two GGC Thorin measures

Volume I, Theorem 15.1.13, page 414 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(b_n=\pi(n^2-1)/12\), let \(E_n\) and \(E'_m\) be independent standard exponential variables, \(S=\sum_{n\text{ odd},\,n\ge3}E_n/b_n\) and \(Z=\sum_{m\ge1}E'_m/b_{3m}\), with \(Z\) independent of \(M_1\), the maximum of independent exponential random variables, one of each rate \(2\pi n\), \(n\ge1\). Then \(S\) and \(Z\) are GGCs (generalised gamma convolutions: laws on \([0,\infty)\) with Laplace transform \(\exp(-d\theta-\int\log(1+\theta/z)\,U(dz))\) for a drift \(d\ge0\) and a positive measure \(U\), the Thorin measure), with \(\mathbb{E}e^{-\theta S}=\frac\pi4\,\frac{1-w^2}{\cos(\pi w/2)}\) and \(\mathbb{E}e^{-\theta Z}=\frac{\sqrt3\,w/2}{\sin(\pi w/3)}\), where \(w=\sqrt{1-12\theta/\pi}\), and \(M_1+Z\overset{d}{=}S\). Equivalently, the signed Thorin measure of \(M_1\), the signed measure \(U_1=\sum_{n\ge2}a(n)\delta_{b_n}\) with \(-\log\mathbb{E}e^{-\theta M_1}=\int\log(1+\theta/z)\,U_1(dz)\), where \(a(n)=1\) for \(n\equiv\pm1\pmod6\), \(a(n)=-1\) for \(6\mid n\) and \(a(n)=0\) otherwise, equals \(U_S-U_Z\) with \(U_S=\sum_{n\text{ odd},\,n\ge3}\delta_{b_n}\) and \(U_Z=\sum_{m\ge1}\delta_{b_{3m}}\), the atoms at \(b_{3m}\) with \(m\) odd cancelling.

Earlier editions: Theorem 8.5.13, page 279 (edition 1).

Source: ch/ch16.tex, label thm:maxexp:ggc.

Certified zeros of the Laplace transform of the maximum M* of exponentials

Volume I, Proposition 15.1.14, page 415 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M^*\) be the maximum of independent exponential random variables, \(24\) of each rate \(2\pi n\), \(n\ge1\), let \(\tau(n)\) be Ramanujan’s tau function (the coefficients of \(\Delta=\eta^{24}\)), and let \(\psi^*(\theta)=\mathbb{E}e^{-\theta M^*}\), which extends to a meromorphic function of \(\theta\in\mathbb{C}\) with poles only at \(\theta=-2\pi m\), \(m\ge1\) (simple, with residue \(-2\pi m\,\tau(m+1)\)). (1) For each of the \(25\) values \(m\le60\) with \(\tau(m+1)\tau(m+2)>0\), \(\psi^*\) changes sign in an interval of length \(2\times10^{-20}\) inside the gap \((-2\pi(m+1),-2\pi m)\); the zeros begin \(\theta/2\pi=-5.48830041\), \(-8.66373325\), \(-11.4203774\), \(-13.2533839\), \(-14.7973058\), and the last is \(-60.7670708\). (2) \(\psi^*\) has exactly one zero in the disc of radius \(10^{-7}\) about each of the ten points \(-155.5940822775511402804\pm2.7631188861212609656\,i\), \(-215.6211277570694988944\pm2.9647292148515237348\,i\), \(-307.4008549057072052743\pm3.2283287024583906394\,i\), \(-322.6135369899940182007\pm3.0449254921259967464\,i\), \(-347.8816009468099010890\pm2.7171007701068929577\,i\), and it lies within \(10^{-19}\) of that point.

Earlier editions: Proposition 8.5.14, page 280 (edition 1).

Source: ch/ch16.tex, label prop:maxexp:psizeros.

The maximum M* of exponentials is not in the Bondesson class

Volume I, Corollary 15.1.15, page 416 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M^*\) be the maximum of independent exponential random variables, \(24\) of each rate \(2\pi n\), \(n\ge1\). Then \(M^*\) is not in the Bondesson class BO (the infinitely divisible laws on \([0,\infty)\) with a completely monotone Lévy density, and a drift \(d\ge0\)); in particular \(M^*\) is not a generalised gamma convolution.

Earlier editions: Corollary 8.5.15, page 281 (edition 1).

Source: ch/ch16.tex, label cor:maxexp:notBO.

Open question on infinite divisibility of maxima of exponentials

Volume I, Question 15.1.16, page 418 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is every maximum of independent exponential random variables infinitely divisible? In particular, is \(M^*\) infinitely divisible, where \(M^*\) is the maximum of independent exponential random variables, \(24\) of each rate \(2\pi n\), \(n\ge1\)?

Earlier editions: Question 8.5.16, page 282 (edition 1).

Source: ch/ch16.tex, label q:maxexp:id.

Open question on real zeros of cosh kernels with exponential multipliers

Volume I, Question 15.1.17, page 420 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For which \(a>0\), \(c_m\ge0\) and \(\beta_m>0\) does the Fourier transform of \(e^{-a\cosh u}\exp\Big(-\sum_mc_m\big(e^{-\beta_me^u}+e^{-\beta_me^{-u}}\big)\Big)\) have only real zeros? (The kernels of the completed L functions of Ramanujan’s cusp form, of the character \(\chi_{12}\) and of the elliptic curve of conductor \(11\) have this form.)

Earlier editions: Question 8.5.17, page 285 (edition 1).

Source: ch/ch16.tex, label q:maxexp:mult.

Thorin flow of the de Bruijn Newman deformation of the completed L function of Delta

Volume I, Corollary 15.1.18, page 422 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Delta\) be Ramanujan’s cusp form with \(L(s,\Delta)=\sum\tau(n)n^{-s}\), let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) and let \(K_\Delta\) be the kernel with \(\Lambda(s)=\int_{\mathbb{R}}K_\Delta(u)e^{(s-6)u}du\). For real \(t\) let \(\Xi_t(\theta)=\int_{\mathbb{R}}e^{tu^2}K_\Delta(u)e^{i\theta u}du\), which solves the backward heat equation \(\partial_t\Xi_t=-\partial_\theta^2\Xi_t\) with \(\Xi_0(\theta)=\Lambda(6+i\theta)\); it is real, even and entire of order one for every real \(t\). Let \(t\) be such that all zeros of \(\Xi_t\) are real, as holds for \(t\ge\frac18\) by de Bruijn’s theorem, for \(t\ge\Lambda_\Delta\) (the de Bruijn Newman constant of \(\Delta\)), and at \(t=0\) exactly under the Riemann Hypothesis for \(L(s,\Delta)\). Let \(0<\gamma_1(t)\le\gamma_2(t)\le\cdots\) be the positive zeros with multiplicity. Then \(\Xi_t(0)/\Xi_t(-i\sqrt s)=\prod_{j\ge1}(1+s/\gamma_j(t)^2)^{-1}\), the Laplace transform of a generalised gamma convolution with Thorin measure \(\sum_j\delta_{\gamma_j(t)^2}\). The proof is Hadamard’s factorisation of an even entire function of order one with real zeros, with no exponential factor by evenness.

Source: ch/ch16.tex, label cor:ch16:flowthorin.

Equation of motion of the zeros of the flowed completed L function of Delta, Dyson and Laguerre form

Volume I, Proposition 15.1.19, page 423 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With \(\Xi_t\) the flowed completed L function of Ramanujan’s \(\Delta\) (even, real, entire of order one, solving \(\partial_t\Xi_t=-\partial_\theta^2\Xi_t\) with \(\Xi_0(\theta)=\Lambda(6+i\theta)\), \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\)) and \(\Lambda_\Delta\) its de Bruijn Newman constant, for \(t>\Lambda_\Delta\) the zeros of \(\Xi_t\) are real and simple (Csordas, Smith and Varga), depend analytically on \(t\), and satisfy \(\frac{d\gamma_j}{dt}=2\sum_{k\ne j}\frac1{\gamma_j-\gamma_k}\), the sum over all zeros \(\pm\gamma_k\) in symmetric order. In the squared variables \(x_j=\gamma_j^2\) of the positive zeros, \(\frac{dx_j}{dt}=2+8x_j\sum_{k\ne j,\ k\ge1}\frac1{x_j-x_k}\). The proof differentiates \(\Xi_t(\gamma_j(t))=0\) along the backward heat equation and uses the Hadamard product. The identification of the second equation with a Wishart or squared Bessel eigenvalue flow is formal and is not claimed.

Source: ch/ch16.tex, label prop:ch16:ode.

Polya approximant of the completed L function of Delta, only real zeros and a GGC reciprocal

Volume I, Proposition 15.1.20, page 425 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_s\) denote the modified Bessel function of the second kind and \(\Xi^*(\theta)=\Lambda^*(6+i\theta)=4\operatorname{Re}K_{6+i\theta}(4\pi)\), where \(\Lambda^*(s)=\int_{\mathbb{R}}2e^{-4\pi\cosh u}\cosh(6u)\,e^{(s-6)u}du=2K_s(4\pi)+2K_{12-s}(4\pi)\). Then \(\Xi^*\) has only real zeros; its zero counting function has the leading asymptotics \(\frac T\pi\log\frac T{2\pi e}\) of that of the completed \(L\) function of Ramanujan’s \(\Delta\) on the line \(\operatorname{Re}s=6\); and \(\Lambda^*(6)/\Lambda^*(6+\sqrt s)\) is the Laplace transform of a generalised gamma convolution with Thorin measure \(\sum_j\delta_{\gamma_j^{*2}}\), where \(\pm\gamma^*_j\) are the zeros of \(\Xi^*\). The proof uses Polya’s theorem that \(\theta\mapsto K_{i\theta}(a)\) has only real zeros for \(a>0\) and that \(\cosh(bu)\) is a universal factor. The zeros themselves, the first at \(3.4883146\), were computed.

Source: ch/ch16.tex, label prop:ch16:polya.

Pólya kernel of an imaginary quadratic field

Volume I, Theorem 15.1.21, page 426 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(d<0\) be a fundamental discriminant, \(K=\mathbb{Q}(\sqrt d)\), \(\chi_d\) the Kronecker symbol of \(d\) (a real primitive odd character modulo \(|d|\)), \(h\) the class number of \(K\) and \(w\) the number of roots of unity in \(K\). Let \(\zeta_K(s)=\zeta(s)L(s,\chi_d)=\sum_{n\ge1}a_nn^{-s}\), where \(a_n=\sum_{m\mid n}\chi_d(m)\ge0\) is the number of ideals of norm \(n\), and put \(\Lambda_K(s)=(\sqrt{|d|}/2\pi)^s\Gamma(s)\zeta_K(s)\), \(\xi_K(s)=2s(s-1)\Lambda_K(s)\), \(\vartheta_K(x)=\sum_{n\ge1}a_ne^{-2\pi nx/\sqrt{|d|}}\), \(\Psi(y)=y(y-2)e^{-y}\), \(H_K(x)=(x^2\vartheta_K'(x))'\) and \(\Phi_K(u)=2e^{u/2}H_K(e^u)\). Then (i) \(\vartheta_K(1/x)=x\,\vartheta_K(x)+\frac hw(x-1)\) for \(x>0\). (ii) \(H_K(x)=\sum_{n\ge1}a_n\Psi(2\pi nx/\sqrt{|d|})\) and \(H_K(1/x)=x\,H_K(x)\); the kernel \(\Phi_K\) is even, analytic in the strip \(|\operatorname{Im}u|<\frac\pi2\) and decays doubly exponentially. (iii) \(\xi_K(\frac12+s)=\int_{\mathbb{R}}\Phi_K(u)e^{su}\,du\) for every \(s\in\mathbb{C}\); in particular \(\xi_K(\frac12+i\theta)=\int_{\mathbb{R}}\Phi_K(u)\cos(\theta u)\,du\). (iv) \(\Phi_K>0\) on \(\mathbb{R}\) if and only if \(H_K>0\) on \([1,\sqrt{|d|}/\pi]\); in particular \(\Phi_K>0\) when \(|d|<\pi^2\), that is for \(d=-3,-4,-7,-8\). (v) If \(\Phi_K>0\), then \(\zeta_K(\frac12)<0\) and \(L(\frac12,\chi_d)>0\); in particular \(L(\frac12,\chi_d)>0\) for \(d=-3,-4,-7,-8\).

Earlier editions: Theorem 8.5.18, page 287 (edition 1).

Source: ch/ch16.tex, label thm:sqfree:kernelK.

Pólya kernel of an imaginary quadratic field takes negative values when 2 and 3 are inert

Volume I, Theorem 15.1.22, page 427 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For a fundamental discriminant \(d<0\) let \(K=\mathbb{Q}(\sqrt d)\), \(\chi_d\) the Kronecker symbol of \(d\), \(a_n=\sum_{m\mid n}\chi_d(m)\) the number of ideals of norm \(n\) in \(K\), \(\vartheta_K(x)=\sum_{n\ge1}a_ne^{-2\pi nx/\sqrt{|d|}}\), \(H_K(x)=(x^2\vartheta_K'(x))'\) and \(\Phi_K(u)=2e^{u/2}H_K(e^u)\) the Pólya kernel of \(K\) (the kernel with \(\xi_K(\frac12+s)=\int_{\mathbb{R}}\Phi_K(u)e^{su}\,du\) for the completed Dedekind zeta function \(\xi_K\)), and let \(K_{\chi_d}(u)=2e^{3u/2}\sum_{n\ge1}n\chi_d(n)e^{-\pi n^2e^{2u}/|d|}\) be the kernel of the odd character \(\chi_d\), with \(\xi(\frac12+s,\chi_d)=\int_{\mathbb{R}}K_{\chi_d}(u)e^{su}\,du\) for \(\xi(s,\chi_d)=(|d|/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_d)\). Let \(d<0\) be a fundamental discriminant with \(\chi_d(2)=\chi_d(3)=-1\), that is \(d\equiv5\pmod{24}\). Then \(H_K(x)<0\) at \(x=1.42\sqrt{|d|}/2\pi\), so \(\Phi_K\) takes negative values, and so does the kernel \(K_{\chi_d}\). There are infinitely many such \(d\), for instance \(d=-p\) for every prime \(p\equiv19\pmod{24}\).

Earlier editions: Theorem 8.5.19, page 289 (edition 1).

Source: ch/ch16.tex, label thm:sqfree:inert.

Open question on the density of positive Pólya kernels

Volume I, Question 15.1.23, page 429 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a fundamental discriminant \(d<0\) let \(\Phi_K\) be the Pólya kernel of \(K=\mathbb{Q}(\sqrt d)\), the even kernel with \(\xi_K(\frac12+s)=\int_{\mathbb{R}}\Phi_K(u)e^{su}\,du\) for the completed Dedekind zeta function \(\xi_K(s)=2s(s-1)(\sqrt{|d|}/2\pi)^s\Gamma(s)\zeta_K(s)\). Does the proportion of fundamental discriminants \(-X\le d<0\) with \(\Phi_K>0\) tend to a limit as \(X\to\infty\), and is the limit positive? Is \(\Phi_K\) positive for infinitely many \(d\), for instance among the \(d\) in which \(2\) and \(3\) split (that is, \(\chi_d(2)=\chi_d(3)=1\) for the Kronecker symbol \(\chi_d\)), where the proportion up to \(20000\) is \(676/767\)?

Earlier editions: Question 8.5.20, page 290 (edition 1).

Source: ch/ch16.tex, label q:sqfree:density.

Open question on positivity criteria for real quadratic fields

Volume I, Question 15.1.24, page 429 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a number field \(K\) with \(r_1\) real and \(r_2\) complex places the completed Dedekind zeta function \(\xi_K\) has the gamma factor \(\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\), with \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(\frac s2)\) and \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\), and its Pólya kernel is the kernel \(\Phi_K\) with \(\xi_K(\frac12+s)=\int_{\mathbb{R}}\Phi_K(u)e^{su}\,du\). For real quadratic fields and for fields of higher degree the gamma factor replaces the exponentials in the theta series of an imaginary quadratic field by other kernels, for \(\Gamma_{\mathbb{R}}(s)^2\) by a Bessel function \(K_0\). For an imaginary quadratic field of discriminant \(d\), \(\Phi_K>0\) on \(\mathbb{R}\) if and only if \(H_K>0\) on \([1,\sqrt{|d|}/\pi]\) for an explicit function \(H_K\) of one variable. Is there a criterion for the positivity of the Pólya kernel of \(\xi_K\) for real quadratic fields and for fields of higher degree that is as simple as this one?

Earlier editions: Question 8.5.21, page 290 (edition 1).

Source: ch/ch16.tex, label q:sqfree:realquad.

Even positive analytic kernel of the completed Dirichlet L function modulo four from the cube of the Dedekind eta function

Volume I, Proposition 15.1.25, page 430 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\eta\) be the Dedekind eta function, \(\chi_4\) the nontrivial character modulo \(4\), and \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\). The kernel \(K_4(u)=2e^{3u/2}\eta(ie^{2u})^3\) satisfies \(\xi(\frac12+s,\chi_4)=\int_{\mathbb{R}}K_4(u)e^{su}du\) for every complex \(s\). It is even, positive and analytic in the strip \(|\operatorname{Im}u|<\pi/4\), it has the symmetric product form \(K_4(u)=2\exp\big(-\frac\pi4\cosh2u\big)\prod_{m\ge1}\big(1-e^{-2\pi me^{2u}}\big)^{3/2}\big(1-e^{-2\pi me^{-2u}}\big)^{3/2}\), so that \(K_4(u)=2\exp\big(\frac32|u|-\frac\pi4e^{2|u|}\big)\big(1+O(e^{-2\pi e^{2|u|}})\big)\), and \(K_4(0)=2\eta(i)^3=\Gamma(\frac14)^3/(4\pi^{9/4})=0.906767655167731\).

Source: ch/ch16.tex, label prop:dbeta:kernel.

Lévy form of the Dirichlet beta function at basepoints from one

Volume I, Proposition 15.1.27, page 432 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(L(s,\chi_4)=\sum_{n\ge0}(-1)^n(2n+1)^{-s}\) be the Dirichlet beta function (\(\chi_4\) the nonprincipal character modulo \(4\)), \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completed function (entire, with \(\xi(s,\chi_4)=\xi(1-s,\chi_4)\)) and \(\Lambda\) the von Mangoldt function. For \(\alpha\ge1\), \(\log\frac{\xi(\alpha+s,\chi_4)}{\xi(\alpha,\chi_4)}=b_\alpha s+\int_0^\infty\big(e^{-sx}-1+sx\big)\,\Pi_\alpha(dx)\) with \(b_\alpha=\frac{\xi'}{\xi}(\alpha,\chi_4)\) and the signed Lévy measure \(\Pi_\alpha(dx)=\frac{e^{-\alpha x}}{2x\sinh x}\,dx+\sum_{n\ge2}\frac{\chi_4(n)\Lambda(n)}{n^\alpha\log n}\,\delta_{\log n}(dx)\), whose canonical measure \(x\Pi_\alpha(dx)\) has the positive density \(e^{-\alpha x}/(2\sinh x)\) and the atoms \(\chi_4(n)\Lambda(n)n^{-\alpha}\) at \(\log n\). For \(\alpha>1\) the form holds for \(\operatorname{Re}s>1-\alpha\), with absolute convergence. At \(\alpha=1\) it holds for \(\operatorname{Re}s>0\), the prime sum converging conditionally, as a limit over \(n\le X\).

Earlier editions: Proposition 8.5.24, page 293 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:levy.

Thorin measure of the Dirichlet beta function from basepoint one

Volume I, Proposition 15.1.28, page 435 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(L(s,\chi_4)=\sum_{n\ge0}(-1)^n(2n+1)^{-s}\) be the Dirichlet beta function, \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completed function and \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta,\chi_4)\). For every \(\alpha\ge1\), \(u\mapsto\xi(\alpha,\chi_4)/\xi(\alpha+\sqrt u,\chi_4)\) is the Laplace transform of a generalised gamma convolution whose Thorin measure has, in the chart \(\theta=\sqrt z\), the positive density \(v_\alpha(\theta)/\pi\) (that is, it is the image of \(v_\alpha(\theta)\,d\theta/\pi\) under \(\theta\mapsto\theta^2\)), and \(\log\frac{\xi(\alpha+\sqrt u,\chi_4)}{\xi(\alpha,\chi_4)}=\frac1\pi\int_0^\infty\log\Big(1+\frac u{\theta^2}\Big)v_\alpha(\theta)\,d\theta\) for \(u\ge0\). For \(\alpha>\frac12\) it is the Laplace transform of a generalised gamma convolution if and only if no zero of \(L(s,\chi_4)\) has real part greater than \(\alpha\). GRH for \(\chi_4\) holds if and only if \(v_\alpha\ge0\) for every \(\alpha>\frac12\), and the Thorin measures at resolution \(\varepsilon=\alpha-\frac12\) converge vaguely, as \(\varepsilon\downarrow0\), to the sum of the point masses \(\delta_{\gamma^2}\) over the zeros \(\frac12+i\gamma\) of \(L(s,\chi_4)\) on the critical line (under GRH this sum is the Thorin measure of \(\xi(\frac12,\chi_4)/\xi(\frac12+\sqrt u,\chi_4)\)).

Earlier editions: Proposition 8.5.25, page 295 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:thorin.

Termwise positivity thresholds for the Dirichlet beta function and chi12

Volume I, Proposition 15.1.29, page 435 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) be the completed Dirichlet beta function, \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta,\chi_4)\), \(\psi=\Gamma'/\Gamma\) and \(\Lambda\) the von Mangoldt function. For \(\alpha>1\) and real \(\theta\), \(v_\alpha(\theta)\ge A^\infty_4(\alpha,\theta)-S_4(\alpha)\), where \(A^\infty_4(\alpha,\theta)=\frac12\log\frac4\pi+\frac12\operatorname{Re}\psi\Big(\frac{\alpha+1+i\theta}2\Big)\) and \(S_4(\alpha)=\sum_{n\text{ odd}}\frac{\Lambda(n)}{n^\alpha}=-\frac{\zeta'}\zeta(\alpha)-\frac{\log2}{2^\alpha-1}\), and \(A^\infty_4(\alpha,\theta)\) increases in \(|\theta|\). The bound is positive at every \(\theta\) exactly when \(\alpha>\alpha_c(\chi_4)=2.2823106368\), the root of \(A^\infty_4(\alpha,0)=S_4(\alpha)\). For the real primitive character \(\chi_{12}\) modulo \(12\), with \(\xi(s,\chi_{12})=(12/\pi)^{s/2}\Gamma(\frac s2)L(s,\chi_{12})\) and \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta,\chi_{12})\), the same bound with \(A^\infty_{12}(\alpha,\theta)=\frac12\log\frac{12}\pi+\frac12\operatorname{Re}\psi\Big(\frac{\alpha+i\theta}2\Big)\) and \(S_{12}(\alpha)=S_4(\alpha)-\frac{\log3}{3^\alpha-1}\) is positive at every \(\theta\) exactly when \(\alpha>\alpha_c(\chi_{12})=1.8348743295\).

Earlier editions: Proposition 8.5.26, page 296 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:termwise.

No bound from the sizes of the prime weights beats the absolute values

Volume I, Proposition 15.1.30, page 436 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(L(s,\chi_4)=\sum_{n\ge0}(-1)^n(2n+1)^{-s}\) be the Dirichlet beta function, \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completed function, \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta,\chi_4)\), \(A^\infty_4(\alpha,\theta)=\frac12\log\frac4\pi+\frac12\operatorname{Re}\psi\big(\frac{\alpha+1+i\theta}2\big)\) with \(\psi=\Gamma'/\Gamma\), and \(S_4(\alpha)=\sum_{n\text{ odd}}\Lambda(n)n^{-\alpha}\) with \(\Lambda\) the von Mangoldt function. For \(\alpha>1\) and every \(T>0\), \(\inf_{\theta\ge T}\big(v_\alpha(\theta)-A^\infty_4(\alpha,\theta)\big)=\inf_{\theta\ge T}\operatorname{Re}\frac{L'}L(\alpha+i\theta,\chi_4)=-S_4(\alpha)\), so no lower bound for \(\operatorname{Re}\frac{L'}L(\alpha+i\theta,\chi_4)\) better than \(-S_4(\alpha)\) holds at all large heights. The bound \(v_\alpha\ge A^\infty_4-S_4\) yields \(v_\alpha(\theta)>0\) exactly for \(|\theta|>t_0(\alpha)\), where \(A^\infty_4(\alpha,t_0)=S_4(\alpha)\), for \(1<\alpha\le\alpha_c\) (so that \(t_0(\alpha_c)=0\)), and at every real \(\theta\) for \(\alpha>\alpha_c\), with \(\alpha_c=2.2823106368\) the threshold at which \(A^\infty_4(\alpha,0)=S_4(\alpha)\), and \(\log t_0(\alpha)=\frac2{\alpha-1}-2\gamma_E-\log\frac8\pi+O(\alpha-1)\) as \(\alpha\downarrow1\), where \(\gamma_E\) is Euler’s constant and \(-2\gamma_E-\log\frac8\pi=-2.0891\).

Earlier editions: Proposition 8.5.27, page 297 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:kron.

Thorin density of the Dirichlet beta function as a difference over the Gaussian field

Volume I, Proposition 15.1.31, page 437 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), let \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) be the completed Dirichlet beta function, \(K=\mathbb{Q}(i)\) with \(\zeta_K(s)=\zeta(s)L(s,\chi_4)\), and \(\xi_K(s)=2s(s-1)\pi^{-s}\Gamma(s)\zeta_K(s)=\xi(s)\,\xi(s,\chi_4)\). Write \(v^K_\alpha(\theta)=\operatorname{Re}\frac{\xi_K'}{\xi_K}(\alpha+i\theta)\), \(v^\xi_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\) and \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta,\chi_4)\). On every vertical line, \(v_\alpha=v^K_\alpha-v^\xi_\alpha\). For \(\alpha>1\) the canonical measure of the Lévy form of \(\xi_K\) at the basepoint \(\alpha\) is \(x\Pi^K_\alpha(dx)=\Big(\frac{e^{-(\alpha+1)x}}{1-e^{-x}}-e^{-(\alpha-1)x}\Big)dx+\sum_{n\ge2}\frac{(1+\chi_4(n))\Lambda(n)}{n^\alpha}\,\delta_{\log n}(dx)\), and the canonical measure \(x\Pi_\alpha\) of the Lévy form of \(\xi(\cdot,\chi_4)\) at \(\alpha\) (with density \(e^{-\alpha x}/(2\sinh x)\) and atoms \(\chi_4(n)\Lambda(n)n^{-\alpha}\) at \(\log n\)) equals \(x\Pi^K_\alpha-x\Pi^\xi_\alpha\), where \(x\Pi^\xi_\alpha(dx)=\Big(\frac{e^{-(\alpha+2)x}}{1-e^{-2x}}-e^{-(\alpha-1)x}\Big)dx+\sum_{n\ge2}\frac{\Lambda(n)}{n^\alpha}\,\delta_{\log n}(dx)\) is that of \(\xi\): the pole densities \(-e^{-(\alpha-1)x}\) cancel, the gamma densities differ by \(e^{-\alpha x}/(2\sinh x)\), and the atoms by \(\chi_4(n)\Lambda(n)n^{-\alpha}\). Here \(\Lambda\) is the von Mangoldt function.

Earlier editions: Proposition 8.5.28, page 298 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:factor.

GRH for chi4 and domination of Thorin densities over Q(i)

Volume I, Corollary 15.1.32, page 437 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Part (iii) assumes the Riemann hypothesis for the Riemann zeta function; parts (i) and (ii) hold without any hypothesis.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) the completed Dirichlet beta function, \(\xi_K(s)=\xi(s)\,\xi(s,\chi_4)\) the completed Dedekind zeta function of \(K=\mathbb{Q}(i)\), and \(v^K_\alpha(\theta)=\operatorname{Re}\frac{\xi_K'}{\xi_K}(\alpha+i\theta)\), \(v^\xi_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\). (i) For every \(\alpha\ge1\) there are independent GGC variables (generalised gamma convolutions) \(H^\xi_\alpha\) and \(H^{(4)}_\alpha\) with Laplace transforms \(\xi(\alpha)/\xi(\alpha+\sqrt u)\) and \(\xi(\alpha,\chi_4)/\xi(\alpha+\sqrt u,\chi_4)\), whose sum \(H^K_\alpha=H^\xi_\alpha+H^{(4)}_\alpha\) is the GGC with Laplace transform \(\xi_K(\alpha)/\xi_K(\alpha+\sqrt u)\) and Thorin density \(v^K_\alpha/\pi\) (in the chart \(\theta=\sqrt z\)). (ii) GRH for \(\chi_4\) holds if and only if \(v^K_\alpha\ge v^\xi_\alpha\) on every line \(\operatorname{Re}s=\alpha>\frac12\): the Thorin density of \(\xi_K\) dominates that of \(\xi\) at every basepoint. (iii) Under the Riemann hypothesis, GRH for \(\chi_4\) holds if and only if, for every \(\alpha>\frac12\), \(\xi_K(\alpha)/\xi_K(\alpha+\sqrt u)\) is the Laplace transform of \(H^\xi_\alpha+H\) for some GGC \(H\) independent of \(H^\xi_\alpha\).

Earlier editions: Corollary 8.5.29, page 298 (edition 1).

Source: ch/ch16.tex, label cor:dbeta:dom.

Open question on a coupling construction over the Gaussian primes

Volume I, Question 15.1.33, page 438 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function, \(\xi(s,\chi_4)\) the completed Dirichlet beta function, \(\xi_K=\xi\cdot\xi(\cdot,\chi_4)\) the completed Dedekind zeta function of \(K=\mathbb{Q}(i)\), and for \(\alpha\ge1\) let \(H^\xi_\alpha\) and \(H^{(4)}_\alpha\) be independent generalised gamma convolutions with Laplace transforms \(\xi(\alpha)/\xi(\alpha+\sqrt u)\) and \(\xi(\alpha,\chi_4)/\xi(\alpha+\sqrt u,\chi_4)\), so that \(H^K_\alpha=H^\xi_\alpha+H^{(4)}_\alpha\) has Laplace transform \(\xi_K(\alpha)/\xi_K(\alpha+\sqrt u)\). Is there a construction of \(H^K_\alpha\) from the Gaussian primes themselves, two of norm \(p\) for each prime \(p\equiv1\pmod4\) and one of norm \(p^2\) for each prime \(p\equiv3\pmod4\), that persists below the basepoint one, that is for \(\frac12<\alpha<1\)? Such a construction must see the phases \(p^{-i\theta}\) of the primes, not only the sizes of the weights, and below \(\alpha=1\) its existence is equivalent, under the Riemann hypothesis, to GRH for \(\chi_4\).

Earlier editions: Question 8.5.30, page 299 (edition 1).

Source: ch/ch16.tex, label q:dbeta:coupling.

Thorin mass of the Dirichlet beta function as an argument

Volume I, Proposition 15.1.34, page 439 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(L(s,\chi_4)=\sum_{n\ge0}(-1)^n(2n+1)^{-s}\) be the Dirichlet beta function, \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completed function and \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta,\chi_4)\). Let \(\alpha>\frac12\) be such that no zero of \(\xi(s,\chi_4)\) lies on the line \(\operatorname{Re}s=\alpha\); this holds for every \(\alpha\ge1\). Then for \(\theta>0\), \(\frac1\pi\int_0^\theta v_\alpha(t)\,dt=\frac1\pi\arg\xi(\alpha+i\theta,\chi_4)\), with the argument continuous along the segment from \(\alpha\) to \(\alpha+i\theta\) and zero at \(\alpha\); for \(\alpha\ge1\) the left side is the mass of \((0,\theta^2]\) under the Thorin measure of \(u\mapsto\xi(\alpha,\chi_4)/\xi(\alpha+\sqrt u,\chi_4)\). As \(\theta\to\infty\), with \(a=\frac{\alpha+1}2\), \(\frac1\pi\arg\xi(\alpha+i\theta,\chi_4)=\frac\theta{2\pi}\log\frac{2\theta}{\pi e}+\frac\alpha4+\frac1\pi\arg L(\alpha+i\theta,\chi_4)-\frac{a(a-1)+\frac16}{\pi\theta}+O(\theta^{-3})\), with \(\arg L\) continuous along the same segment. As \(\alpha\downarrow\frac12\), for \(\theta\) not an ordinate, the left side tends to the number of zeros on the critical line with \(0<\gamma<\theta\), which is \(N(\theta)\) under GRH; at \(\alpha=\frac12\) the expansion is the counting formula \(N(T)=\frac T{2\pi}\log\frac{2T}{\pi e}+\frac18+\frac1\pi\arg L(\frac12+iT,\chi_4)+\frac1{48\pi T}+O(T^{-3})\), where \(N(T)\) counts the zeros of \(L(s,\chi_4)\) with \(0<\operatorname{Im}\rho\le T\), for \(T\) not an ordinate, with \(\arg L(\frac12+iT,\chi_4)\) continued from \(L(2,\chi_4)>0\) along \(2\to2+iT\to\frac12+iT\).

Earlier editions: Proposition 8.5.31, page 300 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:arg.

GRH for chi4 and monotonicity of the argument of the completed function

Volume I, Corollary 15.1.35, page 439 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) be the completed Dirichlet beta function. GRH for \(\chi_4\) holds if and only if, for every \(\alpha>\frac12\), \(\theta\mapsto\arg\xi(\alpha+i\theta,\chi_4)\) (continued along the vertical line from the real point \(\alpha\)) is nondecreasing. With no hypothesis, the heights \(\theta>0\) at which it decreases lie in the union of the windows \(|\theta-\gamma|<\sqrt{\delta^2-(\alpha-\frac12)^2}\), each of length less than \(1\), over the zeros \(\frac12+\delta+i\gamma\) of \(L(s,\chi_4)\) with \(\delta>\alpha-\frac12\), and there are none at heights below \(2.5\times10^7-\frac12\).

Earlier editions: Corollary 8.5.32, page 301 (edition 1).

Source: ch/ch16.tex, label cor:dbeta:argmono.

The half kernel of the Dirichlet beta function is not a gamma mixture

Volume I, Proposition 15.1.36, page 441 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_4(u)=2e^{3u/2}\eta(ie^{2u})^3\) be the kernel of the completed Dirichlet beta function, \(\xi(\frac12+s,\chi_4)=\int_{\mathbb{R}}K_4(u)e^{su}\,du\), with \(\eta\) Dedekind’s eta function. For no \(a>0\) is \(u^{1-a}K_4(u)\) completely monotone on \(u>0\); in particular the half kernel \(K_4(u)\mathbf 1\{u>0\}\) is not a positive mixture of Gamma(\(a\)) densities \(u^{a-1}e^{-\lambda u}\), for any shape \(a\). The theta function \(\eta(ix)^3\) is not completely monotone on \(x>0\).

Earlier editions: Proposition 8.5.33, page 302 (edition 1).

Source: ch/ch16.tex, label prop:dbeta:nocm.

Open question on a reciprocal side representation from the eta product

Volume I, Question 15.1.37, page 442 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(q=e^{-2\pi x}\) and let \(\eta\) be Dedekind’s eta function. The function \(e^{-\pi x/4}/\eta(ix)^3=\prod_{m\ge1}(1-q^m)^{-3}=\exp\Big(3\sum_{N\ge1}\sigma_{-1}(N)e^{-2\pi Nx}\Big)\) of \(x>0\), with \(\sigma_{-1}(N)=\sum_{d\mid N}1/d\), is completely monotone, since its logarithm is, and so is the analogue \(\prod_{m\ge1}(1-q^m)^{-1}\) for the kernel of \(L(s,\chi_{12})\). Can this complete monotonicity be turned into a positive representation on the reciprocal side, in the spirit of Thorin’s construction for the reciprocal of a GGC transform?

Earlier editions: Question 8.5.34, page 303 (edition 1).

Source: ch/ch16.tex, label q:dbeta:recip.

Thorin measure of the Dedekind eta analogue at the basepoint one

Volume I, Proposition 15.2.1, page 443 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_{12}\) be the real primitive even Dirichlet character modulo \(12\) (\(\chi_{12}(n)=1\) for \(n\equiv\pm1\) and \(-1\) for \(n\equiv\pm5 \pmod{12}\), and \(0\) if \(\gcd(n,12)>1\)), \(L(s,\chi_{12})\) its Dirichlet \(L\) function and \(\xi(s,\chi_{12})=(12/\pi)^{s/2}\Gamma(s/2)L(s,\chi_{12})\) the completed function, which is entire (there is no pole to remove) and satisfies \(\xi(s,\chi_{12})=\xi(1-s,\chi_{12})\). Put \(v_1(\theta)=\operatorname{Re}(\xi'/\xi)(1+i\theta,\chi_{12})\). Then \(\varphi_1(u)=\xi(1,\chi_{12})/\xi(1+\sqrt{u},\chi_{12})\) is the Laplace transform of a generalised gamma convolution (GGC) with no drift, whose Thorin measure, in the variable \(\theta=\sqrt{z}\), has the density \(\frac{1}{\pi}v_1(\theta)=\frac{1}{\pi}\sum_\rho\frac{1-\beta}{(1-\beta)^2+(\theta-\gamma)^2}>0\), the sum over the nontrivial zeros \(\rho=\beta+i\gamma\) of \(L(s,\chi_{12})\), in conjugate pairs. Positivity is unconditional because \(L(1+it,\chi_{12})\ne0\) for real \(t\).

Earlier editions: Proposition 11.3.1, page 465 (edition 1).

Source: ch/bpeta.tex, label bp:eta:prop:thorin.

Lévy form of the Dedekind eta analogue at the basepoint one

Volume I, Proposition 15.2.2, page 444 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_{12}\) be the real primitive even Dirichlet character modulo \(12\) (\(\chi_{12}(n)=1\) for \(n\equiv\pm1\) and \(-1\) for \(n\equiv\pm5 \pmod{12}\), and \(0\) if \(\gcd(n,12)>1\)), \(\xi(s,\chi_{12})=(12/\pi)^{s/2}\Gamma(s/2)L(s,\chi_{12})\) the completed function of its Dirichlet \(L\) function, \(b_1=(\xi'/\xi)(1,\chi_{12})\) (which equals \(\sum_\rho\rho^{-1}\) over the nontrivial zeros and is \(0.1650833123\)), \(\Lambda\) the von Mangoldt function and \(\Lambda_{\chi_{12}}(n)=\chi_{12}(n)\Lambda(n)\). For \(\operatorname{Re}s>0\), \(\log\frac{\xi(1+s,\chi_{12})}{\xi(1,\chi_{12})}=b_1s+\int_0^\infty(e^{-sx}-1+sx)\,\Pi(dx)\), where \(\Pi(dx)=\frac{e^{-x}}{x(1-e^{-2x})}\,dx+\sum_{n\ge2}\frac{\Lambda_{\chi_{12}}(n)}{n\log n}\,\delta_{\log n}(dx)\), the prime sum converging conditionally, as a limit over \(n\le X\) as \(X\to\infty\).

Earlier editions: Proposition 11.3.2, page 466 (edition 1).

Source: ch/bpeta.tex, label bp:eta:prop:levy.

Volume I, Chapter 16. Dirichlet L functions and the Linnik constant

Thorin property of the tilted clock of a Dedekind zeta function at every tilt: equivalence with the position of the zeros

Volume I, Theorem 16.2.3, page 454 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be a number field with \(r_1\) real and \(r_2\) complex places, discriminant \(d_F\) and Dedekind zeta function \(\zeta_F\). Put \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\) and \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\); this is an entire function of order one, real on the real axis, with \(\xi_F(s)=\xi_F(1-s)\), and for \(F=\mathbb{Q}\) it equals \(2\xi(s)\) with \(\xi\) Riemann’s xi function. Its zeros are the nontrivial zeros of \(\zeta_F\); write them as \(\rho=\frac12+w_\rho\), put \(z_\rho=w_\rho^2\) (one value for each pair \(\{\rho,1-\rho\}\), with multiplicity), and let \(2m_F\) be the order of vanishing of \(\xi_F\) at \(\frac12\). A generalized gamma convolution (GGC) is a probability law on \([0,\infty)\) whose Laplace transform is \(\exp\big(-bs-\int_{(0,\infty)}\log(1+s/t)\,U(dt)\big)\) with drift \(b\ge0\) and a nonnegative Thorin measure \(U\). For \(u_0\ge0\) with \(\xi_F(\frac12+\sqrt{u_0})\ne0\) put \(\varphi_{F,u_0}(s)=\xi_F(\frac12+\sqrt{u_0})/\xi_F(\frac12+\sqrt{u_0+s})\), \(s\ge0\). For \(u_0\ge0\) the following are equivalent. (i) \(\xi_F(\frac12+\sqrt{u_0})\ne0\) and \(\varphi_{F,u_0}\) is the Laplace transform of a GGC. (ii) Every \(z_\rho\) is real with \(z_\rho<u_0\), and \(m_F=0\) if \(u_0=0\); equivalently, every nontrivial zero of \(\zeta_F\) lies on the critical line or is a real zero \(\beta\) with \(|\beta-\frac12|<\sqrt{u_0}\), and \(\zeta_F(\frac12)\ne0\) when \(u_0=0\). When they hold, \(b=0\) and the Thorin measure is \(U_{F,u_0}=\sum_{\gamma>0}\delta_{u_0+\gamma^2}+m_F\,\delta_{u_0}+\sum_\beta\delta_{u_0-(\beta-\frac12)^2}\), where \(\frac12+i\gamma\) runs over the zeros on the critical line with \(\gamma>0\) and \(\beta\) over the real zeros in \((\frac12,1)\), all with multiplicity.

Earlier editions: Theorem 21.2.3, page 760 (edition 1).

Source: ch/gl2.tex, label gl:thm:family.

The tilts: Thorin property of a Dedekind zeta clock at tilt zero and at tilts above one quarter, the Thorin threshold and the shift between tilts

Volume I, Corollary 16.2.4, page 455 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be a number field with \(r_1\) real and \(r_2\) complex places, discriminant \(d_F\) and Dedekind zeta function \(\zeta_F\), and \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\) with \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\) and \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\), an entire function, real on the real axis, with \(\xi_F(s)=\xi_F(1-s)\) (it equals \(2\xi\) for \(F=\mathbb{Q}\)). For \(u_0\ge0\) put \(\varphi_{F,u_0}(s)=\xi_F(\frac12+\sqrt{u_0})/\xi_F(\frac12+\sqrt{u_0+s})\), and say that the Thorin property holds at tilt \(u_0\) if \(\xi_F(\frac12+\sqrt{u_0})\ne0\) and \(\varphi_{F,u_0}\) is the Laplace transform of a generalized gamma convolution (GGC); its Thorin measure is then written \(U_{F,u_0}\) (the drift is \(0\)). (a) At \(u_0=0\): \(\xi_F(\frac12)/\xi_F(\frac12+\sqrt s)\) is a GGC Laplace transform if and only if all nontrivial zeros of \(\zeta_F\) lie on the critical line and \(\zeta_F(\frac12)\ne0\). (b) For every \(u_0\ge\frac14\), in particular at \(u_0=\frac14\) and \(u_0=1\), where \(\xi_F\) is read from \(1\) and from \(\frac32\): \(\varphi_{F,u_0}\) is a GGC Laplace transform if and only if every nontrivial zero of \(\zeta_F\) lies on the critical line or on the real axis. (c) Define the Thorin threshold \(u^*_F=\max(\beta-\frac12)^2\) over the real zeros \(\beta\in[\frac12,1)\) of \(\xi_F\), with \(u^*_F=0\) if there are none. If all nonreal zeros lie on the critical line, the Thorin property holds at tilt \(u_0\) exactly for \(u_0>u^*_F\), and also at \(u_0=0\) when there are no real zeros at all. (d) When the Thorin property holds at two tilts \(u_1<u_0\), the measures are shifts of one another, \(U_{F,u_0}(A)=U_{F,u_1}(A-(u_0-u_1))\) for Borel sets \(A\) (each atom moves to the right by \(u_0-u_1\)), and \(\varphi_{F,u_1}(s)=\varphi_{F,u_0}(s-(u_0-u_1))/\varphi_{F,u_0}(-(u_0-u_1))\), the exponential tilt of a GGC.

Earlier editions: Corollary 21.2.4, page 761 (edition 1).

Source: ch/gl2.tex, label gl:cor:basepoints.

Closed form of the transfer function from the real axis to the critical line under RH, and its odd index power sum expansion

Volume I, Proposition 16.3.6, page 461 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis: every nontrivial zero of the Riemann zeta function has real part \(\frac12\).

Statement. Assume the Riemann Hypothesis. Let \(\xi\) be Riemann’s xi function, let \(\gamma>0\) run over the ordinates of the zeros \(\frac12+i\gamma\) of \(\xi\) with multiplicity, let \(\gamma_1\) be the smallest of them, and put \(p_j=\sum_{\gamma>0}\gamma^{-2j}\). For all real \(t\), \(\mathfrak T(t):=\frac{\xi(\frac12+it)}{\xi(\frac12+|t|)}=\prod_{\gamma>0}\frac{\gamma^2-t^2}{\gamma^2+t^2}\), and \(\mathfrak T\) is real, even, bounded by \(1\) in modulus, and vanishes exactly at \(t=\pm\gamma\). For \(|t|<\gamma_1\), \(\log\mathfrak T(t)=-2\sum_{\gamma>0}\operatorname{artanh}\big(\frac{t^2}{\gamma^2}\big)=-2\sum_{m\ge1}\frac{p_{2m-1}}{2m-1}\,t^{4m-2}\), which involves only the odd index power sums \(p_1,p_3,p_5,\dots\).

Earlier editions: Proposition 21.3.6, page 767 (edition 1).

Source: ch/gl3.tex, label gl:prop:transfer.

The odd index power sums of the inverse squared ordinates determine the zeros under RH

Volume I, Proposition 16.3.7, page 461 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis: every nontrivial zero of the Riemann zeta function has real part \(\frac12\).

Statement. Assume the Riemann Hypothesis. Let \(\gamma>0\) run over the ordinates of the zeros \(\frac12+i\gamma\) of Riemann’s xi function with multiplicity, let \(\gamma_1\) be the smallest of them, and put \(p_j=\sum_{\gamma>0}\gamma^{-2j}\). The series \(\sum_{m\ge1}p_{2m-1}z^{2m-1}\) converges for \(|z|<\gamma_1^2\) and continues to the meromorphic function \(\frac12\sum_{\gamma>0}\Big(\frac{1}{1-z/\gamma^2}-\frac{1}{1+z/\gamma^2}\Big)\), whose poles are the points \(\pm\gamma^2\), each with residue \(-\frac12\gamma^2\) times the multiplicity of \(\gamma\). Hence the odd index power sums \(\{p_{2m-1}\}\) determine the ordinates \(\{\gamma\}\) with multiplicities, and with them every \(p_j\).

Earlier editions: Proposition 21.3.7, page 767 (edition 1).

Source: ch/gl3.tex, label gl:prop:odd.

The transfer function as a Blaschke product for the right half plane under RH

Volume I, Proposition 16.3.8, page 461 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis: every nontrivial zero of the Riemann zeta function has real part \(\frac12\).

Statement. Assume the Riemann Hypothesis. Let \(\gamma>0\) run over the ordinates of the zeros \(\frac12+i\gamma\) of Riemann’s xi function with multiplicity, and put \(\mathfrak B(z)=\prod_{\gamma>0}(\gamma^2-z)/(\gamma^2+z)\), so that the transfer function \(\mathfrak T(t)=\xi(\frac12+it)/\xi(\frac12+|t|)\) equals \(\mathfrak B(t^2)\). Then \(\mathfrak B\) is a Blaschke product for the right half plane: the product converges there, \(|\mathfrak B(z)|<1\) for \(\operatorname{Re}z>0\), \(|\mathfrak B(z)|=1\) for \(\operatorname{Re}z=0\), its zeros are the points \(\gamma^2\), and it has no singular inner factor.

Earlier editions: Proposition 21.3.8, page 768 (edition 1).

Source: ch/gl3.tex, label gl:prop:inner.

Negative moments of the centre clock as a Mellin transform of the reciprocal of xi on the real axis

Volume I, Proposition 16.3.10, page 462 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and let \(T\) be the centre clock, the positive self decomposable random variable with Laplace transform \(\mathbb{E}[e^{-uT}]=\xi(\frac12)/\xi(\frac12+\sqrt u)\) for \(u\ge0\) (it exists without any hypothesis on the zeros). For \(\operatorname{Re}s>0\), with no hypothesis on the zeros, \(\mathbb{E}[T^{-s}]=\frac{1}{\Gamma(s)}\int_0^\infty u^{s-1}\,\frac{\xi(\frac12)}{\xi(\frac12+\sqrt u)}\,du=\frac{2\,\xi(\frac12)}{\Gamma(s)}\int_0^\infty\frac{w^{2s-1}}{\xi(\frac12+w)}\,dw\).

Earlier editions: Proposition 21.3.10, page 768 (edition 1).

Source: ch/gl3.tex, label gl:prop:negmom.

Growth of the gamma function times the reciprocal of xi on vertical strips under RH: the growth hypothesis of the master theorem class

Volume I, Lemma 16.4.1, page 465 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis, used through the bound \(1/\zeta(w)=O(|\operatorname{Im}w|^\eta)\) on \(\operatorname{Re}w\ge\frac12+\sigma_1\) for every \(\eta>0\).

Statement. Assume the Riemann Hypothesis. Let \(\xi\) be Riemann’s xi function, \(F(s;0)=\xi(\frac12)/\xi(\frac12+s)\), \(0<\sigma_1\le\sigma_2<\frac12\) and \(\eta>0\). Then for \(s=\sigma+i\tau\) with \(\sigma_1\le\sigma\le\sigma_2\) and \(|\tau|\ge1\), \(|\Gamma(s)\,F(s;0)|\le C_\eta\,e^{-\pi|\tau|/4}\,|\tau|^{\sigma/2-9/4+\eta}\), uniformly in \(\sigma\) and \(\tau\) in this range. In particular the growth condition \(|\Gamma(s)F(s;0)|\le C|\tau|^{\kappa(\sigma-\delta)}\) with \(\kappa=1\) and \(\delta=\frac12\), the growth condition of the class \(Q_\infty(1,\frac12)\) (the extension by Chaudhry and Qadir (2012) of Hardy’s class for Ramanujan’s master theorem), holds on such strips, with exponential room. By the functional equation the same argument gives, for \(-\sigma_2\le\sigma\le-\sigma_1\), the exponent \(\frac32\sigma-\frac94+\eta\), which is smaller.

Earlier editions: Lemma 21.4.1, page 771 (edition 1).

Source: ch/gl4.tex, label gl:lem:stirling.

Absolute convergence hypothesis of the master theorem route for the reciprocal of xi holds without RH

Volume I, Lemma 16.4.2, page 465 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, let \(T\) be the centre clock, the self decomposable random variable with Laplace transform \(\xi(\frac12)/\xi(\frac12+\sqrt u)\) (it exists without any hypothesis on the zeros), and let \(g\) be its density. Put \(b_k=\frac{1}{\Gamma(k)}\,\frac{\xi(\frac12)}{\xi(\frac12+k)}\) and \(\mathfrak R(w)=\sum_{k\ge1}b_kw^k\). With no hypothesis on the zeros, \(\mathfrak R\) is entire of order \(\frac23\), and \(\int_0^\infty g(y)\,\mathfrak R(z/y)\,dy<\infty\) for every \(z>0\).

Earlier editions: Lemma 21.4.2, page 772 (edition 1).

Source: ch/gl4.tex, label gl:lem:fubini.

The reciprocal Xi function in the shifted variable lies in Hardy’s class with no hypothesis on the zeros, so the master theorem carries no information on them

Volume I, Proposition 16.4.3, page 466 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+\frac s2)\zeta(s)\) be the completed Riemann zeta function and \(0<\delta<1\). Hardy’s class consists of the functions \(\varphi\) analytic on \(\operatorname{Re}w\ge-\delta\) with \(|\varphi(v+i\tau)|\le Ce^{Pv+A|\tau|}\) for some constants \(C,P\) and \(A<\pi\); for them \(\int_0^\infty x^{w-1}\sum_{k\ge0}(-x)^k\varphi(k)/k!\,dx=\Gamma(w)\varphi(-w)\) for \(0<\operatorname{Re}w<\delta\). The function \(\varphi(w)=\xi(\frac32)/\xi(\frac12+\sqrt{1+w})\), with the principal square root, is analytic and bounded on \(\operatorname{Re}w\ge-\delta\). Hence, with no hypothesis on the zeros, it lies in Hardy’s class with \(P=0\) and any \(A\). The image of \(\operatorname{Re}w\ge-\delta\) under \(w\mapsto\frac12+\sqrt{1+w}\) lies in \(\{\sigma+i\tau:\sigma-\frac12\ge\sqrt{1-\delta+\tau^2}\}\), which contains no zero of \(\xi\), so the master theorem identity in this variable carries no information on the zeros.

Earlier editions: Proposition 21.4.3, page 773 (edition 1).

Source: ch/gl4.tex, label tx:prop:hardy.

Why the square root stays: reciprocal Xi function in the variable s is not a Laplace transform of a random variable, and its values on a vertical line are not a characteristic function

Volume I, Theorem 16.4.4, page 467 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+\frac s2)\zeta(s)\) be the completed Riemann zeta function, and let \(L\) be the class of the functions \(s\mapsto\mathbb{E}e^{-sX}\), \(X\) a real random variable, finite on an interval. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the Pólya kernel, so that \(\xi(\frac12+iz)=\int_{\mathbb{R}}\Phi(u)e^{izu}\,du\). (a) For every real \(\alpha\), \(s\mapsto\xi(\alpha)/\xi(\alpha+s)\) is not in \(L\), whether or not RH holds. (b) For every real \(\alpha\), \(t\mapsto\xi(\alpha+it)/\xi(\alpha)\) is, unconditionally, the characteristic function of the probability density \(\Phi(u)e^{(\alpha-\frac12)u}/\xi(\alpha)\), but it is never infinitely divisible, hence never the law of an exponential generalised gamma convolution. At \(\alpha=\frac12\) it is \(\Xi(t)/\Xi(0)\), where \(\Xi(t)=\int_{\mathbb{R}}\Phi(u)\cos(tu)\,du=\xi(\frac12+it)\). (c) For every real \(\alpha\), \(t\mapsto\xi(\alpha)/\xi(\alpha+it)\) is unbounded, so it is not a characteristic function.

Earlier editions: Theorem 21.4.4, page 774 (edition 1).

Source: ch/gl4.tex, label tx:thm:nosqrt.

Thorin representation of the reciprocal of the completed Dirichlet L function of a real primitive character under GRH

Volume I, Proposition 16.5.1, page 468 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The generalized Riemann hypothesis for \(L(s,\chi)\) (every nontrivial zero has real part \(\frac12\)) and \(L(\frac12,\chi)\ne0\) (Chowla’s conjecture for real characters, open).

Statement. Let \(\chi\) be a real primitive character modulo \(q>1\) of parity \(a\in\{0,1\}\) (so \(\chi(-1)=(-1)^a\)) and \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\) the completed Dirichlet \(L\) function, an entire function of order one, real on the real axis, with \(\xi(s,\chi)=\xi(1-s,\chi)\). Assume GRH for \(L(s,\chi)\) and \(L(\frac12,\chi)\ne0\), and write the zeros as \(\frac12\pm i\gamma\) with \(\gamma>0\), counted with multiplicity. Then \(T_\chi\stackrel{d}{=}\sum_{\gamma>0}e_\gamma/\gamma^2\), with \(e_\gamma\) independent standard exponential variables, is a GGC with \(\xi(\frac12,\chi)/\xi(\frac12+\sqrt u,\chi)=\mathbb{E}[e^{-uT_\chi}]\) for \(u\ge0\) and cumulants \((j-1)!\,p_j(\chi)\), \(p_j(\chi)=\sum_{\gamma>0}\gamma^{-2j}\). The analogues for \(\xi(\cdot,\chi)\) of the statements for Riemann’s \(\xi\) under RH on the transfer function from the real axis to the critical line, the odd index power sums of the inverse squared ordinates, the Blaschke product of the transfer function and the negative moments of the clock hold, with GRH for \(L(s,\chi)\) and \(L(\frac12,\chi)\ne0\) in place of RH and \(\xi(\frac12)\ne0\). At the tilt \(u_0=\frac14\), \(\sum_{\gamma>0}\frac{1}{\gamma^2+1/4}=\frac{\xi'}{\xi}(1,\chi)=\frac12\log\frac q\pi+\frac12\,\psi\big(\frac{1+a}{2}\big)+\frac{L'}{L}(1,\chi)\), with \(\psi\) the digamma function, and \(\mathbb{E}[e^{-T_\chi/4}]=\xi(\frac12,\chi)/\xi(1,\chi)\).

Earlier editions: Proposition 21.5.1, page 775 (edition 1).

Source: ch/gl5.tex, label gl:prop:Lreal.

Thorin representation for a complex character through the product of the completed L function with its conjugate, under GRH

Volume I, Proposition 16.5.2, page 469 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The generalized Riemann hypothesis for \(L(s,\chi)\) (every nontrivial zero has real part \(\frac12\)) and \(L(\frac12,\chi)\ne0\).

Statement. Let \(\chi\) be a complex primitive character modulo \(q>1\) of parity \(a\in\{0,1\}\) and \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\) the completed Dirichlet \(L\) function, an entire function of order one with \(\xi(s,\chi)=W_\chi\,\xi(1-s,\bar\chi)\) and \(|W_\chi|=1\). Put \(\xi_{\chi\bar\chi}(s)=\xi(s,\chi)\,\xi(s,\bar\chi)\). Then \(\xi_{\chi\bar\chi}(s)=\xi_{\chi\bar\chi}(1-s)\) and \(\xi_{\chi\bar\chi}(\sigma)=|\xi(\sigma,\chi)|^2\) for real \(\sigma\). If GRH holds for \(L(s,\chi)\) and \(L(\frac12,\chi)\ne0\), then \(\xi_{\chi\bar\chi}(\frac12)/\xi_{\chi\bar\chi}(\frac12+\sqrt u)=\mathbb{E}[e^{-uT_{\chi\bar\chi}}]\) for \(u\ge0\), where \(T_{\chi\bar\chi}\stackrel{d}{=}\sum_{\gamma\in\Gamma_\chi}e_\gamma/\gamma^2\), \(\Gamma_\chi\) is the multiset of all ordinates \(\gamma\), of both signs, of the zeros \(\frac12+i\gamma\) of \(L(s,\chi)\), and the \(e_\gamma\) are independent standard exponential variables.

Earlier editions: Proposition 21.5.2, page 776 (edition 1).

Source: ch/gl5.tex, label gl:prop:Lcomplex.

Factorisation of the completed Dedekind zeta function of a quadratic field and clocks of cyclotomic fields as independent sums

Volume I, Proposition 16.5.3, page 470 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The statement on cyclotomic fields assumes GRH for the Dirichlet \(L\) functions of the nonprincipal primitive characters of conductor dividing \(q\) and that their values at \(\frac12\) are nonzero (the hypotheses of the propositions on real and complex characters). The factorisations of the completed functions hold without any hypothesis, and the statement for quadratic fields is an implication from the Thorin property of the two factors at the tilt considered.

Statement. Let \(\chi\) be a real primitive character of conductor \(q\) and parity \(a\in\{0,1\}\), with \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\), let \(F=\mathbb{Q}(\sqrt D)\) with \(D=\chi(-1)q\) a fundamental discriminant (so \(\zeta_F=\zeta\cdot L(\cdot,\chi)\), and \(F\) is real quadratic for even \(\chi\) and imaginary quadratic for odd \(\chi\)), let \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\) be its completed Dedekind zeta function (\(d_F\) the discriminant, \(r_1\) and \(r_2\) the numbers of real and complex places, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\)) and let \(\xi\) be Riemann’s xi function. Then \(\xi_F(s)=2\,\xi(s)\,\xi(s,\chi)\) when \(\chi\) is even and \(\xi_F(s)=2q^{-1/2}\,\xi(s)\,\xi(s,\chi)\) when \(\chi\) is odd. For a function \(f\) the tilted clock at tilt \(u_0\ge0\) is the random variable with Laplace transform \(f(\frac12+\sqrt{u_0})/f(\frac12+\sqrt{u_0+s})\), and the centre clock is the case \(u_0=0\). Consequently, at any tilt \(u_0\ge0\) at which \(\xi\) and \(\xi(\cdot,\chi)\) both have the Thorin property (this reciprocal is the Laplace transform of a generalized gamma convolution), so does \(\xi_F\), its tilted clock is the independent sum of the tilted clocks of \(\xi\) and of \(\xi(\cdot,\chi)\), and the Thorin measures add, \(U_{F,u_0}=U_{\mathbb{Q},u_0}+U_{\chi,u_0}\), where \(U_{\chi,u_0}\) is the Thorin measure of the tilted clock of \(\xi(\cdot,\chi)\); at \(u_0=0\) this reads \(T_F\stackrel d=T_\zeta+T_\chi\) for the centre clocks, and the cumulants add, \(p_j(F)=p_j(\zeta)+p_j(\chi)\) with \(p_j=\sum_{\gamma>0}\gamma^{-2j}\) over the ordinates of the zeros. For the cyclotomic field \(F_q=\mathbb{Q}(\zeta_q)\), \(q\ge3\), under GRH for the Dirichlet \(L\) functions of the nonprincipal primitive characters of conductor dividing \(q\) and with their values at \(\frac12\) nonzero, the centre clock \(T_{F_q}\) is the independent sum of \(T_\zeta\) (the centre clock of \(\xi\)), of \(T_\chi\) (the centre clock of \(\xi(\cdot,\chi)\)) over the real primitive characters \(\chi\) of conductor \(f\mid q\) with \(f>1\), and of \(T_{\chi\bar\chi}\) (the centre clock of \(\xi(s,\chi)\xi(s,\bar\chi)\)) over the conjugate pairs of complex primitive characters of conductor dividing \(q\).

Earlier editions: Proposition 21.5.3, page 777 (edition 1).

Source: ch/gl5.tex, label gl:prop:dedekind.

Lévy decomposition of the logarithm of the completed Dedekind zeta function of any number field, with signs

Volume I, Proposition 16.5.4, page 471 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be a number field with \(r_1\) real and \(r_2\) complex places and Dedekind zeta function \(\zeta_F\), and \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\) its completed Dedekind zeta function (\(d_F\) the discriminant, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\)). Let \(\Lambda_F(n)=\sum_{N\mathfrak p^r=n}\log N\mathfrak p\ge0\), the sum over prime ideals \(\mathfrak p\) and integers \(r\ge1\) with \(N\mathfrak p^r=n\), and \(h_s(x)=e^{-sx}-1+sx\ge0\). For \(\alpha>1\) and \(s\ge0\), \(\log\frac{\xi_F(\alpha+s)}{\xi_F(\alpha)}-s\,\frac{\xi_F'}{\xi_F}(\alpha)=\int_0^\infty h_s(x)\,e^{-\alpha x}\,\frac{\nu_F(dx)}x\) with the signed measure \(\nu_F(dx)=\Big(\frac{r_1}{1-e^{-2x}}+\frac{r_2}{1-e^{-x}}-1-e^x\Big)dx+\sum_{n\ge2}\Lambda_F(n)\,\delta_{\log n}(dx)\).

Earlier editions: Proposition 21.5.4, page 778 (edition 1).

Source: ch/gl5.tex, label gl:prop:levyF.

The sine squared lemma on the prime side for Dedekind zeta functions and real Dirichlet characters

Volume I, Proposition 16.5.6, page 472 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be either \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\), the completed Dedekind zeta function of a number field \(F\) with discriminant \(d_F\), \(r_1\) real and \(r_2\) complex places, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\) and \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\), or \(\xi(\cdot,\chi)\), the completed Dirichlet \(L\) function \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\) of a real primitive character \(\chi\) modulo \(q>1\) of parity \(a\in\{0,1\}\). Let \(\alpha>1\), \(h_w(x)=e^{-wx}-1+wx\), and let \(\nu\) be the signed Lévy measure of \(f\), defined by \(\log\frac{f(\alpha+w)}{f(\alpha)}-w\,\frac{f'}{f}(\alpha)=\int_0^\infty h_w(x)\,e^{-\alpha x}\,\frac{\nu(dx)}x\) for \(w\ge0\): for \(f=\xi_F\), \(\nu_F(dx)=\Big(\frac{r_1}{1-e^{-2x}}+\frac{r_2}{1-e^{-x}}-1-e^x\Big)dx+\sum_{n\ge2}\Lambda_F(n)\,\delta_{\log n}(dx)\) with \(\Lambda_F(n)=\sum_{N\mathfrak p^r=n}\log N\mathfrak p\); for \(f=\xi(\cdot,\chi)\), \(\nu_\chi(dx)=\sum_{n\ge2}\chi(n)\Lambda(n)\,\delta_{\log n}(dx)+\frac{e^{(1-a)x}}{2\sinh x}\,dx\), where \(\Lambda\) is the von Mangoldt function. Then, for \(u\ge0\), \(\frac{f(\alpha)}{f(\alpha+\sqrt u)}=\exp\Big(-\int_0^\infty\log\Big(1+\frac u{b^2}\Big)\,\frac{v_\alpha(b)}\pi\,db\Big)\), where \(v_\alpha(b)=\frac{f'}{f}(\alpha)+\int_0^\infty2\sin^2\Big(\frac{bx}2\Big)e^{-\alpha x}\,\nu(dx)=\operatorname{Re}\frac{f'}{f}(\alpha+ib)\).

Earlier editions: Proposition 21.5.6, page 779 (edition 1).

Source: ch/gl5.tex, label gl:prop:sinsq.

The basepoint family of the sine squared lemma: Thorin property at a basepoint to the right of all zeros, failure when a zero lies to the right, and the Poisson form under GRH

Volume I, Theorem 16.5.7, page 473 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Part (c) assumes GRH for \(f\) (every zero has real part \(\frac12\)) and \(f(\frac12)\ne0\). Parts (a) and (b) hold without any hypothesis, and part (a) holds unconditionally for every \(\alpha\ge1\).

Statement. Let \(f\) be either \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\), the completed Dedekind zeta function of a number field \(F\) with discriminant \(d_F\), \(r_1\) real and \(r_2\) complex places, where \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\) and \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\), or \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\), the completed Dirichlet \(L\) function of a real primitive character \(\chi\) modulo \(q>1\) of parity \(a\in\{0,1\}\) (each is entire and real on the real axis, with \(f(s)=f(1-s)\)), with zeros \(\rho=\beta+i\gamma\) counted with multiplicity, and for \(\alpha>\frac12\) put \(v_\alpha(b)=\operatorname{Re}\frac{f'}{f}(\alpha+ib)\). Say that \(f\) has the Thorin property at the basepoint \(\alpha>\frac12\) if \(f(\alpha)\ne0\) and \(u\mapsto f(\alpha)/f(\alpha+\sqrt u)\) is the Laplace transform of a generalized gamma convolution (GGC). (a) If every zero has \(\beta<\alpha\), then \(\frac{f(\alpha)}{f(\alpha+\sqrt u)}=\exp\big(-\int_0^\infty\log(1+u/b^2)\,\frac{v_\alpha(b)}\pi\,db\big)\) for \(u\ge0\), with \(\frac{v_\alpha(b)}\pi=\frac1\pi\sum_\rho\operatorname{Re}\frac{1}{\alpha+ib-\rho}=\frac1\pi\sum_\rho\frac{\alpha-\beta}{(\alpha-\beta)^2+(b-\gamma)^2}>0\), so \(f\) has the Thorin property at the basepoint \(\alpha\), with no drift and Thorin measure \(U_\alpha(dt)=v_\alpha(\sqrt t)\,dt/(2\pi\sqrt t)\). Since every zero has \(\beta<1\), this holds unconditionally for every \(\alpha\ge1\). (b) If some zero has \(\beta>\alpha\) and \(f(\alpha)\ne0\), then \(f\) does not have the Thorin property at the basepoint \(\alpha\). Hence \(f\) has it at every basepoint \(\alpha>\frac12\) if and only if every zero lies on the critical line. (c) Under GRH for \(f\) (every zero has real part \(\frac12\)), and if \(f(\frac12)\ne0\), with \(\varepsilon=\alpha-\frac12\), \(\frac{v_\alpha(b)}\pi=\sum_{\gamma>0}\big(P_\varepsilon(b-\gamma)+P_\varepsilon(b+\gamma)\big)\) with \(P_\varepsilon(x)=\frac{\varepsilon}{\pi(\varepsilon^2+x^2)}\), the Poisson smoothing at height \(\varepsilon\) of the zero counting measure. As \(\alpha\downarrow\frac12\), \((v_\alpha(b)/\pi)\,db\to\sum_{\gamma>0}\delta_\gamma\) vaguely on \([0,\infty)\), and \(U_\alpha\) tends to \(\sum_{\gamma>0}\delta_{\gamma^2}\), the Thorin measure of the centre clock.

Earlier editions: Theorem 21.5.8, page 780 (edition 1).

Source: ch/gl5.tex, label gl:thm:sinsq.

GRH and nonvanishing at the centre from the Thorin property of the Dedekind zeta functions of quadratic and cyclotomic fields

Volume I, Theorem 16.6.1, page 477 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi\) be a real primitive character of conductor \(q\), \(F=\mathbb{Q}(\sqrt D)\) with \(D=\chi(-1)q\) the quadratic field of \(\chi\) (so that \(\zeta_F(s)=\zeta(s)L(s,\chi)\)), \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\) its completed Dedekind zeta function (\(d_F\) the discriminant, \(r_1\) and \(r_2\) the numbers of real and complex places, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\)), and \(\varphi_{F,u_0}(s)=\xi_F(\frac12+\sqrt{u_0})/\xi_F(\frac12+\sqrt{u_0+s})\) for \(u_0\ge0\). (a) If \(\varphi_{F,1}\) is the Laplace transform of a generalized gamma convolution (GGC), then RH holds and every nontrivial zero of \(L(s,\chi)\) lies on the critical line or is real. (b) If \(\varphi_{F,0}\) is the Laplace transform of a GGC, equivalently \(\varphi_{F,1}\) is one with Thorin measure in \((1,\infty)\), then \(L(s,\chi)\) satisfies GRH (all its nontrivial zeros lie on the critical line) and \(L(\frac12,\chi)\ne0\). If \(\varphi_{F_q,0}\) is a GGC Laplace transform for every \(q\ge3\), with \(F_q=\mathbb{Q}(\zeta_q)\), then GRH holds for all Dirichlet \(L\) functions, and \(L(\frac12,\chi)\ne0\) for every primitive character \(\chi\).

Earlier editions: Theorem 21.6.1, page 784 (edition 1).

Source: ch/gl6.tex, label gl:thm:grh.

Mixtures of clocks: a mixture of two GGC Laplace transforms with discrete unit atom Thorin measures is not a GGC transform when it vanishes off the atoms, in particular in the odd configuration

Volume I, Proposition 16.6.3, page 479 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\varphi_1,\varphi_2\) be Laplace transforms of generalized gamma convolutions (GGCs: laws on \([0,\infty)\) with Laplace transform \(\exp(-bs-\int_{(0,\infty)}\log(1+s/t)\,U(dt))\), drift \(b\ge0\) and Thorin measure \(U\)) with zero drift and Thorin measures \(U_1,U_2\) that are sums of unit point masses, repeated according to multiplicity, so that \(\varphi_i(s)=\prod_t t/(t+s)\), over the atoms \(t\) of \(U_i\), is meromorphic on \(\mathbb{C}\) with poles at the points \(-t\). Let \(0<\lambda<1\) and \(\varphi=\lambda\varphi_1+(1-\lambda)\varphi_2\). (a) If \(\varphi(-x)=0\) for some \(x>0\) that is not an atom of \(U_1\) or \(U_2\), then \(\varphi\) is not the Laplace transform of a GGC. (b) Let \(t_1<t_2\) be consecutive points of \(\operatorname{supp}U_1\cup\operatorname{supp}U_2\) with \(t_1\) an atom of only one of the measures and \(t_2\) an atom of only the other. If \(U_1+U_2\) has an odd number of atoms in \((0,t_1]\), counted with multiplicity, then \(\varphi\) has a zero in \((-t_2,-t_1)\), so \(\varphi\) is not the Laplace transform of a GGC, for any \(\lambda\).

Earlier editions: Proposition 21.6.3, page 787 (edition 1).

Source: ch/gl6.tex, label gl:prop:mixture.

The family of characters modulo 5: no mixture of the two clocks is a GGC, under GRH

Volume I, Proposition 16.6.4, page 481 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The generalized Riemann hypothesis for \(L(s,\chi_5)\) and for \(L(s,\psi)\) (every nontrivial zero has real part \(\frac12\)). The values \(L(\frac12,\chi_5)\) and \(L(\frac12,\psi)\) are certified nonzero, and the zeros below height \(12\) are certified, in ball arithmetic; neither is assumed.

Statement. Let \(\chi_5\) be the real primitive character modulo \(5\) (the Legendre symbol, even) and \(\psi\) the primitive character modulo \(5\) with \(\psi(2)=i\) (odd). For a primitive character \(\chi\) modulo \(q\) of parity \(a\in\{0,1\}\) let \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\), and put \(\xi_{\psi\bar\psi}(s)=\xi(s,\psi)\,\xi(s,\bar\psi)\). A generalized gamma convolution (GGC) is a probability law on \([0,\infty)\) whose Laplace transform is \(\exp\big(-bs-\int_{(0,\infty)}\log(1+s/t)\,U(dt)\big)\) with drift \(b\ge0\) and a nonnegative Thorin measure \(U\). Assume GRH for \(L(s,\chi_5)\) and \(L(s,\psi)\). Then for every \(\lambda\in(0,1)\) the function \(u\mapsto\lambda\,\xi(\frac12,\chi_5)/\xi(\frac12+\sqrt u,\chi_5)+(1-\lambda)\,\xi_{\psi\bar\psi}(\frac12)/\xi_{\psi\bar\psi}(\frac12+\sqrt u)\), \(u\ge0\), is not the Laplace transform of a GGC. In particular the case \(\lambda=\frac12\), the clock of a uniformly chosen member of the family \(\{\chi_5,\{\psi,\bar\psi\}\}\) of primitive characters modulo \(5\), is not a GGC.

Earlier editions: Proposition 21.6.4, page 788 (edition 1).

Source: ch/gl6.tex, label gl:prop:mod5.

A real zero of a Dirichlet L function makes the first rung of the twisted heat trace fail beyond an explicit time

Volume I, Proposition 16.6.6, page 482 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi\) be a primitive real Dirichlet character of conductor \(q\ge3\), \(L(s,\chi)\) its \(L\) function, and \(\xi(s,\chi)\) the completed \(L\) function, an entire function with \(\xi(s,\chi)=\xi(1-s,\chi)\). With zeros counted with multiplicity and \(z_\rho=(\rho-\tfrac12)^2\), the twisted heat trace is \(W(t;\chi)=\sum_{\operatorname{Im}\rho>0}e^{tz_\rho}+\tfrac12\sum_{\rho\text{ real}}e^{tz_\rho}\) for \(t>0\). Let \(\beta\in(\tfrac12,1)\) be the largest real zero of \(L(s,\chi)\), suppose that every nonreal zero of \(L(s,\chi)\) lies on the critical line, and put \(c=(\beta-\tfrac12)^2\). Then \(W(t;\chi)\to\infty\) as \(t\to\infty\), and \(-W'(t;\chi)<0\) for all \(t>t_1=\max\{1,c^{-1}\log(C'\log q/c)\}\), where \(C'\) is an absolute constant. Hence \(W(\cdot\,;\chi)\) is not completely monotone, the first rung of the ladder \((-1)^kW^{(k)}>0\) failing at order \(k=1\) beyond \(t_1\). The book adds the following consequence, which rests on the unrefereed theorem of Banks and Kerr (October 2026) that there are absolute constants \(q_0\) and \(\delta_0>0\) with \((1-\beta)\log q\ge\delta_0\) for every real zero \(\beta\) of every primitive real character of conductor \(q\ge q_0\): for \(q\ge q_0\) and \(\eta=\delta_0/\log q\) one has \(c\le\tfrac14-\eta(1-\eta)\), so the growth rate \(c\) is at least \(\eta(1-\eta)\) below the rate \(\tfrac14\) of the pole term \(e^{t/4}\) of \(\zeta\). That consequence is conditional on the Banks and Kerr theorem and is not part of the proved status of this entry; the first part (the growth and the sign of \(-W'\)) uses only the bound \(N(T,\chi)\ll(T+1)\log(q(T+2))\) for the number of zeros up to height \(T\).

Earlier editions: Proposition 21.6.6, page 790 (edition 1).

Source: ch/gl6.tex, label gl:prop:rung1.

Linnik constant seven from the Thorin property of all cyclotomic fields (weaker than the known result under GRH)

Volume I, Corollary 16.7.1, page 483 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Thorin property at tilt \(0\) for every cyclotomic field \(\mathbb{Q}(\zeta_q)\), \(q\ge3\), which by the equivalence theorem of the chapter is GRH for all Dirichlet \(L\) functions together with \(L(\frac12,\chi)\ne0\) for every primitive character \(\chi\). The proof uses only the GRH part, through the theorem of Heath-Brown and Puchta (2002) that GRH gives seven powers of two.

Statement. Suppose that for every integer \(q\ge3\) the function \(\varphi_{F,0}(s)=\xi_F(\frac12)/\xi_F(\frac12+\sqrt s)\), with \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{C}}(s)^{\varphi(q)/2}\zeta_F(s)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\), the completed Dedekind zeta function of the cyclotomic field \(F=\mathbb{Q}(\zeta_q)\), a totally complex field of degree \(\varphi(q)\) and discriminant \(d_F\), is the Laplace transform of a generalized gamma convolution. Then every sufficiently large even integer is a sum of two primes and at most seven powers of two.

Earlier editions: Corollary 21.7.1, page 791 (edition 1).

Source: ch/gl7.tex, label gl:cor:K7.

Open problem: the Thorin property of the Riemann clock and of cyclotomic clocks without first knowing the zeros

Volume I, Problem 16.1, page 485 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\) be the completed Dedekind zeta function of a number field \(F\) (\(d_F\) the discriminant, \(r_1\) and \(r_2\) the numbers of real and complex places, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\)) and \(\varphi_{F,u_0}(s)=\xi_F(\frac12+\sqrt{u_0})/\xi_F(\frac12+\sqrt{u_0+s})\) its tilted reciprocal. Prove that \(\varphi_{\mathbb{Q},u_0}\), or \(\varphi_{F,u_0}\) for the cyclotomic fields \(F\), is the Laplace transform of a generalized gamma convolution (the first condition of the equivalence theorem on the Thorin property of the tilted clock), without first knowing the second condition, namely the position of the zeros. In the route through Ramanujan’s master theorem, which expands \(\xi(\frac12)/\xi(\frac12+\sqrt u)\) term by term, settle the parabolic distortion (the Laplace variable is \(u=s^2\), and the squaring map carries the strip \(|\operatorname{Re}s|<1\) onto the region \(\{|\operatorname{Im}u|<2\sqrt{1-\operatorname{Re}u}\}\)), uniformly in the field, and find a substitute for the residue bookkeeping at the zeros. For \(\mathbb{Q}\) at \(u_0=0\) the statement to be proved is the Riemann Hypothesis, so a growth bound that is itself proved under RH cannot be an input to its proof.

Earlier editions: Problem 21.1, page 792 (edition 1).

Source: ch/gl8.tex, label gl:op:direct.

Open problem: seven powers of two from GRH with possible real zeros, that is from the Thorin property at tilt one for cyclotomic fields

Volume I, Problem 16.2, page 486 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Show that every sufficiently large even integer is a sum of two primes and at most seven powers of two (the Linnik constant \(K=7\)) from the hypothesis that for every cyclotomic field \(\mathbb{Q}(\zeta_q)\), \(q\ge3\), the tilted reciprocal \(\varphi_{\mathbb{Q}(\zeta_q),1}\) of the completed Dedekind zeta function at tilt \(u_0=1\) is the Laplace transform of a generalised gamma convolution; by the equivalence theorem of the chapter this hypothesis says that every nontrivial zero of every Dirichlet \(L\) function lies on the critical line or on the real axis, that is, GRH with possible real zeros. The book locates, by a heuristic estimate, the arcs of the circle method on which a real zero can make the minor arc bound of exponent \(\frac34\) for the sum over primes fail; what remains is to move these arcs into the major arcs and to redo the main term of the major arc estimate with several real zeros, checking that its constant \(2.7895\) survives. The major arc estimate of Heath-Brown and Puchta that gives this constant (their Lemma 7) is stated for \(K\ge9\), with zero density estimates in place of GRH, and their paper does not say how it is used for \(K=7\); the computation of the constant itself holds for \(K\ge7\). If it works, real zeros, and with them Siegel zeros, become irrelevant to \(K=7\).

Earlier editions: Problem 21.2, page 792 (edition 1).

Source: ch/gl8.tex, label gl:op:exceptional.

Open problem: the anchored tail of the moment identity of the centre clock at large height and for Dirichlet L functions

Volume I, Problem 16.3, page 486 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(T\) be the centre clock of Riemann’s xi function. Its moments are computed from the power sums \(p_j=\sum_{\gamma>0}\gamma^{-2j}\) of the inverse squared ordinates of the zeros, summed over the first zeros and completed by a tail. With \(N(t)\) the zero count, \(N_0=\vartheta/\pi\) with \(\vartheta\) the Riemann Siegel theta function, and \(S(t)=N(t)-N_0(t)-1\), the tail beyond a height \(H\) satisfies exactly \(\sum_{\gamma>H}\gamma^{-2j}=\int_H^\infty t^{-2j}\,\frac{\vartheta'(t)}{\pi}\,dt+\big(N_0(H)+1-N(H)\big)H^{-2j}+2j\int_H^\infty S(t)\,t^{-2j-1}\,dt\), and the anchored tail keeps the first two terms. Carry the anchoring to large height with tables of zeros of the Riemann zeta function, where the remaining term, the integral of \(S(t)\,t^{-2j-1}\) over the tail, sets the limit and calls for an explicit formula for \(\int S\), and to Dirichlet \(L(s,\chi)\) with tables of their zeros. The book reproduces the anchoring only for the first ordinates; large heights and the characters are open.

Earlier editions: Problem 21.3, page 793 (edition 1).

Source: ch/gl8.tex, label gl:op:numerics.

Open problem: the negative moment of the clock averaged over real characters of conductor in [Q,2Q] against the symplectic random matrix prediction

Volume I, Problem 16.4, page 486 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a real primitive character \(\chi\) let \(T_\chi\) be the clock (under GRH for \(L(s,\chi)\) and \(L(\frac12,\chi)\ne0\)) with Laplace transform \(\xi(\frac12,\chi)/\xi(\frac12+\sqrt u,\chi)\), and \(T\) the centre clock of Riemann’s xi function. The negative moment \(\mathbb{E}[T^{-1/2}]=\Gamma(\frac12)^{-1}\int_0^\infty u^{-1/2}\prod_{\gamma>0}(1+u/\gamma^2)^{-1}\,du\), the product being over the ordinates \(\gamma>0\) of the zeros, is computed from \(\xi\) alone, and the book finds it to be mostly a smooth statistic of the zeros, so that a single function says little and the natural comparison with random matrix theory is in families. Compute the average of \(\mathbb{E}[T_\chi^{-1/2}]\) over the real primitive characters \(\chi\) of conductor in \([Q,2Q]\) and compare it with the prediction of the symplectic ensemble. Since \(\mathbb{E}[T^{-1/2}]\) is not a linear statistic of the zeros, the one level density of the family (Katz and Sarnak 1999) alone does not give this prediction, and all the correlation functions enter. The single function is computed in the book; the family average is open.

Earlier editions: Problem 21.4, page 793 (edition 1).

Source: ch/gl8.tex, label gl:op:negmoment.

Open problem: infinite divisibility of the family mixture of the clocks of the characters modulo q

Volume I, Problem 16.5, page 487 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\mathcal F_q\) consist of the real primitive characters modulo \(q\) together with one representative of each conjugate pair of complex primitive characters modulo \(q\). Assume GRH for the members and that their central values are nonzero, so that their clocks exist: \(T_\chi\stackrel d=\sum_{\gamma>0}e_\gamma/\gamma^2\) for a real member (the ordinates \(\gamma>0\) of \(L(s,\chi)\), with independent standard exponential variables \(e_\gamma\)) and \(T_{\chi\bar\chi}\stackrel d=\sum_{\gamma\in\Gamma_\chi}e_\gamma/\gamma^2\) for a complex pair (all ordinates of both signs). Let \(T^{(q)}\) be the clock of a uniformly chosen member. Is \(T^{(q)}\) infinitely divisible? Does any \(q\) for which \(\mathcal F_q\) has two or more members avoid, for every pair of its members, the odd configuration of the mixture criterion (two clocks whose Thorin measures are sums of unit point masses, with consecutive atoms \(t_1<t_2\) of the union belonging to different clocks and an odd number of atoms of the union in \((0,t_1]\))? It is settled that \(T^{(q)}\) is not a GGC for \(q=5\) and every weight, under GRH for the two members (their values at \(\frac12\) are nonzero numerically), and, under the same hypotheses, for every \(q\) whose family has exactly two members, when these are in the odd configuration; this does not decide whether \(T^{(5)}\) is infinitely divisible.

Earlier editions: Problem 21.5, page 794 (edition 1).

Source: ch/gl8.tex, label gl:op:family.

Open problem: a sign definite explicit formula for primes in the progressions a modulo q with a not congruent to 1

Volume I, Problem 16.6, page 487 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(F_q=\mathbb{Q}(\zeta_q)\) and \(\psi(x;q,a)=\sum_{n\le x,\ n\equiv a\ (q)}\Lambda(n)\) with \(\Lambda\) the von Mangoldt function. Suppose that the centre clock of the completed Dedekind zeta function \(\xi_{F_q}(s)=s(s-1)|d_{F_q}|^{s/2}\Gamma_{\mathbb{C}}(s)^{\varphi(q)/2}\zeta_{F_q}(s)\), with \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\) and \(d_{F_q}\) the discriminant, is a generalized gamma convolution (equivalently, GRH for the Dirichlet \(L\) functions of conductor dividing \(q\) with nonzero central values). Then the explicit formula for \(\psi(x;q,1)\), which up to the factor \(\varphi(q)\) counts the prime ideals of \(F_q\), becomes an integral against the nonnegative zero counting Thorin measure \(U_{F_q,0}\), and the positive family \(\pi^{-1}\operatorname{Re}(\xi_{F_q}'/\xi_{F_q})(\alpha+ib)\) of the basepoint family joins it to the primes. For \(a\not\equiv1\pmod q\) the characters enter the explicit formula for \(\psi(x;q,a)\) with the weights \(\bar\chi(a)\), which are complex or negative, and no sign is available. These are zero counting measures and their Poisson smoothings, so any gain must come from structure beyond counting, such as monotonicity or log concavity of the associated GGC densities, transported to bounds on \(L'/L\) near the line. Is there a sign definite explicit formula for \(\psi(x;q,a)\) with \(a\not\equiv1\pmod q\), in terms of such measures?

Earlier editions: Problem 21.6, page 794 (edition 1).

Source: ch/gl8.tex, label gl:op:signdefinite.

Open problem: certified backward continuation of the Thorin density of the basepoint family from the primes into the critical strip

Volume I, Problem 16.7, page 487 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(f\) be the completed Dedekind zeta function \(\xi_F(s)=s(s-1)|d_F|^{s/2}\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_F(s)\) of a number field \(F\) (\(d_F\) the discriminant, \(r_1\) and \(r_2\) the numbers of real and complex places, \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)\)) or the completed Dirichlet \(L\) function \(\xi(s,\chi)=(q/\pi)^{(s+a)/2}\Gamma(\frac{s+a}2)L(s,\chi)\) of a real primitive character \(\chi\) modulo \(q>1\) of parity \(a\in\{0,1\}\). The Thorin density \(\pi^{-1}\operatorname{Re}(f'/f)(\alpha+ib)\) of the basepoint family is computable from the primes, with rigorous error bounds, for \(\alpha>1\), and its positivity on every line \(\operatorname{Re}s=\alpha'\) with \(\alpha'>\alpha\) is a zero free half plane \(\operatorname{Re}s>\alpha\). How far into the critical strip (that is, to which \(\alpha<1\)) can positivity be certified from the primes alone, using smoothed prime sums of the kind that locate the zeros of a Dirichlet \(L\) function, before the ill posedness of the backward step takes over? The backward step is a deconvolution of the Poisson smoothing, which is unbounded, and the number of primes it needs grows without bound as \(\alpha\downarrow\frac12\); for the heat flow the corresponding step is stable but cannot carry real zeros. What smoothed prime sums can certify below \(\alpha=1\) is open; the book finds the output of the sine squared lemma not computable from the primes alone in practice for \(\alpha\le1\) (a computation for a character modulo \(4\) with the prime powers up to a finite cutoff), so that the minima it computes there come from the zeros.

Earlier editions: Problem 21.7, page 794 (edition 1).

Source: ch/gl8.tex, label gl:op:backward.

Open problem: the exponential sum over primes on the minor arcs as an expectation against the GGC laws of the clocks

Volume I, Problem 16.8, page 488 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(F=\mathbb{Q}\) let \(\nu_F\) be the signed Lévy measure of the completed zeta function \(\xi_F\), defined by \(\log\frac{\xi_F(\alpha+w)}{\xi_F(\alpha)}-w\frac{\xi_F'}{\xi_F}(\alpha)=\int_0^\infty(e^{-wx}-1+wx)\,e^{-\alpha x}\,\frac{\nu_F(dx)}x\) for \(\alpha>1\) and \(w\ge0\), that is \(\nu_F(dx)=\big(\frac1{1-e^{-2x}}-1-e^x\big)dx+\sum_{n\ge2}\Lambda(n)\,\delta_{\log n}(dx)\) with \(\Lambda\) the von Mangoldt function, and let the GGC laws of the clocks be the laws with Laplace transform \(\xi_F(\alpha)/\xi_F(\alpha+\sqrt u)\), \(u\ge0\), for \(\alpha>1\). The atoms of \(\nu_F\) at the primes \(p\) have mass \(\log p\), the weights that enter the exponential sum \(\mathcal S(\alpha)=\sum_{\varpi N<p\le N}e(p\alpha)\) of the circle method for Linnik’s problem (with \(e(x)=\exp(2\pi ix)\) and \(\varpi>0\) small); the prime powers in \(\nu_F\) do not enter \(\mathcal S(\alpha)\). Can \(\mathcal S(\alpha)\) on the minor arcs be written as an expectation against the GGC laws of the clocks? Such a representation would be a different route to the supremum bound for \(\mathcal S\) on the minor arcs; nothing in the chapter shows that it exists.

Earlier editions: Problem 21.8, page 795 (edition 1).

Source: ch/gl8.tex, label gl:op:arith.

Open problem: the best minor arc exponent in Linnik’s problem from a zero free half plane for all Dirichlet L functions

Volume I, Problem 16.9, page 488 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Consider the circle method for Linnik’s problem of writing large even \(N\) as \(p_1+p_2+2^{k_1}+\cdots+2^{k_K}\), with \(\mathcal S(\alpha)=\sum_{\varpi N<p\le N}e(p\alpha)\), \(e(x)=\exp(2\pi ix)\). Assume \(L(s,\chi)\ne0\) for \(\operatorname{Re}s>\theta_0\) and every Dirichlet character \(\chi\). What is the best exponent \(\theta(\theta_0)\) in the minor arc bound \(\mathcal S(\alpha)\ll N^{\theta+o(1)}\) obtainable by combining the explicit formula with Vaughan’s identity? A one line heuristic balance gives \(\theta=\frac12(1+\theta_0)\), which is better than the unconditional exponent \(\frac{263}{308}\) exactly when \(\theta_0<\frac{109}{154}\), so it does nothing at \(\theta_0=\frac78\); it is not shown to be optimal. Does a hybrid bound, using the half plane only for the characters of small conductor, improve the parameter \(\lambda\) of the large deviation estimate for the sum over powers of two (the measure of the set where \(|\sum_{1\le\nu\le L}e(2^\nu\alpha)|\ge\lambda L\))? And does the half plane, which excludes exceptional zeros of large conductor, raise the constant of the major arc main term? The paper of Heath-Brown and Puchta (2002) suggests that it does not by itself: they state that zeros close to \(\sigma=1\) play no role there; the constant is a sum over small divisors, and the restriction \(q\le P=N^{45/154-4\varpi}\) of the major arcs comes from zero density estimates whose critical zeros have \(\sigma\le\frac34\) and are not removed by a half plane at \(\frac78\).

Earlier editions: Problem 21.9, page 795 (edition 1).

Source: ch/gl8.tex, label gl:op:quasi.

Open problem: does a finite order of total positivity of the density force the zeros onto the line, in a class of even entire functions containing xi

Volume I, Problem 16.10, page 488 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(T\) be the centre clock of Riemann’s xi function, \(Z\) an independent standard normal variable and \(Y=\sqrt{2T}\,Z\). RH holds if and only if the density of \(Y\) is a Pólya frequency function of infinite order, that is, every determinant \(\det[f(x_i-y_j)]\) with \(x_1<\dots<x_n\) and \(y_1<\dots<y_n\) is nonnegative. For this one density the question whether a finite order already implies RH has a trivial answer: the orders decrease, so if RH holds every such implication is true, and if RH fails the property fails at some finite order, where the implication is vacuous. The question has content for a class of functions. Let \(\mathcal E\) be the class of even entire functions \(F\) with \(F(0)=1\) and the order, type and reflection symmetry of \(\xi(\frac12+s)/\xi(\frac12)\) for which \(1/F\) is the characteristic function of a probability density \(f_F\); the density of \(Y\) is \(f_F\) for \(F=\xi(\frac12+\cdot)/\xi(\frac12)\). Is there a finite \(k\) such that, for every \(F\in\mathcal E\), the property that \(f_F\) is a Pólya frequency function of order \(k\) implies that every zero of \(F\) lies on the imaginary axis?

Earlier editions: Problem 21.10, page 795 (edition 1).

Source: ch/gl8.tex, label gl:op:pf.

Volume I, Chapter 17. Completeness of the secant dilates and the Möbius problem

The integer dilates of the hyperbolic secant are complete in L1 under an integrability condition at the origin and at infinity

Volume I, Theorem 17.1.4, page 493 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\operatorname{sech}=1/\cosh\) and let \(g\) be a measurable function on \((0,\infty)\) with \(\int_0^1|g(x)|x^{-1}dx+\int_1^\infty|g(x)|e^{-x}dx<\infty\). If \(\int_0^\infty g(x)\operatorname{sech}(nx)\,dx=0\) for all \(n\in\mathbb N\), then \(g=0\) almost everywhere. The same holds with \(\mathbb N\) replaced by a set \(\Lambda\subset(0,\infty)\) containing an interval, provided the integrals \(\int g\operatorname{sech}(\lambda x)\,dx\), \(\lambda\in\Lambda\), converge absolutely and vanish. The proof uses \(e^{-kx}=\frac12\sum_{n\ge1}\mu(n)\chi_4(n)\operatorname{sech}(nkx)\) for \(k\in\mathbb N\), with \(\mu\) the M"obius function and \(\chi_4\) the nonprincipal character modulo \(4\), together with Lerch’s theorem that \(\int_0^\infty e^{-kx}h(x)\,dx=0\) for all \(k\in\mathbb N\) forces \(h=0\) when \(\int|h|e^{-x}<\infty\).

Earlier editions: Theorem 29.1.4, page 1025 (edition 1).

Source: ch/lv1.tex, label lv:thm:L1.

The Möbius secant series converges to the exponential in every weight W_c with c at least one

Volume I, Theorem 17.2.2, page 494 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\), let \(\beta(s)=L(s,\chi_4)=1-3^{-s}+5^{-s}-\cdots\) be the Dirichlet beta function, with \(\chi_4\) the nonprincipal Dirichlet character modulo \(4\), let \(\mu\) be the M"obius function, \(M_\chi(x)=\sum_{n\le x}\mu(n)\chi_4(n)\) and \(S_N(x)=\frac12\sum_{n\le N}\mu(n)\chi_4(n)\operatorname{sech}(nx)\). For \(c\ge1\), \(S_N\to e^{-x}\) in \(\mathcal W_c\), that is \(\|S_N-e^{-x}\|_c\to0\). For \(c>1\) this follows from \(|\sum_{n\le N}\mu(n)\chi_4(n)n^{-s}|\le\zeta(c)\) and dominated convergence in the Mellin Plancherel formula; for \(c=1\) it follows from the prime number theorem \(M_\chi(x)\ll x\exp(-a\sqrt{\log x})\) for some \(a>0\) and partial summation.

Earlier editions: Theorem 29.2.2, page 1027 (edition 1).

Source: ch/lv1.tex, label lv:thm:c1.

In one weight the secant dilates over any set of dilations with divergent reciprocal sum are complete, whatever the zeros of the beta function do

Volume I, Theorem 17.3.2, page 495 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\). Let \(c>0\) and let \(\Lambda\subset[1,\infty)\) satisfy \(\sum_{\lambda\in\Lambda}1/\lambda=\infty\). Then \(\{\operatorname{sech}(\lambda x)\}_{\lambda\in\Lambda}\) is complete in \(\mathcal W_c\), that is, a function of \(\mathcal W_c\) orthogonal to every \(\operatorname{sech}(\lambda x)\) is zero. In particular the integer dilates \(\operatorname{sech}(nx)\) and the prime dilates \(\operatorname{sech}(px)\), \(p\) prime, are complete in every \(\mathcal W_c\), including \(c=\frac12\). The proof writes the Mellin transform of the orthogonal function against \(\Gamma(c+it)\beta(c+it)\) and applies Carleman’s formula to an analytic function on the right half plane that vanishes on \(\Lambda\).

Earlier editions: Theorem 29.3.2, page 1028 (edition 1).

Source: ch/lv1.tex, label lv:thm:single.

Completeness of the exponential in two weights forces a zero free strip for the beta function

Volume I, Proposition 17.5.1, page 498 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\), let \(\beta(s)=L(s,\chi_4)=1-3^{-s}+5^{-s}-\cdots\) be the Dirichlet beta function, with \(\chi_4\) the nonprincipal Dirichlet character modulo \(4\), and let \(0<a<b\) with \(\mathcal W_a\cap\mathcal W_b\) normed by \(\|f\|_a+\|f\|_b\). If \(e^{-x}\) is a limit in \(\mathcal W_a\cap\mathcal W_b\) of finite linear combinations of the functions \(\operatorname{sech}(nx)\), \(n\in\mathbb N\), then \(\beta(s)\ne0\) for \(a<\Re s<b\).

Earlier editions: Proposition 29.5.1, page 1030 (edition 1).

Source: ch/lv1.tex, label lv:prop:nec.

Two weight completeness of the secant dilates is equivalent to a zero free half plane for L(s,chi_4) and to convergence of the Möbius secant series

Volume I, Theorem 17.5.2, page 498 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\), let \(\beta(s)=L(s,\chi_4)=1-3^{-s}+5^{-s}-\cdots\) be the Dirichlet beta function, with \(\chi_4\) the nonprincipal Dirichlet character modulo \(4\), and let \(\mu\) be the M"obius function, \(M_\chi(x)=\sum_{n\le x}\mu(n)\chi_4(n)\) and \(S_N(x)=\frac12\sum_{n\le N}\mu(n)\chi_4(n)\operatorname{sech}(nx)\). For \(\frac12\le\theta<1\) the following are equivalent: (i) \(L(s,\chi_4)\ne0\) for \(\Re s>\theta\); (ii) \(\{\operatorname{sech}(nx)\}_{n\in\mathbb N}\) is complete in \(\mathcal W_a\cap\mathcal W_b\) (normed by \(\|f\|_a+\|f\|_b\)) for all \(\theta<a<b<1\); (iii) \(S_N\to e^{-x}\) in \(\mathcal W_c\) for every \(c\in(\theta,1)\). With \(\theta=\frac12\), (i) is the generalised Riemann hypothesis for \(L(s,\chi_4)\). When these hold, \(\|S_N-e^{-x}\|_c=O_\varepsilon(N^{\theta-c+\varepsilon})\) for \(c\in(\theta,1)\).

Earlier editions: Theorem 29.5.2, page 1030 (edition 1).

Source: ch/lv1.tex, label lv:thm:two.

The zeta analogue: dilates of the Bose kernel with its pole removed are complete in one weight and in two weights exactly when zeta has the corresponding zero free half plane

Volume I, Theorem 17.6.1, page 499 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\), let \(\varphi(x)=1/(e^x-1)-1/x\) (so \(\varphi\in\mathcal W_c\) exactly for \(0<c<1\) and \(\int_0^\infty x^{s-1}\varphi(x)\,dx=\Gamma(s)\zeta(s)\) for \(0<\Re s<1\)), let \(\mu\) be the M"obius function, \(M(x)=\sum_{n\le x}\mu(n)\) and \(T_N(x)=\sum_{n\le N}\mu(n)\varphi(nx)\). (1) For every \(c\in(0,1)\) the dilates \(\{\varphi(nx)\}_{n\in\mathbb N}\) are complete in \(\mathcal W_c\), unconditionally. (2) Let \(\frac12\le\theta<1\) and suppose \(M(x)=O_\varepsilon(x^{\theta+\varepsilon})\) for every \(\varepsilon>0\). Then \(T_N\to e^{-x}\) in \(\mathcal W_c\) with \(\|T_N-e^{-x}\|_c=O_\varepsilon(N^{\theta-c+\varepsilon})\) for \(\theta<c<1\) and \(0<\varepsilon<c-\theta\). (3) For \(\frac12\le\theta<1\), completeness of \(\{\varphi(nx)\}\) in \(\mathcal W_a\cap\mathcal W_b\) for all \(\theta<a<b<1\) is equivalent to \(\zeta(s)\ne0\) for \(\Re s>\theta\) (the pole at \(1\) being allowed); with \(\theta=\frac12\) it is equivalent to the Riemann hypothesis.

Earlier editions: Theorem 29.6.1, page 1031 (edition 1).

Source: ch/lv1.tex, label lv:thm:zeta.

Under a polynomial bound for the reciprocal of the beta function on the line, the secant dilates over a set with convergent reciprocal sum are not complete

Volume I, Theorem 17.10.2, page 502 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\) and let \(\beta(s)=L(s,\chi_4)=1-3^{-s}+5^{-s}-\cdots\) be the Dirichlet beta function, with \(\chi_4\) the nonprincipal Dirichlet character modulo \(4\). Let \(c>0\) and assume \((\mathrm N_c)\): \(1/\beta(c+it)\ll(1+|t|)^{A}\) for some \(A\ge0\) and all real \(t\). If \(\Lambda\subset[1,\infty)\) and \(\sum_{\lambda\in\Lambda}1/\lambda<\infty\), then \(\{\operatorname{sech}(\lambda x)\}_{\lambda\in\Lambda}\) is not complete in \(\mathcal W_c\). So under \((\mathrm N_c)\) the dilates are complete in \(\mathcal W_c\) if and only if \(\sum_{\lambda\in\Lambda}1/\lambda=\infty\). The hypothesis \((\mathrm N_c)\) holds unconditionally for \(c\ge1\) and fails at \(c=\frac12\), where \(\beta\) has zeros on the line.

Earlier editions: Theorem 29.10.2, page 1035 (edition 1).

Source: ch/lv2.tex, label lv:thm:nec.

The secant dilates are complete in W_c if and only if the reciprocal sum of the dilations diverges, for c at least one, for c above 7/8 given a zero free half plane, and for c not one half under GRH

Volume I, Corollary 17.10.3, page 503 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The zero free half plane \(\Re s>\frac78\) for \(L(s,\chi_4)\), which holds if the quasi Riemann hypothesis preprints (OpenAI, announced for Dirichlet \(L\) functions, unrefereed and not checked for this book) cover the character modulo \(4\); the part for \(c\ge1\) is unconditional, and the part for \(c\ne\frac12\) needs the generalised Riemann hypothesis for \(\beta\).

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\) and let \(\beta(s)=L(s,\chi_4)=1-3^{-s}+5^{-s}-\cdots\) be the Dirichlet beta function, with \(\chi_4\) the nonprincipal Dirichlet character modulo \(4\). For \(\Lambda\subset[1,\infty)\), \(\{\operatorname{sech}(\lambda x)\}_{\lambda\in\Lambda}\) is complete in \(\mathcal W_c\) if and only if \(\sum_{\lambda\in\Lambda}1/\lambda=\infty\), in each of these cases: for \(c\ge1\) (unconditionally); for \(c>\frac78\) given that \(L(s,\chi_4)\ne0\) for \(\Re s>\frac78\); and for every \(c>0\) with \(c\ne\frac12\) under the generalised Riemann hypothesis for \(\beta\).

Earlier editions: Corollary 29.10.3, page 1036 (edition 1).

Source: ch/lv2.tex, label lv:cor:iff.

Salem’s criterion in L1: dilates of the Fermi or secant kernel over a divergent set are complete if and only if the Mellin transform has no zero on the line

Volume I, Theorem 17.11.1, page 504 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\sigma>0\), let \(k\) be \(1/(e^x+1)\) or \(\operatorname{sech}x\), and let \(\Lambda\subset[1,\infty)\) satisfy \(\sum_{\lambda\in\Lambda}1/\lambda=\infty\). Then \(\{x^{\sigma-1}k(\lambda x)\}_{\lambda\in\Lambda}\) is complete in \(L^1(0,\infty)\) if and only if the Mellin transform of \(k\) has no zero on the line \(\Re s=\sigma\). In particular, for \(\frac12<\sigma<1\) the integer dilates of the Fermi kernel \(1/(e^x+1)\) are complete in \(L^1\) if and only if the Riemann zeta function has no zero on \(\Re s=\sigma\), and the integer dilates of \(\operatorname{sech}\) if and only if the Dirichlet beta function \(\beta=L(\cdot,\chi_4)\) has none.

Earlier editions: Theorem 29.11.1, page 1037 (edition 1).

Source: ch/lv2.tex, label lv:thm:salem.

The weighted error of an arbitrary combination of secant dilates is an integral over the Mellin line with the factor 2 in 2D_N - 1/beta, split at a segment

Volume I, Proposition 17.12.1, page 507 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\), let \(\beta(s)=L(s,\chi_4)=1-3^{-s}+5^{-s}-\cdots\) be the Dirichlet beta function, with \(\chi_4\) the nonprincipal Dirichlet character modulo \(4\), \(c>0\), \(T\ge1\) and \(a_n\) arbitrary complex coefficients, with \(D_N(s)=\sum_{n\le N}a_nn^{-s}\), \(\|a\|_{N,c}=\sum_{n\le N}|a_n|n^{-c}\) and \(K_c(t)=\Gamma(c+it)\beta(c+it)\). Then \(\|e^{-x}-\sum_{n\le N}a_n\operatorname{sech}(nx)\|_c^2=\frac1{2\pi}\int_{\mathbb R}|\Gamma(c+it)|^2|1-2\beta(c+it)D_N(c+it)|^2dt\), and this is at most \(\frac1{2\pi}\int_{|t|\le T}|\Gamma(c+it)|^2|1-2\beta D_N|^2dt+C_ce^{-\pi T}T^{2c}(1+\|a\|_{N,c})^2\) for a constant \(C_c\) depending only on \(c\). Where \(\beta\ne0\) the integrand of the first term is \(|K_c(t)|^2|2D_N-1/\beta|^2\). The Mellin transform of \(\operatorname{sech}(nx)\) carries the factor \(2\), which is why \(2D_N-1/\beta\) appears and not \(D_N-1/\beta\).

Earlier editions: Proposition 29.12.1, page 1040 (edition 1).

Source: ch/lv2.tex, label lv:prop:segment.

The Mellin transform of the symmetric Meixner density of order t is an explicit multiple of an entire Dirichlet type series Z_t

Volume I, Theorem 17.13.1, page 508 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(t>0\) let \(f_t(x)=|\Gamma(\frac t2+\frac{ix}2)|^2\), \(x>0\). For \(0<\Re s<1\), \(M_t(s)=\int_0^\infty x^{s-1}f_t(x)\,dx=2^s\pi\,\Gamma(t)Z_t(1-s)/\sin(\pi s/2)\), where \(Z_t\) is the entire function \(Z_t(w)=\Gamma(w)^{-1}\int_0^\infty u^{w-1}(2\cosh(u/2))^{-t}\,du=\sum_{k\ge0}(-1)^k\binom{t+k-1}{k}(k+\frac t2)^{-w}\) for \(\Re w>t-1\). With \(\beta\) the Dirichlet beta function, \(\eta(w)=(1-2^{1-w})\zeta(w)\) the Dirichlet eta function and \(\zeta\) the Riemann zeta function, \(Z_1(w)=2^w\beta(w)\), \(Z_2(w)=\eta(w-1)=(1-2^{2-w})\zeta(w-1)\) and \(Z_3(w)=2^{w-3}(\beta(w)-\beta(w-2))\). For integer \(t\), \(Z_t\) is a linear combination of \(\eta(w-j)\), \(j\) odd, when \(t\) is even, and of \(2^{w-2i}\beta(w-2i)\) when \(t\) is odd.

Earlier editions: Theorem 29.13.1, page 1041 (edition 1).

Source: ch/lv2.tex, label lv:thm:meixner.

The dilates of every symmetric Meixner density are complete in every weight over a set of dilations with divergent reciprocal sum

Volume I, Corollary 17.13.2, page 509 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\) and \(f_t(x)=|\Gamma(\frac t2+\frac{ix}2)|^2\) for \(t>0\). For every \(t>0\) and every \(c>0\), \(\{f_t(\lambda x)\}_{\lambda\in\Lambda}\) with \(\Lambda\subset[1,\infty)\) is complete in \(\mathcal W_c\) whenever \(\sum_{\lambda\in\Lambda}1/\lambda=\infty\); in particular the integer dilates are complete. The proof checks that \(f_t\in\mathcal W_c\) and that its Mellin transform on \(\Re s=c\) is nonzero almost everywhere with the bound \(C(1+|\tau|)^Ae^{-\pi|\tau|/2}\), the only properties used in the proof of single weight completeness.

Earlier editions: Corollary 29.13.2, page 1042 (edition 1).

Source: ch/lv2.tex, label lv:cor:meixner.

For the cosecant kernel the Möbius series for x e^{-x} converges in every weight, so completeness holds unconditionally in one and two weights

Volume I, Theorem 17.13.3, page 510 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(\mathcal W_c=L^2((0,\infty),x^{2c-1}dx)\) and \(\operatorname{sech}=1/\cosh\) and let \(\mu\) be the M"obius function, \(c_n=\mu(n)\) for odd \(n\) and \(c_n=0\) for even \(n\), and \(\operatorname{csch}=1/\sinh\). Then for every \(x>0\), with absolute convergence, \(e^{-x}=\frac12\sum_{n\ge1}c_n\operatorname{csch}(nx)\) and \(xe^{-x}=\frac12\sum_{n\ge1}(c_n/n)(nx)\operatorname{csch}(nx)\), and \(\sum_n(c_n/n)n^{-s}=1/((1-2^{-s-1})\zeta(s+1))\) converges absolutely for \(\Re s>0\). Consequently the series for \(xe^{-x}\) converges absolutely in every \(\mathcal W_c\), \(c>0\), and \(\{(nx)\operatorname{csch}(nx)\}_{n\in\mathbb N}\) is complete in every \(\mathcal W_c\) and in every \(\mathcal W_a\cap\mathcal W_b\), \(0<a<b\), unconditionally.

Earlier editions: Theorem 29.13.3, page 1043 (edition 1).

Source: ch/lv2.tex, label lv:thm:t2.

A zero free half plane for Xi is equivalent to F_theta being a GGC Laplace transform, with Thorin measure a sum of squared folded Cauchy laws centred at the ordinates

Volume I, Theorem 17.14.2, page 512 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi\) be either the Riemann function \(\xi\) or \(\xi(\cdot,\chi_4)\) with \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)\beta(s)\); both are entire of order one, real and positive on the real axis, satisfy \(\Xi(s)=\Xi(1-s)\) and have the Hadamard product \(\Xi(s)=\Xi(0)\prod_\rho(1-s/\rho)\) over the nontrivial zeros \(\rho\), paired with \(\bar\rho\). For \(\theta\ge\frac12\) let \(F_\theta(s)=\Xi(\theta)/\Xi(\theta+\sqrt s)\) for \(s\) in the plane cut along \((-\infty,0]\), with the principal square root. A probability law on \([0,\infty)\) is a generalised gamma convolution (GGC) if its Laplace transform is \(\exp(-as-\int_0^\infty\log(1+s/u)U(du))\) with \(a\ge0\) and a locally finite positive Thorin measure \(U\) on \((0,\infty)\) with \(\int_{(0,1)}|\log u|U(du)<\infty\) and \(\int_1^\infty u^{-1}U(du)<\infty\). For \(a\ge0\) and \(\gamma>0\) let \(Q_{0,\gamma}=\delta_{\gamma^2}\) and, for \(a>0\), \(Q_{a,\gamma}\) be the probability measure on \((0,\infty)\) that is the image under \(v\mapsto v^2\) of the law of \(|\gamma+aC|\) with \(C\) standard Cauchy, with density \(\frac1{2\pi\sqrt u}[\frac a{a^2+(\sqrt u-\gamma)^2}+\frac a{a^2+(\sqrt u+\gamma)^2}]\). Let \(\theta\ge\frac12\). The following are equivalent: (i) \(\Xi\) has no zeros in \(\Re s>\theta\); (ii) \(F_\theta\) is the Laplace transform of a GGC; (iii) \(F_\theta\) is analytic in the cut plane. When they hold, the Thorin measure of \(F_\theta\) is \(U_\theta=\sum_{\rho,\ \Im\rho>0}m_\rho Q_{\theta-\Re\rho,\ \Im\rho}\) over the distinct zeros \(\rho\) in the upper half plane with multiplicity \(m_\rho\), and \(F_\theta(s)=\exp(-\int_0^\infty\log(1+s/u)U_\theta(du))\) with no linear term. For \(\Xi=\xi(\cdot,\chi_4)\), (i) is equivalent to two weight completeness of the secant dilates on \((\theta,1)\) (for \(\theta<1\), the two weight criterion for \(\beta\)); for \(\Xi=\xi\), to two weight completeness of the dilates of \(\varphi(x)=1/(e^x-1)-1/x\) (the zeta version of the two weight criterion).

Earlier editions: Theorem 29.14.2, page 1045 (edition 1).

Source: ch/lv2.tex, label lv:thm:thorin.

Reciprocal xi and reciprocal xi of chi_4 are GGC Laplace transforms from theta equal to one unconditionally and from 7/8 given the zero free half plane

Volume I, Corollary 17.14.3, page 513 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The zero free half plane \(\Re s>\frac78\) for the Riemann zeta function and, for the statement about \(\xi(\cdot,\chi_4)\), for \(L(s,\chi_4)\), from the quasi Riemann hypothesis preprints (OpenAI, unrefereed, announced for Dirichlet \(L\) functions; that the theorem covers the character modulo \(4\) has not been checked); the part for \(\theta\ge1\) is unconditional.

Statement. Let \(\Xi\) be either the Riemann function \(\xi\) or \(\xi(\cdot,\chi_4)\) with \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)\beta(s)\); both are entire of order one, real and positive on the real axis, satisfy \(\Xi(s)=\Xi(1-s)\) and have the Hadamard product \(\Xi(s)=\Xi(0)\prod_\rho(1-s/\rho)\) over the nontrivial zeros \(\rho\), paired with \(\bar\rho\). For \(\theta\ge\frac12\) let \(F_\theta(s)=\Xi(\theta)/\Xi(\theta+\sqrt s)\) for \(s\) in the plane cut along \((-\infty,0]\), with the principal square root. A probability law on \([0,\infty)\) is a generalised gamma convolution (GGC) if its Laplace transform is \(\exp(-as-\int_0^\infty\log(1+s/u)U(du))\) with \(a\ge0\) and a locally finite positive Thorin measure \(U\) on \((0,\infty)\) with \(\int_{(0,1)}|\log u|U(du)<\infty\) and \(\int_1^\infty u^{-1}U(du)<\infty\). Let \(U_\theta=\sum_{\Im\rho>0}m_\rho Q_{\theta-\Re\rho,\Im\rho}\), over the distinct zeros \(\rho\) with \(\Im\rho>0\) and with \(m_\rho\) the multiplicity of \(\rho\), be the Thorin measure of \(F_\theta\) when \(\Xi\) has no zeros in \(\Re s>\theta\), where for \(a\ge0\) and \(\gamma>0\), \(Q_{0,\gamma}=\delta_{\gamma^2}\) and, for \(a>0\), \(Q_{a,\gamma}\) is the image under \(v\mapsto v^2\) of the law of \(|\gamma+aC|\) with \(C\) standard Cauchy. For every \(\theta\ge1\), \(F_\theta\) for \(\xi\) and for \(\xi(\cdot,\chi_4)\) is the Laplace transform of a GGC with the absolutely continuous Thorin measure \(U_\theta\), since the zeros have \(\Re\rho<1\). Conditionally on the zero free half plane \(\Re s>\frac78\) the GGC representation holds for every \(\theta\ge\frac78\), and the Thorin measure is absolutely continuous for \(\theta>\frac78\), and at \(\theta=\frac78\) if no zero lies on \(\Re s=\frac78\); a zero \(\rho\) on that line contributes an atom \(m_\rho\delta_{\gamma_\rho^2}\), \(\gamma_\rho=\Im\rho\). For \(\theta=\frac12\) the statement for \(\xi\) is the Riemann Hypothesis and the Thorin measure is then \(\sum_{\gamma>0}m_\gamma\delta_{\gamma^2}\), over the ordinates \(\gamma>0\) of the zeros \(\frac12+i\gamma\) of \(\xi\) with multiplicity \(m_\gamma\).

Earlier editions: Corollary 29.14.3, page 1045 (edition 1).

Source: ch/lv2.tex, label lv:cor:uncond.

The symmetrised Thorin measures of F_theta form a Cauchy convolution semigroup in theta

Volume I, Theorem 17.14.4, page 513 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi\) be either the Riemann function \(\xi\) or \(\xi(\cdot,\chi_4)\) with \(\xi(s,\chi_4)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)\beta(s)\); both are entire of order one, real and positive on the real axis, satisfy \(\Xi(s)=\Xi(1-s)\) and have the Hadamard product \(\Xi(s)=\Xi(0)\prod_\rho(1-s/\rho)\) over the nontrivial zeros \(\rho\), paired with \(\bar\rho\). For \(\theta\ge\frac12\) let \(F_\theta(s)=\Xi(\theta)/\Xi(\theta+\sqrt s)\) for \(s\) in the plane cut along \((-\infty,0]\), with the principal square root. A probability law on \([0,\infty)\) is a generalised gamma convolution (GGC) if its Laplace transform is \(\exp(-as-\int_0^\infty\log(1+s/u)U(du))\) with \(a\ge0\) and a locally finite positive Thorin measure \(U\) on \((0,\infty)\) with \(\int_{(0,1)}|\log u|U(du)<\infty\) and \(\int_1^\infty u^{-1}U(du)<\infty\). Let \(\mathrm{Cauchy}(\gamma,a)\) be the Cauchy law of centre \(\gamma\) and scale \(a>0\) and \(\mathrm{Cauchy}(\gamma,0)=\delta_\gamma\), and let \(U_\theta=\sum_{\Im\rho>0}m_\rho Q_{\theta-\Re\rho,\Im\rho}\), over the distinct zeros \(\rho\) with \(\Im\rho>0\) and with \(m_\rho\) the multiplicity of \(\rho\), be the Thorin measure of \(F_\theta\) when \(\Xi\) has no zeros in \(\Re s>\theta\), where for \(a\ge0\) and \(\gamma>0\), \(Q_{0,\gamma}=\delta_{\gamma^2}\) and, for \(a>0\), \(Q_{a,\gamma}\) is the image under \(v\mapsto v^2\) of the law of \(|\gamma+aC|\) with \(C\) standard Cauchy. Suppose \(\Xi\) has no zeros in \(\Re s>\theta_0\). For \(\theta\ge\theta_0\) let \(\widetilde U_\theta=\frac12\sum_\rho m_\rho\,\mathrm{Cauchy}(\Im\rho,\theta-\Re\rho)\) be the symmetric measure on \(\mathbb R\) in the variable \(v\), summed over the distinct zeros \(\rho\) of both half planes with multiplicity \(m_\rho\) (the factor \(\frac12\) is needed: without it the image is \(2U_\theta\)). Then \(U_\theta\) is the image of \(\widetilde U_\theta\) under \(v\mapsto v^2\), and for \(\theta_0\le\theta<\theta'\), \(\widetilde U_{\theta'}=\widetilde U_\theta*\mathrm{Cauchy}(0,\theta'-\theta)\). In particular, if all zeros lie on \(\Re s=\theta_0\) (which for the completed functions forces \(\theta_0=\frac12\)), then \(\widetilde U_\theta\) is the Poisson smoothing at width \(\theta-\theta_0\) of \(\frac12\sum_\rho m_\rho\delta_{\Im\rho}\) and converges vaguely to it as \(\theta\downarrow\theta_0\). For \(\xi\) this is the Cauchy flow of the book’s Poisson structure result, credited to Nicholas Polson (Chapter 8 of his monograph On Hilbert’s 8th Problem, 2026).

Earlier editions: Theorem 29.14.4, page 1046 (edition 1).

Source: ch/lv2.tex, label lv:thm:semigroup.

Volume I, Chapter 18. Hitting times, Krein strings and the Pólya Bessel clock

Laplace transform of Bessel hitting times with curvature

Volume I, Proposition 18.1.6, page 525 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For real \(\kappa\) let \(s_\kappa(z)=\sin(\sqrt\kappa z)/\sqrt\kappa\) (so that \(s_{-1}=\sinh\), \(s_0(z)=z\) and \(s_1=\sin\)), and let \(J^{(\kappa)}_a\) be the hitting time of the radius \(a\) from the centre by the radial part of Brownian motion on the three dimensional space form of curvature \(\kappa\), the diffusion with generator \(\frac12\frac{d^2}{dx^2}+\sqrt\kappa\cot(\sqrt\kappa x)\frac d{dx}\), with \(a<\pi/\sqrt\kappa\) when \(\kappa>0\). For \(\kappa=0\) this is the hitting time \(J_a\) of \(a\) by the three dimensional Bessel process started at \(0\). For \(u\ge0\), \(\mathbb{E}e^{-uJ^{(\kappa)}_a}=\frac{\sqrt{2u-\kappa}\;s_\kappa(a)}{\sinh\big(a\sqrt{2u-\kappa}\big)}\) (read through \(\sinh(ix)/i=\sin x\) when \(\kappa>2u\)), the Laplace transform of a GGC with Thorin measure \(\sum_{n\ge1}\delta_{\pi^2n^2/2a^2-\kappa/2}\). Equivalently, the law of \(J^{(\kappa)}_a\) is the exponential tilt \(e^{\kappa t/2}\Pr(J_a\in dt)/\mathbb{E}e^{\kappa J_a/2}\) of the law of \(J_a\).

Earlier editions: Proposition 8.7.6, page 313 (edition 1).

Source: ch/biane.tex, label prop:biane:skappa.

Exact quantisation rule for the zeros of Pólya’s Bessel function

Volume I, Proposition 18.1.8, page 529 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_\mu\) and \(I_\mu\) be the modified Bessel functions of the second and first kind, \(P(\theta)=K_{i\theta/2}(2\pi)\), \(\nu_1<\nu_2<\cdots\) the positive zeros of \(\nu\mapsto K_{i\nu}(2\pi)\) (so that the zeros of \(P\) are \(\pm2\nu_n\)), \(N_P(T)\) the number of zeros of \(P\) in \((0,T]\) and \(N_0(T)=\frac T{2\pi}\log\frac T{2\pi e}\). For real \(\nu\) put \(\vartheta_P(\nu)=\operatorname{Im}\log\Gamma(1+i\nu)-\nu\log\pi-\arg{}_0F_1(;1+i\nu;\pi^2)\), with \({}_0F_1(;b;z)=\sum_{k\ge0}z^k/((b)_kk!)\) and the branches continuous in \(\nu\) and equal to zero at \(\nu=0\), where \({}_0F_1(;1;\pi^2)=I_0(2\pi)>0\), and \(x(T)=\frac1\pi\vartheta_P(T/2)\). Then (i) for \(\nu>0\), \(I_{i\nu}(2\pi)\ne0\) and \(K_{i\nu}(2\pi)=\frac{\pi\,|I_{i\nu}(2\pi)|\,\sin\vartheta_P(\nu)}{\sinh\pi\nu}\). (ii) \(\vartheta_P(\nu)>0\) for \(\nu>0\), \(\vartheta_P(\nu_n)=n\pi\) for every \(n\), and the number of zeros of \(\nu\mapsto K_{i\nu}(2\pi)\) in \((0,\nu)\) is \(\lceil\vartheta_P(\nu)/\pi\rceil-1\); equivalently \(N_P(T)=\lfloor x(T)\rfloor\) for every \(T>0\) that is not a zero of \(P\). (iii) As \(\nu\to\infty\), \(\vartheta_P(\nu)=\nu\log\frac\nu\pi-\nu+\frac\pi4+\frac{\pi^2-\frac1{12}}\nu+O(\nu^{-3})\), so that \(x(T)=N_0(T)+\frac14+\big(2\pi-\frac1{6\pi}\big)\frac1T+O(T^{-3})\) as \(T\to\infty\).

Earlier editions: Proposition 8.7.8, page 316 (edition 1).

Source: ch/biane.tex, label prop:biane:phase.

No finite rank selfadjoint perturbation of a Gram regular operator has the zeta spectrum

Volume I, Theorem 18.1.9, page 531 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\vartheta\) be the Riemann Siegel theta function. Call a selfadjoint operator \(A\), bounded below with compact resolvent, Gram regular if \(N_A(t)=\#\{\mu\le t^2/4\}\), counted over the eigenvalues \(\mu\) of \(A\), satisfies \(N_A(t)=\vartheta(t)/\pi+O(1)\). The target is the multiset \(\{\gamma^2/4\}\), with \(\gamma\) running over the ordinates of the nontrivial zeros of \(\zeta\) with \(\operatorname{Re}\gamma>0\) (real exactly under the Riemann Hypothesis). Let \(A\) be Gram regular. (i) If \(F\) is a bounded selfadjoint operator of finite rank, the spectrum of \(A+F\) is not \(\{\gamma^2/4\}\) with multiplicity, even when a finite set of eigenvalues and of ordinates is discarded. (ii) The same holds for every selfadjoint \(B\) such that \((A+c)^{-1}-(B+c)^{-1}\) has finite rank for some \(c\) with \(A+c>0\); this covers finite rank perturbations of \(A^{-1}\) when \(0\notin\operatorname{spec}A\) and changes of boundary condition. (iii) The same holds for the target \(\{\gamma^2/4+c'\}\) with \(c'\in\mathbb{R}\). The statement is unconditional. It rests on Selberg’s theorem that \(S(t)=\frac1\pi\arg\zeta(\frac12+it)\) is unbounded above and below, and it offers no candidate operator.

Source: ch/biane.tex, label thm:biane:nogo.

Morse type half line operators are Gram regular

Volume I, Proposition 18.1.10, page 531 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. (a) For real \(k\) let \(B_k=-\frac{d^2}{dx^2}+e^{2x}+2ke^x\) on \([\log2\pi,\infty)\) with a Dirichlet condition at \(\log2\pi\). Its eigenvalues are \(\mu=\nu^2\) where, for large \(\nu\), \(\vartheta(2\nu)+\frac{3\pi}8-\frac{k\pi}2+O_k(1/\nu)\equiv0 \bmod \pi\), with \(\vartheta\) the Riemann Siegel theta function, so \(B_k\) is Gram regular (its counting function in the variable \(t=2\sqrt\mu\) is \(\vartheta(t)/\pi+O(1)\)). (b) Let \(V\) be real and locally bounded on \([\log2\pi,\infty)\) with \(|V(x)|\le Ce^x\). Then \(-\frac{d^2}{dx^2}+e^{2x}+V\) on that half line with the same Dirichlet condition is Gram regular, so no finite rank selfadjoint perturbation of it has the spectrum \(\{\gamma^2/4\}\) of the squared half ordinates of the zeta zeros. Consequently any real potential whose levels reach the zeros up to finite rank satisfies \(\limsup_{x\to\infty}|V(x)|e^{-x}=\infty\).

Source: ch/biane.tex, label prop:biane:morse.

Discrepancy span against a constant phase, rank one excluded on the zeros up to height 680

Volume I, Proposition 18.1.11, page 532 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\vartheta\) be the Riemann Siegel theta function, \(N\) the zero counting function, \(S(t)=N(t)-\vartheta(t)/\pi-1\), and for a constant \(c\) let the reference levels be the solutions of \(\vartheta(t)=(m+c)\pi\) with \(m\ge m_0\), with counting function \(N_c\) and discrepancy \(D_c=N-N_c\). Then \(D_c=S+\{\vartheta/\pi-c\}+c+m_0\), where \(\{\cdot\}\) is the fractional part (proved), so over any range \(\sup S-\inf S-1<\operatorname{span}D_c<\sup S-\inf S+1\) once the range holds enough levels. Computed, not proved for every height: over \([30,680]\) one has \(S\in[-1.1455,1.1642]\), with the supremum \(1.1642\) just after the zero \(\gamma_{379}=650.669\) and the infimum \(-1.1455\) just before \(\gamma_{289}=527.904\), hence \(2\le\operatorname{span}D_c\le3\) for every real \(c\), so a rank one perturbation is excluded against every constant phase on this range; the span is \(3\) exactly for \(c\in[0,0.1455)\cup(0.8358,1)\) modulo \(1\) and \(2\) otherwise. Weak interlacing for a positive or negative rank one perturbation holds exactly when \(D_c\) takes at most two adjacent values.

Source: ch/biane.tex, label prop:biane:span.

Operators with levels given by a prime truncated phase are Gram regular

Volume I, Proposition 18.1.13, page 533 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\vartheta\) be the Riemann Siegel theta function, fix \(p_0\), and put \(\Theta_{p_0}(t)=\vartheta(t)-\sum_{p\le p_0}\sum_{k\ge1}\sin(kt\log p)/(kp^{k/2})\) with \(p\) running over primes. Any selfadjoint operator whose levels are, up to a finite set, the solutions of \(\Theta_{p_0}(t)\equiv c\pi \bmod \pi\) is Gram regular (its counting function is \(\vartheta(t)/\pi+O_{p_0}(1)\) in the variable \(t=2\sqrt\mu\)), so no finite rank selfadjoint perturbation of it has the spectrum \(\{\gamma^2/4\}\) of the squared half ordinates of the zeta zeros.

Source: ch/biane.tex, label prop:biane:euler.

Index side Thorin measure of Pólya’s Bessel function

Volume I, Theorem 18.1.14, page 534 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_\mu(x)=\int_0^\infty e^{-x\cosh t}\cosh(\mu t)\,dt\) be the Macdonald function and \(\nu_1<\nu_2<\cdots\) the positive zeros of \(\nu\mapsto K_{i\nu}(2\pi)\) (all zeros of \(\mu\mapsto K_\mu(2\pi)\) are purely imaginary and simple). Let \(H\) be the first passage time to the level \(\sqrt{2\pi}\) of Brownian motion with drift \(\sqrt{2\pi}\), with density \(\tau^{-3/2}e^{2\pi}e^{-\pi(\tau+1/\tau)}\) on \((0,\infty)\) and \(\mathbb{E}H^s=2e^{2\pi}K_{s-1/2}(2\pi)\). Then for \(u\ge0\), \(\frac{K_0(2\pi)}{K_{\sqrt u}(2\pi)}=\frac{\mathbb{E}H^{1/2}}{\mathbb{E}H^{1/2+\sqrt u}}=\prod_{n\ge1}\Big(1+\frac u{\nu_n^2}\Big)^{-1}\). This is the Laplace transform of the GGC \(T_P=\sum_{n\ge1}e_n/\nu_n^2\), with independent standard exponential variables \(e_n\), no drift and Thorin measure \(\sum_n\delta_{\nu_n^2}\).

Earlier editions: Theorem 8.7.9, page 318 (edition 1).

Source: ch/biane.tex, label thm:biane:indexP.

Clock of Pólya’s Bessel function as a hitting time of an explicit diffusion

Volume I, Theorem 18.1.15, page 536 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_0\), \(K_1\) and \(K_\mu\) be Macdonald functions and \(R\) the diffusion on \((0,\infty)\) with generator \(\mathcal{G}=\frac12\frac{d^2}{dr^2}+\Big(\frac1{2r}-\frac{K_1(r)}{K_0(r)}\Big)\frac d{dr}\), the radial part of planar Brownian motion killed at rate \(\frac12\) and Doob transformed by \(K_0\). For \(r>b>0\) let \(T_b\) be the hitting time of \(b\) and \(\tau_r=\frac12\int_0^{T_b}dt/R_t^2\) under the law of \(R\) started at \(r\). Then \(T_b<\infty\) almost surely and, for \(u\ge0\), \(\mathbb{E}e^{-u\tau_r}=\frac{K_{\sqrt u}(r)}{K_0(r)}\cdot\frac{K_0(b)}{K_{\sqrt u}(b)}\). As \(r\to\infty\), \(\tau_r\) converges in law to \(\tau_\infty\) with \(\mathbb{E}e^{-u\tau_\infty}=K_0(b)/K_{\sqrt u}(b)\). For \(b=2\pi\), \(\tau_\infty\overset{d}{=}T_P\), the GGC \(\sum_{n\ge1}e_n/\nu_n^2\) with independent standard exponential variables \(e_n\) and Thorin measure \(\sum_n\delta_{\nu_n^2}\), where \(\nu_n\) are the positive zeros of \(\nu\mapsto K_{i\nu}(2\pi)\).

Earlier editions: Theorem 8.7.10, page 320 (edition 1).

Source: ch/biane.tex, label thm:biane:clock.

Sine squared forms of the Bessel clock, the drifted clock and Pólya’s clock

Volume I, Proposition 18.1.18, page 539 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_\mu\) be the Macdonald function, \(\nu_n\) the positive zeros of \(\nu\mapsto K_{i\nu}(2\pi)\) and \(\Theta_P(v)=\sum_{n\ge1}e^{-\nu_nv}=\operatorname{tr}e^{-v\sqrt{A_P}}\), where \(A_P=-\frac{d^2}{dx^2}+e^{2x}\) acts in \(L^2([\log2\pi,\infty))\) with a Dirichlet condition at \(\log2\pi\) and has the eigenvalues \(\nu_n^2\). For real \(\lambda\) and \(a,c,\ell>0\), \(2\log\frac{\sinh\lambda a}{\lambda a}=\int_0^\infty(1-\cos\lambda v)\,\frac{4\,dv}{v\,(e^{\pi v/a}-1)}\), \(\ell\big(\sqrt{c^2+\lambda^2}-c\big)=\int_0^\infty(1-\cos\lambda v)\,\frac{2\ell c}\pi K_1(cv)\,\frac{dv}v\), and \(\log\frac{K_\lambda(2\pi)}{K_0(2\pi)}=\int_0^\infty4\sin^2\Big(\frac{\lambda v}2\Big)\Theta_P(v)\,\frac{dv}v\). The left sides are \(-\log\mathbb{E}e^{-\lambda^2W/2}\) for \(W=J_a+J'_a\), the sum of two independent hitting times of the level \(a\) by the three dimensional Bessel process started at \(0\), and for \(W=H^{(c)}_\ell\), the first passage time to the level \(\ell\) of Brownian motion with drift \(c\), and \(-\log\mathbb{E}e^{-\lambda^2T_P}\) for the GGC \(T_P=\sum_ne_n/\nu_n^2\) with independent standard exponential variables \(e_n\).

Earlier editions: Proposition 8.7.13, page 323 (edition 1).

Source: ch/biane.tex, label prop:biane:sin2.

Klein Gordon criterion for real zeros

Volume I, Theorem 18.1.19, page 540 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g\) be even, positive and continuous on \(\mathbb{R}\) with \(\int g(w)e^{c|w|}\,dw<\infty\) for every \(c\), and \(G(t)=\int g(w)e^{itw}\,dw\). Suppose that there are \(x_0\in\mathbb{R}\), a real continuous function \(V\) on \([x_0,\infty)\) bounded below, and \(F\in C^2([x_0,\infty)\times\mathbb{R})\) such that (a) \((\partial_x^2-\partial_w^2-V(x))F=0\) and \(F(x_0,\cdot)=g\); (b) \(F\) and its partial derivatives of order at most two are \(O(e^{-c|w|})\) for every \(c>0\), uniformly for \(x\) in compact subsets of \([x_0,\infty)\), and \(\Psi_\mu(x)=\int F(x,w)e^{\mu w}\,dw\) lies in \(L^2((x_0,\infty))\) for every \(\mu\in\mathbb{C}\); (c) \(\Psi_\mu\) is not identically zero for any \(\mu\in\mathbb{C}\). Then every zero of \(G\) is real, and its square is an eigenvalue of \(-\frac{d^2}{dx^2}+V\) on \([x_0,\infty)\) with a Dirichlet condition at \(x_0\).

Earlier editions: Theorem 8.7.14, page 324 (edition 1).

Source: ch/biane.tex, label thm:biane:kg.

De Bruijn Newman constant of Pólya’s function is finite

Volume I, Proposition 18.1.20, page 541 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For real \(t\) let \(P_t(\theta)=\int_{\mathbb{R}}e^{tu^2/4}e^{-2\pi\cosh2u}\,e^{i\theta u}\,du\), so that \(P_0(\theta)=K_{i\theta/2}(2\pi)\) is Pólya’s Macdonald function. There is a constant \(\Lambda_P\) with \(-\infty<\Lambda_P\le0\) such that \(P_t\) has only real zeros if and only if \(t\ge\Lambda_P\).

Earlier editions: Proposition 8.7.15, page 325 (edition 1).

Source: ch/biane.tex, label prop:biane:lambdaP.

A property that survives Gaussian damping cannot imply real zeros

Volume I, Proposition 18.1.21, page 545 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}\big(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\big)\exp(-\pi n^2e^{2u})\) be the Pólya kernel of Riemann’s xi function, with \(\xi(\frac12+i\theta)=\int_{\mathbb{R}}\Phi(u)\cos(\theta u)\,du\), and \(\Phi^*(u)=8\pi^2\cosh\frac{9u}2\,e^{-2\pi\cosh2u}\) Pólya’s asymptotic replacement. Let \(\mathcal{P}\) be a property of even, positive, integrable kernels such that (i) if \(\mathcal{P}(g)\) holds, then \(\int g(u)e^{i\theta u}\,du\) has only real zeros, and (ii) there is \(\varepsilon>0\) such that \(\mathcal{P}(g)\) implies \(\mathcal{P}(e^{-\varepsilon u^2/4}g)\). (a) If (ii) holds for \(g=\Phi\), then \(\mathcal{P}(\Phi)\) is false. (b) If (ii) holds for all \(g\) with the same \(\varepsilon\), then \(\mathcal{P}(g)\) is false for every kernel \(g\) that is \(O(e^{-|u|^b})\) for some \(b>2\) and satisfies \(g(u)e^{Cu^4}\to0\) as \(|u|\to\infty\) for every \(C\); in particular \(\mathcal{P}\) fails for \(\Phi\), for \(\Phi^*\) and for \(e^{-2\pi\cosh2u}\).

Earlier editions: Proposition 8.7.16, page 329 (edition 1).

Source: ch/biane.tex, label prop:biane:damping.

Open question on a diffusion for the clock of the Riemann xi function

Volume I, Question 18.1.22, page 546 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) and let \(T\) be the clock of \(\xi\), the random time with Laplace transform \(\mathbb{E}e^{-uT}=\xi(\frac12)/\xi(\frac12+\sqrt u)\) (it exists without any hypothesis). Is there a diffusion on a half line, with an entrance boundary, whose hitting time of a level from the entrance boundary has the Laplace transform \(\xi(\frac12)/\xi(\frac12+2\sqrt u)\), that of \(4T\), as the clock \(\tau_\infty\) of Pólya’s Bessel function has the transform \(K_0(2\pi)/K_{\sqrt u}(2\pi)\)? By Kent’s theorem such a hitting time is \(\sum_ne_n/\lambda_n\) over the Dirichlet eigenvalues \(\lambda_n>0\) of the generator, so a positive answer implies the Riemann hypothesis; under the Riemann hypothesis and the simplicity of the zeros, Krein’s inverse spectral theory gives a generalised diffusion with the eigenvalues \(\gamma^2/4\), \(\frac12+i\gamma\) running over the zeros with \(\gamma>0\). The question asks for one constructed without the Riemann hypothesis, for example as a perturbation of the diffusion for Pólya’s function, with generator \(\frac{d^2}{dx^2}+2\frac{\psi'}\psi\frac d{dx}\) and \(\psi(x)=K_0(e^x)\), whose potential \(e^{2x}\) has the right Weyl law.

Earlier editions: Question 8.7.17, page 330 (edition 1).

Source: ch/biane.tex, label q:biane:radial.

Open question on a Klein Gordon extension of Pólya’s kernel

Volume I, Question 18.1.23, page 546 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}\big(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\big)\exp(-\pi n^2e^{2u})\) be the Pólya kernel of Riemann’s xi function and \(g(w)=\Phi(w/2)\). Does \(g\) admit a Klein Gordon extension, that is: a real continuous potential \(V\) on some \([x_0,\infty)\), bounded below and asymptotic to \(e^{2x}\) at \(+\infty\), and a function \(F\in C^2([x_0,\infty)\times\mathbb{R})\) with \((\partial_x^2-\partial_w^2-V(x))F=0\) and \(F(x_0,\cdot)=g\), such that \(F\) and its partial derivatives of order at most two are \(O(e^{-c|w|})\) for every \(c>0\), uniformly for \(x\) in compact sets, and \(\Psi_\mu(x)=\int F(x,w)e^{\mu w}\,dw\) lies in \(L^2((x_0,\infty))\) for every complex \(\mu\) and is not identically zero for any \(\mu\)? By the Klein Gordon criterion a positive answer implies the Riemann hypothesis. Under the Riemann hypothesis the question asks whether \(\mu\mapsto2\xi(\frac12+2\mu)\) is, up to normalisation, the Jost function of such a potential whose Klein Gordon solution has the boundary value \(g\).

Earlier editions: Question 8.7.18, page 331 (edition 1).

Source: ch/biane.tex, label q:biane:kg.

Open question on the de Bruijn Newman constants of Pólya’s functions

Volume I, Question 18.1.24, page 547 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For real \(t\) let \(P_t(\theta)=\int_{\mathbb{R}}e^{tu^2/4}e^{-2\pi\cosh2u}\,e^{i\theta u}\,du\) (so that \(P_0(\theta)=K_{i\theta/2}(2\pi)\) is Pólya’s Macdonald function) and \(\Xi^*_t(\theta)=\int_{\mathbb{R}}e^{tu^2/4}\Phi^*(u)e^{i\theta u}\,du\) with Pólya’s kernel \(\Phi^*(u)=8\pi^2\cosh\frac{9u}2\,e^{-2\pi\cosh2u}\). The de Bruijn Newman constant \(\Lambda_P\) of \(P\), the least \(t\) from which all zeros of \(P_t\) are real, satisfies \(-\infty<\Lambda_P\le0\), and the same holds for the constant \(\Lambda_{\Xi^*}\) of \(\Xi^*\). What are \(\Lambda_P\) and \(\Lambda_{\Xi^*}\)? At what height do the first nonreal zeros of \(P_t\) appear as \(t\) decreases through \(\Lambda_P\), and do they appear, as for the de Bruijn Newman flow of Riemann’s \(\Xi\), through the collision of two real zeros?

Earlier editions: Question 8.7.19, page 331 (edition 1).

Source: ch/biane.tex, label q:biane:lambda.

Open question on a family joining the two clocks

Volume I, Question 18.1.25, page 547 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(H\) be the first passage time of Brownian motion with drift \(\sqrt{2\pi}\) to the level \(\sqrt{2\pi}\), whose Mellin transform is \(\mathbb{E}H^s=2e^{2\pi}K_{s-1/2}(2\pi)\) and for which \(1/H\) has the law of \(H\) size biased by \(H\), a reciprocal symmetry that encodes the functional equation \(\mathbb{E}H^s=\mathbb{E}H^{1-s}\). Let \(S_2=J_1+J'_1\) be the sum of two independent hitting times of the level \(1\) by the three dimensional Bessel process started at \(0\), for which \(2\xi(s)=\mathbb{E}\big[(\pi S_2/2)^{s/2}\big]\) with Riemann’s xi function \(\xi\), the functional equation coming from Jacobi’s theta transformation. Is there a one parameter family of laws joining the drifted first passage time and the sum of two Bessel hitting times, each with a reciprocal symmetry that encodes a functional equation, along which the index side Thorin measure can be followed? The curvature family of Brownian hitting times, an exponential tilt of the Bessel hitting time, breaks the symmetry. Positivity of the index side Thorin measure at the Bessel end is the Riemann hypothesis.

Earlier editions: Question 8.7.20, page 331 (edition 1).

Source: ch/biane.tex, label q:biane:curv.

Selfadjoint corrections of the wall operator defined from the primes (open question)

Volume I, Question 18.1.26, page 547 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(A_P\) be the selfadjoint operator of the book’s Pólya wall, whose levels \(\nu_n^2\) have the counting function of the Riemann Siegel phase up to a bounded error. Open question: define, without using the zeros, a sequence of selfadjoint finite rank corrections of \(A_P\), for instance by a prime sum, whose spectra converge to \(\{\gamma^2/4\}\), with \(\gamma\) running over the ordinates of the nontrivial zeros of \(\zeta\) with positive real part. The book proves that the ranks must be unbounded (no single finite rank correction works), so the question is open as stated. Any solution gives the Riemann Hypothesis, because limits of real numbers are real and no ordinate is purely imaginary. Under the Riemann Hypothesis and with the zeros as input a solution exists trivially, as a sequence of rank one moves of the eigenvalues of \(A_P\) onto \(\gamma_n^2/4\) in the diagonal basis; the content of the question lies in the clause that the corrections be defined from the primes.

Source: ch/biane.tex, label q:biane:infrank.

Krein string masses of the zeros against the wall levels (open question)

Volume I, Question 18.1.27, page 547 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Fix one construction that assigns a Krein string to a finite list of levels and norming weights, for instance the Stieltjes continued fraction of \(\sum_nw_n\delta_{\gamma_n^2/4}\), and apply it to the first \(N\) ordinates \(\gamma_n\) of the zeta zeros and to the first \(N\) levels of the wall operator \(A_P\) with the weights given by the norming constants of its eigenfunctions. Open question: how do the masses of the two strings compare as \(N\) grows? A string is determined by its spectral measure, the eigenvalues together with the norming constants, and not by the ordinates alone, so the comparison must fix the weights of the zeros; a difference of masses is not a perturbation of any rank. With the zeros as input this tests an inverse algorithm and is no arithmetic mechanism.

Source: ch/biane.tex, label q:biane:string.

Volume I, Chapter 19. The Pólya density as a maximum and as an extreme value law

Pólya kernel from a sum of two maxima of exponential variables

Volume I, Proposition 19.2.1, page 552 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function and \(Y\) the random variable with \(\mathbb{E}e^{sY}=2\xi(s)\) for every complex \(s\). Let \(M_4\) be the maximum of independent exponential random variables with one variable of each rate \(\pi n\), \(n\ge1\), and one more of each rate \(\pi(2m-1)\), \(m\ge1\), so that \(\Pr(M_4\le t)=\prod_{n\ge1}(1-e^{-\pi nt})\prod_{m\ge1}(1-e^{-\pi(2m-1)t})=\sum_{n\in\mathbb{Z}}(-1)^ne^{-\pi n^2t}\) for \(t>0\), and let \(M_4'\) be an independent copy of \(M_4\). Then (i) \(e^{2Y}\) has the same law as \(M_4+M_4'\). (ii) The density of \(e^Y=\sqrt{M_4+M_4'}\) is \(f(y)=4\sum_{n\ge1}\big(2\pi^2n^4y^3-3\pi n^2y\big)e^{-\pi n^2y^2}\) for \(y>0\), and the Pólya kernel \(\Phi(u)=2\sum_{n\ge1}\big(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\big)\exp(-\pi n^2e^{2u})\), the even function with \(\int_{\mathbb{R}}\Phi(u)e^{i\theta u}\,du=\xi(\frac12+i\theta)\), satisfies \(\Phi(u)=\frac12e^{3u/2}f(e^u)\).

Earlier editions: Proposition 14.2.1, page 584 (edition 1).

Source: ch/maxwald.tex, label mw:prop:sum.

Khinchin factorisation of the tilted maximum of exponential variables

Volume I, Theorem 19.3.1, page 553 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Fix \(\sigma>1\). Let \(M_4\) be the maximum of independent exponential random variables with one variable of each rate \(\pi n\), \(n\ge1\), and one more of each rate \(\pi(2m-1)\), \(m\ge1\), and let \(\widetilde M_4\) have the tilted law \(\Pr(\widetilde M_4\in dm)\propto m^{\sigma/2}\Pr(M_4\in dm)\). Let \(K\), \(N_\sigma\) and \(\Gamma_{1+\sigma/2}\) be independent of each other and of \(\widetilde M_4\), where \(\Pr(K=k)=(1-a)a^k\) for \(k\ge0\) with \(a=2^{1-\sigma}\); \(N_\sigma=\prod_pp^{G_p}\) is zeta distributed with parameter \(\sigma\), that is \(\Pr(N_\sigma=n)=n^{-\sigma}/\zeta(\sigma)\) for \(n\ge1\), the product running over the primes \(p\) with independent geometric variables \(G_p\), \(\Pr(G_p=k)=(1-p^{-\sigma})p^{-k\sigma}\); and \(\Gamma_{1+\sigma/2}\) is gamma distributed with shape \(1+\sigma/2\) and scale \(1\). Then \(\widetilde M_4\,4^{-K}\) has the same law as \(\Gamma_{1+\sigma/2}/(\pi N_\sigma^2)=\frac{\Gamma_{1+\sigma/2}}\pi\prod_pp^{-2G_p}\).

Earlier editions: Theorem 14.3.1, page 586 (edition 1).

Source: ch/maxwald.tex, label mw:thm:khinchin.

Wald couples of the maximum of exponential variables and the logistic law

Volume I, Theorem 19.4.2, page 554 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Call a pair \((\tau,\varsigma)\) of independent random variables with \(\tau\ge0\) a Wald couple if \(\mathbb{E}e^{s\varsigma}<\infty\) and \(\mathbb{E}e^{s\varsigma}\,\mathbb{E}e^{-s^2\tau/2}=1\) for all real \(s\); equivalently, with \(Z\) standard normal and independent of \(\tau\), the characteristic functions \(\varphi_1\) of \(\varsigma\) and \(\varphi_2\) of \(\sqrt\tau\,Z\) form a van Dantzig pair, \(\varphi_2(t)=1/\varphi_1(it)\). Let \(M_4\) be the maximum of independent exponential random variables with one variable of each rate \(\pi n\), \(n\ge1\), and one more of each rate \(\pi(2m-1)\), \(m\ge1\), so that \(\mathbb{E}e^{-\lambda M_4}=\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda}\), let \(M_4'\) be an independent copy, let \(\upsilon,\upsilon_1,\upsilon_2\) be uniform on \((-1,1)\), independent of each other and of \(M_4,M_4'\), and let \(c=\sqrt{\pi/2}\). Then \((M_4,c\upsilon)\) and \((M_4+M_4',c(\upsilon_1+\upsilon_2))\) are Wald couples. The associated van Dantzig pairs are \((\sin(ct)/(ct),\ ct/\sinh(ct))\), the characteristic functions of \(c\upsilon\) and of \(\sqrt{M_4}\,Z\) with \(Z\) standard normal and independent of \(M_4\), and their squares, and \(\sqrt{M_4}\,Z\) has the logistic law, with characteristic function \(ct/\sinh(ct)\).

Earlier editions: Theorem 14.4.2, page 586 (edition 1).

Source: ch/maxwald.tex, label mw:thm:wald.

Stieltjes criterion for RH through the smoothed Thorin density of the clock

Volume I, Theorem 19.5.1, page 555 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), let \(\rho\) run over the nontrivial zeros of \(\zeta\) with \(\operatorname{Im}\rho>0\), counted with multiplicity, put \(a_\rho=-(\rho-\frac12)^2\) and \(G(\lambda)=\xi(\frac12+\sqrt\lambda)/\xi(\frac12)=\prod_\rho(1+\lambda/a_\rho)\). Let \(T\) be the clock, the random variable with \(\mathbb{E}e^{-\lambda T}=1/G(\lambda)\), which exists with no hypothesis on the zeros. Put \(W(x)=\sum_\rho e^{-a_\rho x}\) for \(x>0\), so that \((G'/G)(\lambda)=\sum_\rho(\lambda+a_\rho)^{-1}=\int_0^\infty e^{-\lambda x}W(x)\,dx\), and \(r_\eta(x)=\frac1\pi\operatorname{Im}\frac{G'}G(-x-i\eta)\) for \(x\in\mathbb{R}\) and \(\eta>0\), wherever \(-x-i\eta\) is not a pole of \(G'/G\). The following are equivalent. (a) The Riemann Hypothesis (RH). (b) \(T\) is a generalised gamma convolution (GGC), that is \(\mathbb{E}e^{-\lambda T}=\exp\big(-a\lambda-\int_{(0,\infty)}\log(1+\lambda/x)\,U(dx)\big)\) for some \(a\ge0\) and positive measure \(U\), the Thorin measure; then \(T\) has the same law as \(\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\), and its Thorin measure is \(U_{\mathrm{RH}}=\sum_{\gamma>0}\delta_{\gamma^2}\), the ordinates \(\gamma=\operatorname{Im}\rho\) counted with multiplicity. (c) \(G'/G\) is a Stieltjes function, that is of the form \(b+\int_{[0,\infty)}\sigma(dx)/(\lambda+x)\) with \(b\ge0\) and \(\sigma\) a positive measure. (d) \(r_\eta(x)\ge0\) for all \(x\in\mathbb{R}\) and \(\eta>0\) for which \(-x-i\eta\) is not a pole of \(G'/G\). (e) \(W\) is completely monotone. In that case \(r_\eta(x)=\sum_{\gamma>0}\eta/(\pi((x-\gamma^2)^2+\eta^2))\) is the Poisson smoothing of \(U_{\mathrm{RH}}\) at scale \(\eta\) and \(U_{\mathrm{RH}}((0,x])=N(\sqrt x)\), where \(N(t)\) is the number of zeros \(\rho\) with \(0<\operatorname{Im}\rho\le t\), counted with multiplicity.

Earlier editions: Theorem 14.6.1, page 587 (edition 1).

Source: ch/maxwald.tex, label mw:thm:stieltjes.

An off line zero makes the smoothed Thorin density of the clock negative

Volume I, Lemma 19.5.2, page 556 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), let \(\rho'\) run over the nontrivial zeros of \(\zeta\) with \(\operatorname{Im}\rho'>0\), counted with multiplicity, put \(a_{\rho'}=-(\rho'-\frac12)^2\), \(G(\lambda)=\xi(\frac12+\sqrt\lambda)/\xi(\frac12)=\prod_{\rho'}(1+\lambda/a_{\rho'})\), so that \(1/G\) is the Laplace transform of the clock, and \(r_\eta(x)=\frac1\pi\operatorname{Im}\frac{G'}G(-x-i\eta)\) for \(x\in\mathbb{R}\) and \(\eta>0\). Let \(\rho=\beta+i\gamma\) be a zero of multiplicity \(m\) with \(\beta>\frac12\) and \(\gamma>0\), and put \(\delta=\beta-\frac12\), \(x_\rho=\gamma^2-\delta^2\) and \(b=2\delta\gamma\), so that \(a_\rho=x_\rho-ib\). Then \(r_\eta(x_\rho)\to-\infty\) as \(\eta\uparrow b\). More precisely, for \(0<\eta<b\) such that \(-x_\rho-i\eta\) is not a pole of \(G'/G\), \(\pi r_\eta(x_\rho)=\eta\Big(A_\eta-\frac{2m}{b^2-\eta^2}\Big)\), where \(A_\eta=\frac1\eta\sum_{\rho'}\operatorname{Im}\,(a_{\rho'}-x_\rho-i\eta)^{-1}\), the sum running over the zeros \(\rho'\) with \(\operatorname{Im}\rho'>0\), counted with multiplicity, whose \(a_{\rho'}\) is neither \(a_\rho\) nor \(\overline{a_\rho}\). Hence \(r_\eta(x_\rho)<0\) at every \(\eta\in(0,b)\) with \(A_\eta<2m/b^2\). If all the other zeros lie on the critical line, \(A_\eta\to A_0=\sum_{\gamma'}(\gamma'^2-x_\rho)^{-2}\) as \(\eta\downarrow0\), the sum running over the ordinates \(\gamma'>0\) of those zeros, counted with multiplicity.

Earlier editions: Lemma 14.6.2, page 588 (edition 1).

Source: ch/maxwald.tex, label mw:lem:detect.

An explicit GGC within a stated Laplace distance of the clock

Volume I, Proposition 19.6.2, page 558 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), let \(\rho_k=\beta_k+i\gamma_k\), \(k\ge1\), be the nontrivial zeros of \(\zeta\) with \(\gamma_k>0\) in nondecreasing order of \(\gamma_k\), counted with multiplicity, and put \(a_k=-(\rho_k-\frac12)^2\), \(G(\lambda)=\xi(\frac12+\sqrt\lambda)/\xi(\frac12)=\prod_k(1+\lambda/a_k)\) and \(G_K(\lambda)=\prod_{k\le K}(1+\lambda/\gamma_k^2)\). Let \(T\) be the clock, with \(\mathbb{E}e^{-\lambda T}=1/G(\lambda)\). For laws \(P,Q\) on \([0,\infty)\) with Laplace transforms \(L_P,L_Q\) let \(d_L(P,Q)=\sup_{\lambda\ge0}|L_P(\lambda)-L_Q(\lambda)|\). Let \(H=3\times10^{12}\) and let \(N\) be the number of zeros with \(0<\gamma_k\le H\), all of which lie on the critical line by the verification of RH up to height \(H\) of Platt and Trudgian. Let \(T^{[N]}=\sum_{k\le N}e_k/\gamma_k^2\) with independent standard exponential variables \(e_k\), the GGC with Laplace transform \(1/G_N\), and put \(\varepsilon_H=\sum_{k>N}1/|a_k|\) and \(S_K=\sup_{\lambda\ge0}\lambda/G_K(\lambda)\) for \(K\le N\). Then \(d_L\big(T,T^{[N]}\big)\le S_K\,\varepsilon_H\) for every \(K\le N\), with \(\varepsilon_H<1.4798\times10^{-12}\) and \(S_{1700}<16.85\), so that \(d_L\big(T,T^{[N]}\big)<2.5\times10^{-11}\). The proof uses only the verification of RH up to height \(H\) and makes no assumption on the zeros above \(H\).

Earlier editions: Proposition 14.7.2, page 590 (edition 1).

Source: ch/maxwald2.tex, label mw:prop:approx.

Term of one zero in the Thorin density of the basepoint family in the lambda chart

Volume I, Proposition 19.7.1, page 559 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\alpha>\frac12\), \(P_\delta(\theta)=\delta/(\pi(\delta^2+\theta^2))\) the Poisson kernel of width \(\delta\) (so that \(P_{-a}=-P_a\)), and \(u_\alpha(\lambda)=\operatorname{Re}\frac{\xi'}\xi(\alpha+i\sqrt\lambda)/(2\pi\sqrt\lambda)\) for \(\lambda>0\), the density in the variable \(\lambda\) of the Thorin measure of the shifted clock \(T_\alpha\), the law with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt\lambda)\), when every zero has real part below \(\alpha\). Let \(\rho=\beta+i\gamma\) be a nontrivial zero of \(\zeta\) with \(\gamma>0\) and put \(\delta=\alpha-\beta\ne0\) and \(c=\alpha-\rho=\delta-i\gamma\). The term of \(\rho\) in \(u_\alpha=\sum_{\gamma>0}u_\rho\), where the zeros \(\rho\) and \(\bar\rho\) are paired, is \(u_\rho(\lambda)=\big(P_\delta(\sqrt\lambda-\gamma)+P_\delta(\sqrt\lambda+\gamma)\big)/(2\sqrt\lambda)\) for \(\lambda>0\), and it satisfies the following. (a) \(u_\rho(\lambda)=\kappa_\rho(\lambda)C_\rho(\lambda)=\delta(\lambda+|c|^2)/(\pi\sqrt\lambda\,|c^2+\lambda|^2)\), where \(\kappa_\rho(\lambda)=(\lambda+\gamma^2+\delta^2)/(2\gamma\sqrt\lambda)\) and \(C_\rho(\lambda)=\frac1\pi\cdot2\delta\gamma/\big((\lambda-\gamma^2+\delta^2)^2+4\delta^2\gamma^2\big)\); for \(\delta>0\), \(C_\rho\) is the Cauchy density centred at \(\gamma^2-\delta^2\) with half width \(2\delta\gamma\). (b) \(\int_0^\infty u_\rho(\lambda)\,d\lambda=\operatorname{sign}\delta\). (c) \(\kappa_\rho(\lambda)\ge\sqrt{1+\delta^2/\gamma^2}>1\) for every \(\lambda>0\), \(\kappa_\rho(\gamma^2-\delta^2)=(1-\delta^2/\gamma^2)^{-1/2}\) when \(|\delta|<\gamma\), and for \(K\ge1\) and \(|\delta|\le\gamma/(2K)\), \(|\kappa_\rho(\lambda)-1|\le(K^2/2+1)\,\delta^2/\gamma^2\) for \(|\lambda-\gamma^2|\le K|\delta|\gamma\). Hence near its peak the term is a Lorentzian of full width \(4|\delta|\gamma\) at half height, with the sign of \(\delta\), up to a relative error \(O(\delta^2/\gamma^2)\), while \(u_\rho(\lambda)\sim\delta/(\pi(\gamma^2+\delta^2)\sqrt\lambda)\) as \(\lambda\downarrow0\) and \(u_\rho(\lambda)\sim\delta/(\pi\lambda^{3/2})\) as \(\lambda\to\infty\).

Earlier editions: Proposition 14.8.1, page 591 (edition 1).

Source: ch/maxwald2.tex, label mw:prop:lambda.

The clock is not a maximum of independent exponential variables

Volume I, Proposition 19.8.1, page 562 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(T\) the clock, the random variable with \(\mathbb{E}e^{-\lambda T}=\xi(\frac12)/\xi(\frac12+\sqrt\lambda)\), and \(F_T(t)=\Pr(T\le t)\). Then \(-\log F_T\) is not completely monotone on \((0,\infty)\). Hence \(T\) does not have the law of a maximum of independent exponential variables \(E_i/r_i\), with \(E_i\) standard exponential and deterministic rates \(r_i>0\) with \(\sum_ie^{-r_it}<\infty\) for \(t>0\), for any choice of the rates, nor of the maximum of a Poisson cloud of exponential clocks, that is of the variables \(E_i/x_i\) attached to the points \(x_i\) of a Poisson process of intensity \(\nu\) on \((0,\infty)\); for both of these laws \(-\log F(t)=\int e^{-xt}\,\nu(dx)\) for a positive measure \(\nu\), which is completely monotone in \(t\). The proof uses the verification of RH up to height \(3\times10^{12}\) and the simplicity of the zeros of height below \(200\).

Earlier editions: Proposition 14.9.1, page 595 (edition 1).

Source: ch/maxwald2.tex, label mw:prop:nomax.

Gumbel limit for maxima of independent copies of the clock and of the maximum M_4

Volume I, Proposition 19.9.1, page 563 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(\theta)=\xi(\frac12+i\theta)\), \(\gamma_1\) the least positive ordinate of a zero of \(\zeta\) (the zero \(\frac12+i\gamma_1\) is simple) and \(R_1=-2\gamma_1\Xi(0)/\Xi'(\gamma_1)\), the residue at \(\lambda=-\gamma_1^2\) of \(\xi(\frac12)/\xi(\frac12+\sqrt\lambda)\), the Laplace transform of the clock \(T\). Let \(M_4\) be the maximum of independent exponential random variables with one variable of each rate \(\pi n\), \(n\ge1\), and one more of each rate \(\pi(2m-1)\), \(m\ge1\), and let \(M_4^{(j)}\) and \(T^{(j)}\), \(j\le n\), be independent copies of \(M_4\) and of \(T\). (i) \(\Pr(M_4>t)\sim2e^{-\pi t}\) as \(t\to\infty\), and \(\Pr\big(\pi\max_{j\le n}M_4^{(j)}-\log(2n)\le x\big)\to\exp(-e^{-x})\) as \(n\to\infty\), for every real \(x\). (ii) \(\Pr(T>t)\sim(R_1/\gamma_1^2)\,e^{-\gamma_1^2t}\) as \(t\to\infty\), \(\Pr\Big(\gamma_1^2\max_{j\le n}T^{(j)}-\log\frac{R_1n}{\gamma_1^2}\le x\Big)\to\exp(-e^{-x})\) for every real \(x\), and \(\max_{j\le n}T^{(j)}/\log n\to1/\gamma_1^2\) in probability.

Earlier editions: Proposition 14.10.1, page 596 (edition 1).

Source: ch/maxwald2.tex, label mw:prop:gnedenko.

Convergence of the finite maxima and of the finite stages of xi

Volume I, Theorem 19.11.1, page 566 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \((E_n)_{n\ge1}\) and \((E'_m)_{m\ge1}\) be independent standard exponential random variables, for \(N\ge1\) let \(M_4^{(N)}=\max\big\{E_n/(\pi n)\ (n\le N),\ E'_m/(\pi(2m-1))\ (m\le N)\big\}\), and let \(M_4=\lim_NM_4^{(N)}\) be the maximum of all the variables \(E_n/(\pi n)\) and \(E'_m/(\pi(2m-1))\), \(n,m\ge1\). Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, and define the finite stage \(\xi_N(s)=\frac{(s-1)\,\mathbb{E}(M_4^{(N)})^{s/2}}{2(1-2^{1-s})}\) and its symmetrisation \(\xi_N^{\mathrm{sym}}(s)=\frac12\big(\xi_N(s)+\xi_N(1-s)\big)\), so that the same expression with \(M_4\) in place of \(M_4^{(N)}\) is \(\xi(s)\) (here \(\xi_N\) is the finite stage, not \(\xi\) evaluated at \(N\)). Then (i) \(M_4^{(N)}\uparrow M_4\) almost surely, and \(\Pr(M_4^{(N)}\ne M_4)\le C\sqrt N\,e^{-\pi\sqrt N/2}\) for a constant \(C\) independent of \(N\). (ii) For every compact set \(K\) in the open strip \(0<\operatorname{Re}s<1\) there is a constant \(C_K\) with \(\sup_K|\xi_N-\xi|\le C_K\,N^{1/4}e^{-\pi\sqrt N/4}\) and \(\sup_K|\xi_N^{\mathrm{sym}}-\xi|\le C_K\,N^{1/4}e^{-\pi\sqrt N/4}\) for all \(N\).

Earlier editions: Theorem 14.12.1, page 599 (edition 1).

Source: ch/maxwald3.tex, label mw:thm:finite.

Zeros of the symmetrised finite stages and a sufficient condition for RH

Volume I, Corollary 19.11.2, page 567 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M_4^{(N)}\) be the maximum of the independent exponential variables \(E_n/(\pi n)\) (\(n\le N\)) and \(E'_m/(\pi(2m-1))\) (\(m\le N\)), with \(E_n,E'_m\) independent standard exponential variables, let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, \(\xi_N(s)=\frac{(s-1)\,\mathbb{E}(M_4^{(N)})^{s/2}}{2(1-2^{1-s})}\) the finite stage and \(\xi_N^{\mathrm{sym}}(s)=\frac12\big(\xi_N(s)+\xi_N(1-s)\big)\) its symmetrisation. (a) Every zero of \(\xi\) in the open strip \(0<\operatorname{Re}s<1\) is a limit of zeros of \(\xi_N\), and of zeros of \(\xi_N^{\mathrm{sym}}\), with multiplicity, and every limit of zeros of \(\xi_N\) inside the open strip is a zero of \(\xi\). (b) If \(\rho=\frac12+i\gamma\) is a simple zero of \(\xi\), then for all large \(N\) the function \(\xi_N^{\mathrm{sym}}\) has exactly one zero in a small disc about \(\rho\), and that zero lies on the critical line. (c) Suppose that there are \(N_j\to\infty\) and heights \(h_j\to\infty\) such that \(\xi^{\mathrm{sym}}_{N_j}\) has no zeros in \(\{\frac12<\operatorname{Re}s<1,\ |\operatorname{Im}s|<h_j\}\) for every \(j\). Then the Riemann Hypothesis holds. In particular this is so if there are \(c>0\) and \(N_0\) such that for every \(N\ge N_0\) the function \(\xi^{\mathrm{sym}}_N\) has no zeros in \(\{\frac12<\operatorname{Re}s<1,\ |\operatorname{Im}s|<c\sqrt N\}\).

Earlier editions: Corollary 14.12.2, page 599 (edition 1).

Source: ch/maxwald3.tex, label mw:cor:hurwitz.

Finite order GGC, Bondesson and HCM classes of the clock, and RH as the infinite order case

Volume I, Theorem 19.12.2, page 571 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) and let \(T\) be the clock, with Laplace transform \(\varphi(\lambda)=\xi(\frac12)/\xi(\frac12+\sqrt\lambda)\) and Lévy density \(W(x)/x\), where \(W(x)=\sum_\rho e^{(\rho-1/2)^2x}\) is summed over the nontrivial zeros \(\rho\) of \(\zeta\) with \(\operatorname{Im}\rho>0\), counted with multiplicity. A function \(f\) on an interval is \(k\) monotone if \((-1)^jf^{(j)}\ge0\) for \(0\le j\le k\), and completely monotone if it is \(k\) monotone for every \(k\). For \(k\ge1\) let \(\mathrm{GGC}_k\) be the class of laws on \([0,\infty)\) with a Lévy density \(\ell\) such that \(x\ell(x)\) is \(k\) monotone, and \(\mathrm{BO}_k\) the class with \(\ell\) itself \(k\) monotone, so that \(\mathrm{GGC}_k\subset\mathrm{BO}_k\), \(\mathrm{GGC}_1\) is the self decomposable class, \(\bigcap_k\mathrm{GGC}_k\) is the class GGC of generalised gamma convolutions and \(\bigcap_k\mathrm{BO}_k=\mathrm{BO}\) is the Bondesson class. A positive function \(\varphi\) on \((0,\infty)\) is HCM (hyperbolically completely monotone) of order \(k\) if for every \(u>0\) the function \(w\mapsto\varphi(uv)\varphi(u/v)\), \(w=v+1/v\ge2\), is \(k\) monotone on \([2,\infty)\), and HCM if it is HCM of every order. Let \(H=3\times10^{12}\). Statements (i) and (ii) are unconditional: they use the verification of RH up to height \(H\) of Platt and Trudgian. (i) \(T\in\mathrm{GGC}_k\), and hence \(T\in\mathrm{BO}_k\), for every \(k\le1.16\times10^{25}\). (ii) \(\varphi\) is HCM of order \(k\) for every \(k\le9\times10^{12}\). (iii) With no hypothesis, the following are equivalent: RH; \(T\) is a GGC; \(T\in\mathrm{BO}\); \(\varphi\) is HCM; \(\varphi\) is HCM of every order.

Earlier editions: Theorem 14.13.2, page 604 (edition 1).

Source: ch/maxwald3.tex, label mw:thm:bondesson.

Finite arithmetic form of the basepoint density below basepoint one

Volume I, Proposition 19.13.1, page 573 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Part (b) assumes the Riemann hypothesis. Part (a) holds without any hypothesis, and part (c) holds without any hypothesis when delta is at least one half and otherwise needs that every zero lies within delta of the critical line, which the Riemann hypothesis gives for every delta above zero.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Lambda\) the von Mangoldt function, \(\psi\) the digamma function, \(\alpha=\frac12+\delta\) with \(\delta>0\), \(t\in\mathbb{R}\) and \(X>0\). Put \(g_X(x)=e^{-\delta|x|}\cos(tx)\,(1-|x|/X)_+\) and \(h_X(r)=\int_{\mathbb{R}}g_X(x)e^{irx}\,dx\), so that \(h_X\ge0\) on \(\mathbb{R}\): it is \((2\pi)^{-1}\) times the convolution of the two sided Poisson kernel \(h_\infty(r)=\frac\delta{\delta^2+(t-r)^2}+\frac\delta{\delta^2+(t+r)^2}\) with the Fejér kernel \(4\sin^2(Xr/2)/(Xr^2)\). Define \(A_X(\delta,t)=\frac12\Big[h_X(\tfrac i2)+h_X(-\tfrac i2)-\log\pi+\frac1{2\pi}\int_{\mathbb{R}}h_X(r)\operatorname{Re}\psi\big(\tfrac14+\tfrac{ir}2\big)\,dr-2\sum_{n<e^X}\frac{\Lambda(n)}{n^{\alpha}}\cos(t\log n)\Big(1-\frac{\log n}X\Big)\Big]\), where \(h_X(\tfrac i2)+h_X(-\tfrac i2)=2\operatorname{Re}\big[J(-\delta+\tfrac12+it)+J(-\delta-\tfrac12+it)\big]\) with \(J(c)=-1/c+(e^{cX}-1)/(Xc^2)\). For a nontrivial zero \(\rho=\beta+i\gamma\) of \(\zeta\) put \(\gamma_\rho=(\rho-\frac12)/i\). Then (a) unconditionally, \(A_X(\delta,t)=\frac12\sum_\rho h_X(\gamma_\rho)\), the sum over all nontrivial zeros. (b) Under RH, \(A_X(\delta,t)\ge0\) for every \(X\), \(\delta\) and \(t\); in particular a single triple \((\delta,t,X)\) with \(A_X(\delta,t)<0\) would disprove RH. (c) If every zero has \(|\beta-\frac12|<\delta\) (under RH, or unconditionally when \(\delta\ge\frac12\)), then \(A_X(\delta,t)\to\operatorname{Re}\frac{\xi'}\xi(\alpha+it)\) as \(X\to\infty\).

Earlier editions: Proposition 14.14.1, page 606 (edition 1).

Source: ch/maxwald3.tex, label mw:prop:AX.

Zero freeness of the symmetric finite stages of the maximum

Volume I, Problem 19.1, page 575 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function. For \(N\ge1\) let \(M_4^{(N)}\) be the maximum of the independent exponential variables \(E_n/(\pi n)\) (\(n\le N\)) and \(E'_m/(\pi(2m-1))\) (\(m\le N\)), with \(E_n,E'_m\) independent standard exponential variables, so that \(\Pr(M_4^{(N)}\le t)=\sum_{m\ge0}p^{(N)}_me^{-\pi mt}\) where \(\sum_{m\ge0}p^{(N)}_mq^m=\prod_{n\le N}(1-q^n)\prod_{m\le N}(1-q^{2m-1})\), and put \(D_N(s)=\sum_{m\ge1}p^{(N)}_mm^{-s/2}\), a Dirichlet polynomial with integer coefficients. Let \(F_N(s)=\frac{s-1}2\,\mathbb{E}\big(M_4^{(N)}\big)^{s/2}=-\frac{s-1}2\,\Gamma(1+\frac s2)\pi^{-s/2}D_N(s)\) and \(F^{\mathrm{sym}}_N(s)=\frac12\big(F_N(s)+F_N(1-s)\big)\); as \(N\to\infty\), \(F^{\mathrm{sym}}_N\to\mathfrak q\,\xi\) uniformly on compact subsets of the open strip \(0<\operatorname{Re}s<1\), with \(\mathfrak q(s)=1-2^{-s}-2^{s-1}\). Find heights \(h_N\to\infty\) such that for infinitely many \(N\) the function \(F^{\mathrm{sym}}_N\) has no zeros with \(\frac12<\operatorname{Re}s<1\) and \(|\operatorname{Im}s|<h_N\), for instance \(h_N=c\sqrt N\) with some \(c>0\). Either statement implies the Riemann Hypothesis, by Hurwitz’s theorem, because a zero of \(\xi\) in that region would be a zero of the limit \(\mathfrak q\,\xi\). Numerical counts of zeros by the argument principle show, for \(N=8\), \(16\), \(32\), \(48\), \(64\) and \(96\), that all zeros of \(F^{\mathrm{sym}}_N\) with \(-1\le\operatorname{Re}s\le2\) and \(0<\operatorname{Im}s\le T^*(N)\) lie on the critical line, for a height \(T^*(N)\) of order \(N\) (at \(N=128\) a pair outside the strip \(0<\operatorname{Re}s<1\) appears at height \(60.7\), while the strip has no off line zero below \(130\)), and the sharper conjecture is \(T^*(N)\ge\kappa N\) for some \(\kappa>0\) and all \(N\), which would also imply RH. A mechanism for \(T^*(N)\), presumably through the ratio \(|D_N(\sigma+it)/D_N(1-\sigma+it)|\), is the first thing to find.

Earlier editions: Problem 14.1, page 608 (edition 1).

Source: ch/maxwald3.tex, label mw:prob:hurwitz.

A termwise positive continuation of the prime form below basepoint one

Volume I, Problem 19.2, page 576 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function. For \(\alpha>1\) the density \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}\xi(\alpha+i\theta)\) has the sine squared prime form, through a signed canonical measure (atoms \(\Lambda(n)n^{-\alpha}\) at \(\log n\), with \(\Lambda\) the von Mangoldt function, an archimedean density and a pole density) whose prime and gamma terms are nonnegative and whose pole terms are negative; its positivity, which is Lagarias positivity for \(\alpha>1\), comes from the absence of zeros with real part at least one, not from the terms one by one. The book continues this form below \(\alpha=1\) by compensation, and shows that the compensated sine squared sum converges for every \(\alpha'>\alpha\) exactly when \(\zeta\) has no zero with \(\operatorname{Re}s>\alpha\); a smoothed finite form, built from a Fejér tapered sum over the primes with explicit pole terms and a gamma integral, continues the identity below \(\alpha=1\) but is not positive term by term. Is there a continuation of the prime form below \(\alpha=1\) that is positive term by term? The prime sum diverges there, but smoothed versions of it, with \(n^{-\alpha}\) replaced by a weight that cuts off at \(n\approx e^{1/\varepsilon}\), remain positive termwise, and the question is whether the loss from the cutoff can be controlled by \((\xi'/\xi)(\alpha)\), the gamma sum and the pole terms. A positive answer on \(\alpha\in(\alpha_0,1]\) would prove the zero free region \(\beta\le\alpha_0\), that is, that \(\zeta\) has no zero with real part above \(\alpha_0\).

Earlier editions: Problem 14.2, page 608 (edition 1).

Source: ch/maxwald3.tex, label mw:prob:lagarias.

Existence of the half flow clock for small positive times

Volume I, Problem 19.3, page 576 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)\exp(-\pi n^2e^{4u})\) the Pólya kernel in the rescaled variable of Rodgers and Tao, so that \(H_0(z)=\int_0^\infty\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du=\frac18\xi(\frac12+\frac{iz}2)\), and \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\) its de Bruijn Newman deformation, with de Bruijn Newman constant \(\Lambda\), the least \(t\) from which all zeros of \(H_t\) are real, known to satisfy \(0\le\Lambda\le0.2\) by Rodgers and Tao, Platt and Trudgian, and the Polymath project. Let \(R_t(u)=H_t(0)/H_t(2i\sqrt u)\), the Laplace transform of the clock of the flow, with \(R_0(u)=\xi(\frac12)/\xi(\frac12+\sqrt u)\). For \(t\ge\Lambda\), \(R_t\) is the transform of a GGC with Thorin measure \(\sum_k\delta_{(z_k(t)/2)^2}\), where \(z_k(t)>0\) are the positive zeros of \(H_t\), each pair \(\pm z\) giving one atom; so the law \(T_t\) with Laplace transform \(\lambda\mapsto R_t(\lambda/2)\) exists and is a GGC for \(t\ge0.2\) unconditionally and, under RH, for every \(t\ge0\), and \(T_0\) is half the clock, which exists unconditionally. For \(0<t<0.2\), does \(T_t\) exist as a probability law, and is it infinitely divisible? This is the analogue along the de Bruijn Newman flow of the existence of the clock; it would need an explicit formula for the heat trace of the flow, which the Gaussian tilt destroys.

Earlier editions: Problem 14.3, page 609 (edition 1).

Source: ch/maxwald3.tex, label mw:prob:flow.

An analogue of the Khinchin factorisation at one half

Volume I, Problem 19.4, page 577 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(\theta)=\xi(\frac12+i\theta)\). For \(\sigma>1\) the maximum \(M_4\) of independent exponential variables with one variable of each rate \(\pi n\) and one more of each rate \(\pi(2m-1)\), tilted to the law \(\propto x^{\sigma/2}\Pr(M_4\in dx)\) and multiplied by an independent factor \(4^{-K}\) with \(\Pr(K=k)=(1-a)a^k\) for \(k\ge0\) and \(a=2^{1-\sigma}\), has the law of \(\Gamma_{1+\sigma/2}/(\pi N_\sigma^2)\), where \(\Gamma_{1+\sigma/2}\) is a gamma variable of shape \(1+\sigma/2\) and scale \(1\) and \(N_\sigma\) is an independent zeta distributed integer, \(\Pr(N_\sigma=n)=n^{-\sigma}/\zeta(\sigma)\), built from the primes by the Euler product. Is there an analogue of this factorisation at \(\sigma=\frac12\), that is, a factorisation of the tilted law of the clock \(T\) (the random variable with Laplace transform \(\xi(\frac12)/\xi(\frac12+\sqrt\lambda)\)) or of the Pólya variable (the variable with characteristic function \(\Xi(\theta)/\Xi(0)\)) through a law built from the primes, with the explicit formula replacing the Euler product?

Earlier editions: Problem 14.4, page 609 (edition 1).

Source: ch/maxwald3.tex, label mw:prob:khinchin.

Each term of the theta series kernel is an explicit combination of two Gumbel kernels of fractional order

Volume I, Theorem 19.16.1, page 578 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\) and let \(g_k(x)=e^{-kx}\exp(-e^{-x})/\Gamma(k)\) be the density of \(-\log G_k\) for a gamma variable \(G_k\) of shape \(k\). With \(x=-4u-\log(\pi n^2)\), as densities in \(x\), \(\phi_n(u)\,du=\pi^{-1/4}n^{-1/2}\Gamma(\frac54)[\frac58g_{9/4}(x)-\frac34g_{5/4}(x)]\,dx\). The signed mass of the term is \(-\frac18\pi^{-1/4}\Gamma(\frac54)n^{-1/2}\), and the formal sum of the masses, regularised by \(\zeta\), is \(-\frac18\pi^{-1/4}\Gamma(\frac54)\zeta(\frac12)=\frac14\xi(\frac12)=\int_{\mathbb R}\Phi\). With \(s=(1+iz)/2\), \(\int_{\mathbb R}\phi_n(u)e^{izu}\,du=\frac14n^{-s}\xi(s)/\zeta(s)\).

Earlier editions: Theorem 25.1.1, page 916 (edition 1).

Source: ch/ev1.tex, label ev:thm:decomp.

Zeros of the Dirichlet polynomial of the Dyck chain at n equal to 25600 below height 60 (computed)

Volume I, Result 19.16, page 579 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(H_n\) be the height of a uniform Dyck path of length \(2n\), \(Y_n=\sqrt{2/\pi}\,H_n/\sqrt{2n}\), \(F_n(s)=\mathbb{E}Y_n^s\) (a Dirichlet polynomial in \(s\)), and \(\gamma_k\) the positive ordinates of the zeros \(\frac12+i\gamma_k\) of the Riemann zeta function on the critical line. The number of Dyck paths of length \(2n\) and height at most \(h\) is \(\sum_{k\in\mathbb{Z}}\big[\binom{2n}{n+k(h+2)}-\binom{2n}{n-1+k(h+2)}\big]\), out of \(\binom{2n}{n}/(n+1)\). At \(n=25600\), with these probabilities as balls at \(256\) bits, the argument principle on the boundary of \(-4<\operatorname{Re}s<6\), \(0<\operatorname{Im}s<60\) counts \(13\) zeros of \(F_n\), in the bands \(0<\operatorname{Im}s<20\), \(20<\operatorname{Im}s<40\) and \(40<\operatorname{Im}s<60\) the numbers \(1\), \(5\) and \(7\) that the ordinates \(\gamma_k\) have (there are \(13\) of them below \(60\)). There is no zero with \(\frac12<\operatorname{Re}s<6\) and \(0<\operatorname{Im}s<60\), and the real parts of the \(13\) zeros run from \(0.0907\) (the highest, at height \(59.126\)) to \(0.4553\) (the lowest, at height \(14.072\)). Followed in \(n\) from \(25600\) through \(12800\), \(6400\) and \(3200\) down to \(1600\), with Newton’s method started at the zero found for the next larger \(n\), the \(13\) zeros stay distinct and to the left of the line, and the zero of lowest height lies at distance \(0.3082\), \(0.1544\) and \(0.0773\) from \(\frac12+i\gamma_1\) for \(n=1600\), \(6400\) and \(25600\). The count is for \(n=25600\) only: at \(n=1600\) the same contour holds \(14\) zeros, and Newton’s method started at \(\frac12+i\gamma_k\) does not stay near it for \(k=4,\dots,12\) (the starts \(k=7\) and \(8\) converge to one zero), so the zeros are identified by following them in \(n\). This is a computation and no proof for other \(n\). The computation follows a note of Nicholas Polson merged into the book.

Earlier editions: Result 25.1, page 918 (edition 1).

Source: ch/ev1.tex, label ev:res:dyckcensus.

Lattice bias expansion of the Dirichlet polynomial of the Dyck chain (conjecture)

Volume I, Conjecture 19.16.2, page 580 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(H_n\) be the height of a uniform Dyck path of length \(2n\), \(Y_n=\sqrt{2/\pi}\,H_n/\sqrt{2n}\), \(F_n(s)=\mathbb{E}Y_n^s\), and \(\xi\) the Riemann xi function. Conjecture (open): uniformly on every compact subset of \(\operatorname{Re}s>0\), \(F_n(s)=2\xi(s)-\frac{3s}{\sqrt{\pi n}}\xi(s-1)+O(1/n)\). At \(s=1\) it says \(\mathbb{E}H_n=\sqrt{\pi n}-\frac32+O(n^{-1/2})\), which agrees with the expansion of de Bruijn, Knuth and Rice, and it implies that \(F_n\to2\xi\) locally uniformly on \(\operatorname{Re}s>0\). The zeros computed for the chain agree with its consequence (computed). The conjecture follows a heuristic about the lattice of heights in a note of Nicholas Polson merged into the book.

Earlier editions: Conjecture 25.1.2, page 918 (edition 1).

Source: ch/ev1.tex, label ev:conj:lbias.

First order displacement of the zeros of the Dyck chain polynomial from the zeros of xi (conditional)

Volume I, Proposition 19.16.3, page 581 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The lattice bias conjecture, that \(F_n(s)=2\xi(s)-\frac{3s}{\sqrt{\pi n}}\xi(s-1)+O(1/n)\) locally uniformly on \(\operatorname{Re}s>0\).

Statement. Let \(F_n(s)=\mathbb{E}Y_n^s\) be the Dirichlet polynomial of the Dyck chain (\(Y_n=\sqrt{2/\pi}\,H_n/\sqrt{2n}\), \(H_n\) the height of a uniform Dyck path of length \(2n\)), \(\xi\) the Riemann xi function and \(\delta_n=3/(2\sqrt{2n})\). Assume that, uniformly on compact subsets of \(\operatorname{Re}s>0\), \(F_n(s)=2\xi(s)-\frac{3s}{\sqrt{\pi n}}\xi(s-1)+O(1/n)\). Let \(\rho\) be a simple zero of \(\xi\) and put \(c_{\rm D}(\rho)=\sqrt{2/\pi}\,\rho\,\xi(\rho-1)/\xi'(\rho)=\sqrt{2/\pi}\,\rho\,\xi(2-\rho)/\xi'(\rho)\). There is a constant \(C\) such that for all large \(n\), \(F_n\) has exactly one zero \(z_n(\rho)\) in the disc \(|s-\rho-\delta_nc_{\rm D}(\rho)|<C/n\); thus \(z_n(\rho)=\rho+\delta_nc_{\rm D}(\rho)+O(1/n)\). The conditional statement is proved by Rouch'e’s theorem from the conjecture.

Earlier editions: Proposition 25.1.3, page 919 (edition 1).

Source: ch/ev1.tex, label ev:prop:lbias.

Phase formula for the lattice bias coefficient at a zero on the critical line

Volume I, Proposition 19.16.4, page 581 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the Riemann xi function, \(\rho=\frac12+i\gamma\), \(\gamma>0\), a simple zero of \(\zeta\) on the critical line, \(Z\) the Hardy function with Riemann Siegel phase \(\theta\) (\(\zeta(\frac12+it)=e^{-i\theta(t)}Z(t)\)), and \(c_{\rm D}(\rho)=\sqrt{2/\pi}\,\rho\,\xi(\rho-1)/\xi'(\rho)\). Then \(c_{\rm D}(\rho)=-C_{\rm D}(\gamma)e^{-i\Theta(\gamma)}/Z'(\gamma)\) with \(C_{\rm D}(\gamma)=\frac{\sqrt2}{\pi}\,|\frac32-i\gamma|\,\big|\Gamma(\frac34+\frac{i\gamma}2)/\Gamma(\frac14+\frac{i\gamma}2)\big|\,|\zeta(\frac32+i\gamma)|>0\) and \(\Theta(\gamma)=\theta(\gamma)+\arctan\tanh\frac{\pi\gamma}2+\arg\zeta(\frac32+i\gamma)-\arctan\frac3{2\gamma}\), so that \(\operatorname{Re}c_{\rm D}(\rho)\) has the sign of \(-\cos\Theta(\gamma)/Z'(\gamma)\).

Earlier editions: Proposition 25.1.4, page 919 (edition 1).

Source: ch/ev1.tex, label ev:prop:lbphase.

Lattice bias displacement of the Dyck chain zeros, computed for 200 zeros of xi and the zero near the 64th (computed)

Volume I, Result 19.16, page 582 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\rho_k=\frac12+i\gamma_k\) be the zeros of \(\zeta\) on the critical line in order of height, \(c_{\rm D}(\rho)=\sqrt{2/\pi}\,\rho\,\xi(\rho-1)/\xi'(\rho)\), \(\delta_n=3/(2\sqrt{2n})\), and \(F_n(s)=\mathbb{E}Y_n^s\), with \(Y_n=\sqrt{2/\pi}\,H_n/\sqrt{2n}\) and \(H_n\) the height of a uniform Dyck path of length \(2n\), the Dirichlet polynomial of the Dyck chain with zero \(z_n(\rho_k)\) near \(\rho_k\). In ball arithmetic, from the first \(200\) zeros of a stored list, \(\xi'(\rho_k)\ne0\) for every \(k\le200\), \(\operatorname{Re}c_{\rm D}(\rho_k)<0\) for every \(k\le63\), \(c_{\rm D}(\rho_1)=-6.72018-9.54303i\), \(c_{\rm D}(\rho_{64})=53.08757-266.82638i\) (radius below \(10^{-55}\)), and among \(k\le200\), \(\operatorname{Re}c_{\rm D}(\rho_k)>0\) exactly for \(k=64\), \(92\), \(115\), \(126\), \(136\), \(160\), \(165\), \(183\) and \(187\). At \(n=6400\) and \(25600\), Newton’s method started at \(\rho_k+\delta_nc_{\rm D}(\rho_k)\), \(k\le13\), converges to the zeros of the census of \(F_n\), each alone in a square of half side \(0.25\) by the argument principle, and the distance \(|z_n(\rho_k)-\rho_k-\delta_nc_{\rm D}(\rho_k)|\) falls by a factor between \(3.77\) and \(4.64\) from \(n=6400\) to \(n=25600\), for \(11\) of the \(13\) zeros between \(3.94\) and \(4.14\), as a remainder \(O(1/n)\) predicts; for \(\rho_1\) at \(n=25600\) the law gives the real part \(\frac12-0.04455\) against \(\frac12-0.04470\). The zero of \(F_n\) near \(\rho_{64}\), followed from \(n=10^8\) down to \(10^5\), lies to the right of the line at every computed \(n\): \(\operatorname{Re}z_n(\rho_{64})-\frac12\) is \(0.1615\), \(0.1348\), \(0.0837\), \(0.0456\), \(0.0223\), \(0.0118\) and \(0.0061\) at \(n=10^5\), \(3\cdot10^5\), \(10^6\), \(3\cdot10^6\), \(10^7\), \(3\cdot10^7\) and \(10^8\), against the first order values \(\delta_n\operatorname{Re}c_{\rm D}(\rho_{64})\) from \(0.1781\) down to \(0.00563\), and for \(10^6\le n\le10^8\) the product \(n\,|z_n(\rho_{64})-\rho_{64}-\delta_nc_{\rm D}(\rho_{64})|\) lies between \(6.33\cdot10^4\) and \(6.61\cdot10^4\). The first order regime needs \(\delta_n|c_{\rm D}(\rho_{64})|\) small: \(|c_{\rm D}(\rho_{64})|=272.1\), so this is \(0.91\) at \(n=10^5\) and \(0.029\) at \(n=10^8\). The neighbours \(\rho_{63}\) and \(\rho_{65}\) stay to the left, with real parts \(\frac12-0.0283\) and \(\frac12-0.0227\) at \(n=10^8\). The computation uses Newton’s method on ball enclosures of the exact law with the terms below \(10^{-110}\) dropped and not bounded, checked against a second law with another cut, and uniqueness by the argument principle, sampled; it is a computation. The result follows a note of Nicholas Polson merged into the book.

Earlier editions: Result 25.1, page 920 (edition 1).

Source: ch/ev1.tex, label ev:res:lbias.

All zeros of Xi are real if and only if the reciprocal xi law is a GGC, and real and simple if and only if it is the hitting time of a string

Volume I, Corollary 19.18.1, page 584 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi\) be the Riemann Xi function, \(\xi\) the Riemann xi function, \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\) the theta series kernel, and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\). Write \(\pm\gamma_k\) for the zeros of \(\Xi\), so that \(\frac12+i\gamma_k\) are the nontrivial zeros of \(\zeta\), ordered by real part (under the Riemann Hypothesis (RH) they are all real and \(0<\gamma_1\le\gamma_2\le\cdots\)), \(L(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)=H(0)/H(2i\sqrt s)\), which under RH equals \(\prod_k(1+s/\gamma_k^2)^{-1}\), and \(T\) for the random variable with Laplace transform \(L\). Then the following are equivalent: all zeros of \(\Xi\) are real (RH); \(L\) is the Laplace transform of a generalised gamma convolution (GGC) whose Thorin measure \(\sum_k\delta_{\gamma_k^2}\) is positive. Moreover, all zeros of \(\Xi\) are real and simple if and only if there is a Krein string with positive mass whose hitting time \(T\) satisfies \(\mathbb Ee^{-sT}=L(s)\). The same statements hold with \(\widehat H\) and \(\widehat L(s)=\widehat H(0)/\widehat H(2i\sqrt s)\) in place of \(H\) and \(L\), for \(\widehat H\) an even real entire function of order less than two with \(\widehat H(0)>0\).

Earlier editions: Corollary 25.3.1, page 922 (edition 1).

Source: ch/ev1.tex, label ev:cor:rh.

Under RH with simple zeros the tail of the hitting time is a sum of exponentials and its excess is exponential with rate the first squared ordinate

Volume I, Proposition 19.18.2, page 584 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on RH, simple zeros, and \(|\xi'(\rho)|\ge e^{-C|\gamma|}\) for some constant \(C\) and all zeros \(\rho=\frac12+i\gamma\).

Statement. Let \(\Xi\) be the Riemann Xi function, \(\xi\) the Riemann xi function, \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\) the theta series kernel, and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\). Write \(\pm\gamma_k\) for the zeros of \(\Xi\), so that \(\frac12+i\gamma_k\) are the nontrivial zeros of \(\zeta\), ordered by real part (under the Riemann Hypothesis (RH) they are all real and \(0<\gamma_1\le\gamma_2\le\cdots\)), \(L(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)=H(0)/H(2i\sqrt s)\), which under RH equals \(\prod_k(1+s/\gamma_k^2)^{-1}\), and \(T\) for the random variable with Laplace transform \(L\). If RH holds, the zeros are simple and \(|\xi'(\rho)|\ge e^{-C|\gamma|}\) for some constant \(C\) and all zeros \(\rho\), then \(\mathbb P(T>x)=\sum_kc_ke^{-\gamma_k^2x}\) with \(c_k=\prod_{j\ne k}(1-\gamma_k^2/\gamma_j^2)^{-1}=2i\xi(\frac12)/(\gamma_k\xi'(\frac12+i\gamma_k))\). In particular the excess of \(T\) over a high threshold is asymptotically exponential with rate \(\lambda_1=\gamma_1^2\), the bottom of the spectrum of the string; this last clause needs only the two term tail, which has no hypothesis on \(\xi'\).

Earlier editions: Proposition 25.3.2, page 923 (edition 1).

Source: ch/ev1.tex, label ev:prop:tail.

Hankel test: the zeros of an even entire kernel with infinitely many zeros are real if and only if two families of Hankel determinants are positive

Volume I, Proposition 19.21.1, page 588 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\widehat H\) be an even real entire function of order less than two with \(\widehat H(0)>0\) and infinitely many zeros \(\pm\widehat\gamma_k\) (counted in the variable \(s\) with \(\widehat H(2i\sqrt s)=\widehat H(0)\prod_k(1+s/\widehat\gamma_k^2)\)), let \(\mu_j=\sum_k\widehat\gamma_k^{-2j}\) be the power sums of the inverse atoms (the cumulants of the candidate hitting time divided by \((j-1)!\)), \(w_k=\widehat\gamma_k^{-2}\), and assume \(\sum_k|w_k|<\infty\). The zeros of \(\widehat H\) are all real (and nonzero) if and only if all the Hankel determinants \(D_n=\det[\mu_{i+j+1}]_{i,j=0}^{n-1}\) and \(D_n'=\det[\mu_{i+j+2}]_{i,j=0}^{n-1}\), \(n\ge1\), are positive.

Earlier editions: Proposition 25.6.1, page 927 (edition 1).

Source: ch/ev1.tex, label ev:prop:hankel.

Crossing symmetric Hankel positivity of the zeta amplitude is equivalent to RH

Volume I, Theorem 19.22.1, page 591 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Xi(z)=\xi(\frac12+iz)\). Write the zeros of \(\Xi\) as \(\pm\gamma_n\) with multiplicities \(g_n\) and \(\operatorname{Re}\gamma_n>0\), and put \(q_n=\gamma_n^2\) (unconditionally \(|\operatorname{Im}\gamma_n|<\frac12\) and \(\operatorname{Re}\gamma_n>14\)). Let \(A(s)=-\frac{d}{ds}\log\Xi(\sqrt s)=\sum_ng_n/(q_n-s)\) and let \(c_{2k}\) be the Taylor coefficients of \(A(s)+A(-s)=\sum_{k\ge0}c_{2k}s^{2k}\), that is \(c_{2k}=2\sum_ng_nq_n^{-(2k+1)}\). Then the Riemann Hypothesis holds if and only if both Hankel matrices \((c_{2(i+j)})_{i,j<n}\) and \((c_{2(i+j+1)})_{i,j<n}\) are positive definite for every \(n\ge1\).

Source: ch/ev1.tex, label ev:thm:amp.

Volume I, Chapter 20. Ferromagnetic random variables

Clock and van Dantzig partner of a ferromagnetic law

Volume I, Theorem 20.1.4, page 595 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A real random variable \(X\) is called ferromagnetic if it is symmetric, \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function \(L_X(z)=\mathbb{E}e^{zX}\), which is then entire, vanishes only on the imaginary axis; equivalently every zero of the characteristic function \(\varphi_X(\theta)=L_X(i\theta)\) is real. By Newman’s theorem such an \(X\) has the Hadamard form \(L_X(s)=e^{bs^2}\prod_k(1+s^2/c_k^2)\) with \(b\ge0\), \(c_k>0\) and \(\sum_kc_k^{-2}<\infty\), each \(c_k\) repeated according to the multiplicity of the zero \(ic_k\). Let \(X\) be ferromagnetic with this form, let \(e_k\) be independent standard exponential variables and \(Z\) a standard normal variable independent of them, and put \(T_X=b+\sum_ke_k/c_k^2\) and \(Y_X=Z\sqrt{2T_X}\); the series converges almost surely and in mean. Then (i) \(\mathbb{E}e^{-uT_X}=1/L_X(\sqrt{u})\) for \(u\ge0\), and \(T_X\) is a generalised gamma convolution (GGC, a law on \([0,\infty)\) with Laplace transform \(\exp\{-b\lambda-\int\log(1+\lambda/t)\,\mu(dt)\}\) for a drift \(b\ge0\) and a positive Thorin measure \(\mu\)) with drift \(b\) and Thorin measure \(\sum_k\delta_{c_k^2}\), one unit atom for each pair of zeros \(\pm ic_k\). (ii) \(\mathbb{E}e^{i\theta Y_X}=1/L_X(\theta)\) for real \(\theta\), so that \(\varphi_{Y_X}(\theta)\,\varphi_X(i\theta)=1\) and \((\varphi_X,\varphi_{Y_X})\) is a van Dantzig pair, and \(Y_X\overset{d}{=}\sqrt{2b}\,Z+\sum_kL_k/c_k\) with independent standard Laplace variables \(L_k\), whose characteristic function is \(1/(1+\theta^2)\). (iii) The moment generating function of \(Y_X\) is \(1/\varphi_X(t)\) for \(|t|<c_1=\min_kc_k\) and infinite for \(|t|\ge c_1\), unless \(X\) is Gaussian. (iv) Unless \(X\) is degenerate, \(Y_X\) has a Pólya frequency density of infinite order. (v) \(\kappa_m(T_X)=(-1)^{m+1}m!\,\kappa_{2m}(X)/(2m)!\) for \(m\ge1\), where \(\kappa_j\) denotes the \(j\)th cumulant; in particular \(\mathbb{E}T_X=\frac12\operatorname{Var}X\) and \(\operatorname{Var}T_X=-\kappa_4(X)/12\).

Earlier editions: Theorem 13.1.4, page 531 (edition 1).

Source: ch/fm1.tex, label fm:thm:dual.

Ferromagnetic laws are closed under convergence in law

Volume I, Theorem 20.1.5, page 597 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A real random variable \(X\) is called ferromagnetic if it is symmetric, \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function \(L_X(z)=\mathbb{E}e^{zX}\), which is then entire, vanishes only on the imaginary axis; Gaussian laws are allowed. Let \(X_n\) be ferromagnetic and suppose that \(X_n\to X\) in law. Then \(X\) is ferromagnetic, \(\operatorname{Var}X_n\to\operatorname{Var}X\), and \(L_{X_n}\to L_X\) locally uniformly on \(\mathbb{C}\).

Earlier editions: Theorem 13.1.5, page 533 (edition 1).

Source: ch/fm1.tex, label fm:thm:closed.

RH holds exactly when the Pólya law is a limit in law of ferromagnetic laws

Volume I, Corollary 20.1.8, page 598 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A real random variable \(X\) is called ferromagnetic if it is symmetric, \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function \(\mathbb{E}e^{zX}\), which is then entire, vanishes only on the imaginary axis. The Pólya law is the law of the symmetric random variable \(X\) with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\); its moment generating function is \(\mathbb{E}e^{sX}=\xi(\frac12+s)/\xi(\frac12)\). The Riemann Hypothesis holds if and only if the Pólya law is a limit in law of ferromagnetic laws.

Earlier editions: Corollary 13.1.8, page 534 (edition 1).

Source: ch/fm1.tex, label fm:cor:limit.

A uniform law mixed with an atom at the origin is not ferromagnetic

Volume I, Theorem 20.2.1, page 599 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: counterexample, proved.

Statement. A real random variable is called ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function, which is then entire, vanishes only on the imaginary axis. Let \(U\) be uniform on \([-1,1]\), so that \(\mathbb{E}e^{sU}=\sinh s/s\). For every \(0<q<1\) the mixture \((1-q)\,\mathrm{Unif}[-1,1]+q\,\delta_0\) of the uniform law on \([-1,1]\) with weight \(1-q\) and the point mass at \(0\) with weight \(q\) is symmetric and unimodal but not ferromagnetic: its moment generating function \((1-q)\sinh s/s+q\) has no real zero, only finitely many zeros on the imaginary axis, and infinitely many zeros in all, so that all but finitely many of its zeros are off the imaginary axis. Consequently, for every \(0<q<1\) and every small enough \(a>0\), the mixture of the uniform laws on \([-1,1]\) and on \([-a,a]\) with weights \(1-q\) and \(q\) is not ferromagnetic. Both components of these mixtures are ferromagnetic, so the class of ferromagnetic laws is not closed under mixtures.

Earlier editions: Theorem 13.2.1, page 535 (edition 1).

Source: ch/fm2.tex, label fm:thm:atom.

The Pólya law is not a scale mixture of the triangular law, and its kernel has one inflection point

Volume I, Proposition 20.2.3, page 600 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_2(x)=(1-|x|)_+\) be the triangular density on \([-1,1]\), so that its characteristic function is \(4\sin^2(\theta/2)/\theta^2\), and call \(X\) a scale mixture of \(f_2\) for a probability law \(\nu\) on \((0,\infty)\) if \(X=rY\) with \(r\sim\nu\) independent of \(Y\sim f_2\). Let \(\Phi(u)=2\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\bigr)\exp(-\pi n^2e^{2u})\) be the Pólya kernel, so that \(\Xi(\theta)=\xi(\frac12+i\theta)=\int_{\mathbb{R}}\Phi(u)\cos(\theta u)\,du\) for Riemann’s completed zeta function \(\xi\). (i) If \(X\) is a scale mixture of \(f_2\) for \(\nu\), then \(\mathbb{E}e^{i\theta X}\ge0\) for real \(\theta\), and for \(\theta\ne0\) it vanishes exactly when \(\nu\) is carried by \(\{2\pi j/|\theta|:j\ge1\}\); in particular if \(\nu\) has a nonzero absolutely continuous part, \(\mathbb{E}e^{i\theta X}\) has no real zeros. (ii) The Pólya law, with density proportional to \(\Phi\), is not a scale mixture of \(f_2\) for any probability law: with \(y_n=\pi n^2e^{2u}\) and \(R(y)=32y^4-224y^3+330y^2-75y\), \(\Phi''(u)=\frac12e^{u/2}\sum_{n\ge1}R(y_n)e^{-y_n}\), \(\Phi''(0)<0\), and \(\Phi''(u)>0\) for \(u\ge\frac12\log(7/\pi)\), whereas a mixture of the convex functions \((1-x/r)_+\) is convex on \((0,\infty)\). (iii) \(\Phi\) has exactly one positive inflection point \(u^*\), with \(\Phi''<0\) on \([0,u^*)\), \(\Phi''>0\) on \((u^*,\infty)\) and \(\Phi'''(u^*)>0\); certified in ball arithmetic, \(u^*=0.23726528718742524911\). The proof of (ii) and (iii) uses three inequalities certified in ball arithmetic.

Earlier editions: Proposition 13.2.3, page 536 (edition 1).

Source: ch/fm2.tex, label fm:prop:triangle.

Shoulder theorem for symmetric laws on an interval

Volume I, Theorem 20.2.4, page 601 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A real random variable is called ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function, which is then entire, vanishes only on the imaginary axis; equivalently every zero of its characteristic function \(\varphi_X\) is real. Let \(p\) be a symmetric probability density on \([-1,1]\) that is absolutely continuous and nonincreasing on \([0,1]\), with \(p(1)=0\), let \([-c,c]\) with \(0<c\le1\) be its support, and let \(X\) have density \(p\). (i) If \(p\) is concave on \([0,1]\), then \(X\) is ferromagnetic. (ii) If \(p\) is convex on \([0,1]\) and not linear on \([0,c]\), then \(X\) is not ferromagnetic; moreover \(\varphi_X\ge0\) on \(\mathbb{R}\), and its real zeros, if any, are double and lie in a lattice \((2\pi/s_0)\mathbb{Z}\) with \(0<s_0<c\).

Earlier editions: Theorem 13.2.4, page 537 (edition 1).

Source: ch/fm2.tex, label fm:thm:shoulder.

Ferromagnetic laws among the kernels 1 minus |x|^b, k monotone kernels and products of uniform laws

Volume I, Corollary 20.2.5, page 602 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A real random variable is called ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function, which is then entire, vanishes only on the imaginary axis. (i) The law with density proportional to \(1-|x|^b\) on \([-1,1]\), \(b>0\), is ferromagnetic if and only if \(b\ge1\). In Khinchine’s form, \(UR\) with \(U\) uniform on \([-1,1]\) and \(R\) independent with the Beta\((a,1)\) law (density \(ar^{a-1}\) on \([0,1]\)) is ferromagnetic if and only if \(a\ge2\). (ii) For an integer \(k\ge1\) the symmetric \(k\) monotone kernel \(f_k(x)=\frac k2(1-|x|)_+^{k-1}\) (the uniform density for \(k=1\) and the triangle density for \(k=2\)) is ferromagnetic if and only if \(k\le2\). (iii) The laws with density proportional to \((1-x^2)^m\) on \([-1,1]\) with \(0<m\le1\), which include the semicircle law, are ferromagnetic. (iv) The product \(U_1U_2\) of two independent uniform variables on \([-1,1]\), with density \(\frac12\log(1/|x|)\) on \([-1,1]\), is not ferromagnetic.

Earlier editions: Corollary 13.2.5, page 538 (edition 1).

Source: ch/fm2.tex, label fm:cor:shoulder.

Densities nondecreasing in |x| and mixtures of uniform laws on shells are ferromagnetic

Volume I, Theorem 20.2.6, page 602 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A real random variable is called ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function, which is then entire, vanishes only on the imaginary axis. Every symmetric probability density on \([-1,1]\) that is nondecreasing in \(|x|\) is ferromagnetic. Equivalently, every mixture of the uniform laws on the shells \(\{a\le|x|\le1\}\), \(0\le a<1\), is ferromagnetic.

Earlier editions: Theorem 13.2.6, page 539 (edition 1).

Source: ch/fm2.tex, label fm:thm:ushaped.

Residue expansion of the Mellin transform of a ferromagnetic clock and of a pair

Volume I, Proposition 20.3.1, page 604 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be a ferromagnetic random variable (a characteristic function \(\varphi_X\) with only real zeros) with \(b=0\) in its Hadamard product and infinitely many simple zeros \(\pm ic_k\), \(0<c_1<c_2<\cdots\), so that \(L_X(s)=\mathbb{E}e^{sX}=\prod_k(1+s^2/c_k^2)\), and let \(T_X=\sum_ke_k/c_k^2\) be its clock, with \(e_k\) independent standard exponential variables. Put \(\mathcal D_X(w)=\Gamma(w)\mathbb{E}T_X^{-w}=\int_0^\infty u^{w-1}(1/L_X(\sqrt u)-1)\,du\) for \(-1<\operatorname{Re}w<0\), and \(A_k=-2/(c_k\varphi_X'(c_k))\), \(B_k=2(c_k\varphi_X''(c_k)-\varphi_X'(c_k))/(c_k^2\varphi_X'(c_k)^3)\). (i) Near \(u=-c_k^2\), \(1/L_X(\sqrt u)=A_k(1+u/c_k^2)^{-1}+O(1)\) and \(1/L_X(\sqrt u)^2=A_k^2(1+u/c_k^2)^{-2}+B_k(1+u/c_k^2)^{-1}+O(1)\). (ii) If \(1/L_X(\sqrt u)=1+\sum_kA_k((1+u/c_k^2)^{-1}-1)\) for \(u\ge0\) and \(\sum_k|A_k|c_k^{2\sigma}<\infty\) for some \(\sigma\in(-1,0)\), then \(\mathcal D_X(w)=\frac{\pi}{\sin\pi w}\sum_kA_kc_k^{2w}\) for \(-1<\operatorname{Re}w\le\sigma\). (iii) If \(1/L_X(\sqrt u)^2=1+\sum_k[A_k^2((1+u/c_k^2)^{-2}-1)+B_k((1+u/c_k^2)^{-1}-1)]\) for \(u\ge0\) and \(\sum_k(A_k^2+|B_k|)c_k^{2\sigma}<\infty\) for some \(\sigma\in(-1,0)\), then for an independent copy \(X'\) of \(X\), \(\mathcal D_{X+X'}(w)=\frac{\pi}{\sin\pi w}\sum_k[(1-w)A_k^2+B_k]c_k^{2w}\) for \(-1<\operatorname{Re}w\le\sigma\). The result is from a note of Nicholas Polson merged into the book.

Earlier editions: Proposition 13.3.1, page 541 (edition 1).

Source: ch/fm2m.tex, label fm:prop:residue.

Mellin transform of the clock of a sum of k uniform spins as a combination of zeta values

Volume I, Proposition 20.3.2, page 606 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(U_1,\dots,U_k\) be independent random variables uniform on \([-1,1]\), \(X_k=\pi(U_1+\dots+U_k)\), and \(\mathcal D_{X_k}(w)=\Gamma(w)\mathbb{E}T^{-w}\) the Mellin transform of the Laplace transform of the clock \(T\) of \(X_k\) (the sum of \(e_j/c_j^2\) over the zeros \(\pm c_j\) of the characteristic function, \(e_j\) independent standard exponential variables). For \(\operatorname{Re}w>0\), \(\mathcal D_{X_k}(w)=2(2\pi)^{-2w}\Gamma(2w+k)\sum_{m\ge0}\binom{m+k-1}{k-1}(m+\frac k2)^{-2w-k}\). For \(k=2r\) the series is \(\sum_{i=0}^{r-1}a_{r,i}\zeta(2w+1+2i)\), where \(a_{r,i}\) is the coefficient of \(j^{2r-1-2i}\) in \(j\prod_{l=1}^{r-1}(j^2-l^2)/(2r-1)!\), and for \(k=2r+1\) it is \(\sum_{i=0}^rb_{r,i}(2^{2w+1+2i}-1)\zeta(2w+1+2i)\), where \(b_{r,i}\) is the coefficient of \(h^{2r-2i}\) in \(\prod_{l=1}^r(h^2-(l-\frac12)^2)/(2r)!\). The first and the last of these coefficients are not \(0\), so for \(k\ge3\) the series has at least two terms. In particular \(\mathcal D_{X_1}(w)=4\pi^{-2w}(1-2^{-2w-1})\Gamma(2w+1)\zeta(2w+1)\), \(\mathcal D_{X_4}(w)=\frac13(2\pi)^{-2w}\Gamma(2w+4)[\zeta(2w+1)-\zeta(2w+3)]\) and \(\mathcal D_{X_6}(w)=\frac1{60}(2\pi)^{-2w}\Gamma(2w+6)[\zeta(2w+1)-5\zeta(2w+3)+4\zeta(2w+5)]\). The result is from a note of Nicholas Polson merged into the book.

Earlier editions: Proposition 13.3.2, page 542 (edition 1).

Source: ch/fm2m.tex, label fm:prop:kspins.

Zeros of the Mellin transform of the clock for one, four and six uniform spins (computed)

Volume I, Result 20.3, page 607 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(U_1,\dots,U_k\) be independent uniform on \([-1,1]\) and \(X_k=\pi(U_1+\dots+U_k)\), and let \(\mathcal D_{X_k}\) be the Mellin transform of the Laplace transform of the clock of \(X_k\) (the clock being the sum of \(e_j/c_j^2\) over the zeros \(\pm c_j\) of the characteristic function). In the variable \(z=2w+1\), for \(k=1\) the nonreal zeros of \(\mathcal D_{X_1}\) are those of \(\zeta(z)\) and the points \(z=2\pi im/\log2\), \(m\ne0\), so they lie on the two lines \(\operatorname{Re}z=\frac12\) (under the Riemann Hypothesis) and \(\operatorname{Re}z=0\); for \(k=2\) they are the zeros of \(\xi(z)\). For \(k=4\) and \(k=6\) they do not lie on one vertical line: \(\zeta(z)-\zeta(z+2)\) vanishes at \(z=1.76891972665+23.3078297413i\) and \(z=0.650238826497+31.7124838244i\), and \(\zeta(z)-5\zeta(z+2)+4\zeta(z+4)\) at \(z=3.18634599215+31.979112615i\) and \(z=1.51363208762+41.215007487i\) (computed in ball arithmetic, each zero alone in a square of half side \(7.5\cdot10^{-9}\) by the argument principle). The statement for \(k=4,6\) is a computation.

Earlier editions: Result 13.3, page 543 (edition 1).

Source: ch/fm2m.tex, label fm:res:kzeros.

The density (1+cos t)/2pi gives a ferromagnetic law whose Mellin transform has zeros off one line (computed)

Volume I, Result 20.3, page 607 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(X\) have the density \((1+\cos t)/2\pi\) on \([-\pi,\pi]\), which is \(C^1\) and not \(C^2\) at \(\pm\pi\), let \(X'\) be an independent copy, and let \(\mathcal D_Y(w)\) be the Mellin transform of the Laplace transform of the clock of \(Y\) (the clock being the sum of \(e_k/c_k^2\) over the zeros \(\pm c_k\) of the characteristic function of \(Y\), \(e_k\) independent standard exponential variables), \(U,U'\) independent uniform on \([-1,1]\). The characteristic function of \(X\) is that of \(\pi U\) divided by \(Q(\theta^2)\) with \(Q(x)=1-x\), the zeros are \(c_k=k+1\), \(k\ge1\), the law is ferromagnetic, and \(\mathcal D_{X+X'}(w)=\mathcal D_{\pi U+\pi U'}(w)+2\mathcal D_{\pi U+\pi U'}(w+1)+\mathcal D_{\pi U+\pi U'}(w+2)\), a combination of \(\zeta(-2w)\), \(\zeta(-2w-2)\) and \(\zeta(-2w-4)\). Its residue weights \(A_k=2(-1)^{k+1}(k^2+2k)\) grow like \(k^2\). In the box \(-1<\operatorname{Re}w<1\), \(\frac12<\operatorname{Im}w<30\), which holds the \(13\) zeros \(\frac{\rho_k-1}2\), \(\gamma_k<60\), of \(\mathcal D_{\pi U+\pi U'}\) (\(\rho_k=\frac12+i\gamma_k\) the zeros of \(\zeta\) on the critical line), \(\mathcal D_{X+X'}\) has no zero, and in \(-3<\operatorname{Re}w<-2\), \(\frac12<\operatorname{Im}w<30\) it has \(6\), at \(-2.443554+15.837821i\), \(-2.976747+19.238797i\), \(-2.479242+21.119434i\), \(-2.273308+24.426843i\), \(-2.862245+26.206760i\) and \(-2.756588+28.553832i\), so not on one vertical line (computed in ball arithmetic).

Earlier editions: Result 13.3, page 543 (edition 1).

Source: ch/fm2m.tex, label fm:res:onecos.

Deformation of the uniform pair, the first two Mellin zeros leave the line at different rates (computed)

Volume I, Result 20.3, page 607 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. For \(\varepsilon\ge0\) let \(X_\varepsilon\) have the density \((1+\varepsilon t^2)/(2\pi+\frac23\pi^3\varepsilon)\) on \([-\pi,\pi]\), which is nondecreasing in \(|t|\), so that \(X_\varepsilon\) and the sum \(X_\varepsilon+X_\varepsilon'\) with an independent copy are ferromagnetic, and let \(\mathcal D_{X_\varepsilon+X_\varepsilon'}\) be the Mellin transform of the Laplace transform of the clock of the pair. At \(\varepsilon=0\) the zeros \(w_k(0)=\frac{\rho_k-1}2\) of \(\mathcal D_{\pi U+\pi U'}\) (\(U,U'\) independent uniform on \([-1,1]\), \(\rho_k\) the zeros of \(\zeta\) on the critical line) move, where \(\rho_k\) is simple, with \(\frac{dw_k}{d\varepsilon}(0)=-\dot{\mathcal D}(w_k(0))/\mathcal D'_{\pi U+\pi U'}(w_k(0))\) by the implicit function theorem, \(\dot{\mathcal D}\) being the derivative of \(\mathcal D_{X_\varepsilon+X_\varepsilon'}\) in \(\varepsilon\) at \(\varepsilon=0\). The computed values are \(\frac{dw_1}{d\varepsilon}(0)=5.89145-13.09321i\) and \(\frac{dw_2}{d\varepsilon}(0)=1.56442-16.25014i\), with real parts different from \(0\) and from each other (computed by quadrature with three rules that agree). So for small \(\varepsilon>0\) the first two zeros leave the line \(\operatorname{Re}w=-\frac14\) at different rates, and the nonreal zeros of \(\mathcal D_{X_\varepsilon+X_\varepsilon'}\) do not lie on one vertical line although every \(X_\varepsilon\) is ferromagnetic. By this computation no theorem whose hypotheses hold for all symmetric densities on an interval that are nondecreasing in \(|t|\) can put the zeros of the Mellin transform of the pair on one line.

Earlier editions: Result 13.3, page 544 (edition 1).

Source: ch/fm2m.tex, label fm:res:deform.

The Pólya law has no Gaussian component and no ferromagnetic summand with finitely many zeros

Volume I, Theorem 20.4.2, page 611 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be a symmetric random variable with \(\mathbb{E}e^{sX}<\infty\) for every real \(s\) and \(\log\mathbb{E}e^{sX}=o(s^2)\) as \(s\to\infty\), and suppose that \(X\overset{d}{=}D+R\) with \(D\) and \(R\) independent. Then \(\log\mathbb{E}e^{sD}=o(s^2)\) as \(|s|\to\infty\). In particular, if \(D\) is ferromagnetic (symmetric, with \(\mathbb{E}e^{b_0D^2}<\infty\) for some \(b_0>0\) and a moment generating function vanishing only on the imaginary axis) and not \(0\), then in its Hadamard form \(\mathbb{E}e^{sD}=e^{bs^2}\prod_k(1+s^2/c_k^2)\) the Gaussian factor has \(b=0\), and the characteristic function of \(D\) has infinitely many zeros. The Pólya variable, the symmetric random variable with density \(\Phi/\xi(\frac12)\) and moment generating function \(\xi(\frac12+s)/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), satisfies the hypothesis with no assumption on the zeros of \(\zeta\). So it has no Gaussian component (it is not the sum of an independent normal variable of positive variance and another variable), and it is not a finite or convergent infinite sum of independent ferromagnetic variables of which one is not \(0\) and has a characteristic function with finitely many zeros.

Earlier editions: Theorem 13.4.1, page 547 (edition 1).

Source: ch/fm3.tex, label fm:thm:nosqrt.

Finite sums of independent spins, uniforms and a Gaussian do not converge to the Pólya law

Volume I, Proposition 20.4.3, page 612 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X_n=\sum_i\lambda_{n,i}S_{n,i}+\sum_k\mu_{n,k}U_{n,k}+\sigma_nZ_n\) be finite sums of independent symmetric Ising spins \(S\in\{\pm1\}\), independent uniform variables \(U\) on \([-1,1]\) and an independent standard Gaussian \(Z\), with nonnegative coefficients. Then \(X_n\) does not converge in law to the Pólya law \(X\), the symmetric law with moment generating function \(\xi(\frac12+s)/\xi(\frac12)\) where \(\xi\) is the completed zeta function. The proof uses the computed value of the first ordinate \(\gamma_1=14.1347\ldots\) and that no ordinate equals \(2\gamma_1\) or \(3\gamma_1\) (the ordinates below \(50\) being known), and no hypothesis on the other ordinates.

Source: ch/fm3.tex, label fm:prop:indep.

Independent ferromagnetic blocks of bounded total weight do not converge to the Pólya law

Volume I, Theorem 20.4.5, page 613 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A block is \(X_B=\lambda\sum_{i\le k}n_iS_i\) with \(\lambda>0\), positive integers \(n_i\), total weight \(K=\sum_in_i\), and \((S_i)\) a ferromagnetic Ising system with positive couplings. Let \(X_n\) be finite sums of independent blocks, each block with its own \(\lambda\), couplings and integers, plus an independent Gaussian, and suppose every block has total weight \(K\le K_0\). Then \(X_n\) does not converge in law to the Pólya law \(X\) (the symmetric law with moment generating function \(\xi(\frac12+s)/\xi(\frac12)\)) in the following two cases. (i) \(K_0\le287\); this is proved, with the first \(8000\) ordinates computed in ball arithmetic (radius below \(10^{-34}\)). (ii) \(K_0\) is arbitrary; this is proved with the theorem of Li and Radziwiłł that for every \(\alpha>0\) and real \(\beta\) at least one third of the numbers \(\alpha n+\beta\) are not ordinates of zeros of \(\zeta\) on the critical line, and it contains (i).

Source: ch/fm3.tex, label fm:thm:blocks.

The Pólya law has no convolution factor that is a bounded spin with an arithmetic progression of zeros

Volume I, Proposition 20.4.6, page 614 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(\Xi(t)=\xi(\frac12+it)\), and let \(X\) be the Pólya law, the symmetric random variable with characteristic function \(\varphi_X(t)=\Xi(t)/\Xi(0)\). Let \(W\) be a bounded symmetric random variable whose characteristic function vanishes at every point of an infinite arithmetic progression \(\{\alpha n+\beta:n\ge0\}\) with \(\alpha>0\) (for example a uniform, Ising or cosine power spin). Then for no \(a>0\) and no probability measure \(q\) is the law of \(X\) equal to \(\mathrm{law}(aW)*q\). Consequently, in any exact representation of the Pólya law as the law of \(\sum_ka_kW_k\) under a ferromagnetic Gibbs measure with such spins, every spin is coupled to at least one other spin. The proof uses the theorem of Li and Radziwiłł that at least one third of the numbers \(\alpha n+\beta\) are not ordinates of zeros of \(\zeta\) on the critical line, and for a uniform spin the theorem of Putnam that the ordinates contain no progression \(\{d,2d,3d,\dots\}\).

Source: ch/fm3.tex, label fm:prop:indecomp.

Ferromagnetic sums of uniforms with couplings bounded below do not converge to the Pólya law

Volume I, Theorem 20.4.8, page 615 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya law, with characteristic function \(\Xi(t)/\Xi(0)\), \(\Xi(t)=\xi(\frac12+it)\). A ferromagnetic sum of uniforms is \(X_n=\sum_ka_{n,k}V_k\) with \(a_{n,k}>0\) and \((V_k)\) having joint density proportional to \(\exp(\sum_{k<l}J_{n,kl}v_kv_l)\) on \([-1,1]^m\), \(J_{n,kl}\ge0\). Let \(\beta_0>0\) be such that for \(0\le\beta\le\beta_0\) the even measure \(e^{-\beta v^2}dv\) on \([-1,1]\) has a characteristic function with only real zeros (such \(\beta_0\) exists). Suppose the couplings satisfy \(J_{n,kl}\ge2\varepsilon a_{n,k}a_{n,l}\) for all \(k<l\) and all \(n\), with one \(\varepsilon>0\) and \(\varepsilon\max_{n,k}a_{n,k}^2\le\beta_0\). Then \(X_n\) does not converge in law to \(X\). The proof uses the theorem of Rodgers and Tao that the damped densities have a transform with nonreal zeros, that is \(\Lambda\ge0\).

Source: ch/fm3.tex, label fm:thm:dense.

Long range couplings with harmonic weights do not converge to the Pólya law

Volume I, Corollary 20.4.9, page 615 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya law, with characteristic function \(\Xi(t)/\Xi(0)\). Let \(c>0\), \(a_k=1/(2(k+c))\) for \(k\ge0\), and let finite ferromagnetic sums of uniforms \(\sum_ka_kV_k\) with weights \(a_k\) have couplings \(J_{kl}\ge J|k-l|^{-\alpha}\) for all pairs, with \(J>0\) and \(\alpha\le1\) (mean field is \(\alpha=0\)). Then these sums do not converge in law to \(X\). Nearest neighbour chains and long range couplings with \(\alpha>1\) are not excluded.

Source: ch/fm3.tex, label fm:cor:longrange.

Cauchy blocks of the hitting time: GGC property, limits, tail and decomposition

Volume I, Proposition 20.4.11, page 616 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(\theta,\gamma>0\) let \(\mathcal B_{\theta,\gamma}\) be the positive variable with \(\mathbb{E}e^{-s\mathcal B_{\theta,\gamma}}=(\theta^2+\gamma^2)/((\theta+\sqrt s)^2+\gamma^2)\) for \(s\ge0\). (i) \(\mathcal B_{\theta,\gamma}\) is a generalised gamma convolution whose Thorin density in \(w=\sqrt t\) is the folded Cauchy density \(\frac1\pi\bigl[\frac\theta{\theta^2+(w-\gamma)^2}+\frac\theta{\theta^2+(w+\gamma)^2}\bigr]\). (ii) As \(\theta\downarrow0\), \(\mathcal B_{\theta,\gamma}\to e/\gamma^2\) in law, with \(e\) a standard exponential. (iii) \(P(\mathcal B_{\theta,\gamma}>y)\sim\frac{2\theta}{\theta^2+\gamma^2}(\pi y)^{-1/2}\) as \(y\to\infty\). (iv) Let \(\xi\) be the completed zeta function, \(\rho=\beta_\rho+i\gamma_\rho\) its zeros, and \(T_\alpha\) the variable with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt s)\). If no zero has \(\beta_\rho\ge\alpha\), in particular for every \(\alpha\ge1\), then \(T_\alpha\overset d=\sum_{\gamma_\rho>0}\mathcal B_{\alpha-\beta_\rho,\gamma_\rho}\) with independent summands and almost sure convergence, the sum over the zeros with positive ordinate, with multiplicity.

Source: ch/fm3.tex, label fm:prop:block.

Brownian motion at twice the tilted hitting time has a Cauchy component

Volume I, Proposition 20.4.12, page 616 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha\ge1\), \(\xi\) the completed zeta function, \(T_\alpha\) the variable with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt s)\), \(\mathrm{Br}\) a Brownian motion independent of \(T_\alpha\), and \(W_\alpha=\mathrm{Br}(2T_\alpha)\). Then \(W_\alpha\) is symmetric with \(\mathbb{E}e^{iuW_\alpha}=\xi(\alpha)/\xi(\alpha+|u|)=1/\mathbb{E}e^{|u|V_\alpha}\), where \(V_\alpha\) is the Esscher tilt of the Pólya variable at the basepoint \(\alpha\). Moreover \(W_\alpha\overset d=C_\alpha+W_\alpha^\circ\) with independent terms, \(C_\alpha\) a Cauchy variable of scale \(b_\alpha=(\xi'/\xi)(\alpha)\) and \(W_\alpha^\circ=\mathrm{Br}(2T_\alpha^\circ)\), where \(T_\alpha=S_\alpha+T_\alpha^\circ\) is the decomposition of the book into a one half stable part and a remainder. Further \(P(|W_\alpha|>x)=2b_\alpha/(\pi x)+O(x^{-3})\) and \(P(|W_\alpha^\circ|>x)\sim4c(\alpha)/(3\pi x^3)\), with \(c(\alpha)\) the constant of the book’s tail result for \(T_\alpha^\circ\), namely \(P(T_\alpha^\circ>t)\sim c(\alpha)t^{-3/2}/(6\sqrt\pi)\).

Source: ch/fm3.tex, label fm:prop:wcauchy.

Harmonic sum of independent Pólya variables, moments, and RH as a uniform sum

Volume I, Theorem 20.4.13, page 617 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable (symmetric, with moment generating function \(\xi(\frac12+z)/\xi(\frac12)\), \(\xi\) the completed zeta function), \(X_1,X_2,\dots\) independent copies, and \(S=\sum_{n\ge1}X_n/n\), convergent almost surely and in \(L^2\). Then \(\mathbb{E}e^{zS}=\prod_{n\ge1}\xi(\frac12+z/n)/\xi(\frac12)\) for \(z\in\mathbb{C}\) and \(\kappa_{2m}(S)=\zeta(2m)\kappa_{2m}(X)\) for \(m\ge1\), with \(\kappa_{2m}\) the cumulant of order \(2m\). Moreover the Riemann Hypothesis holds if and only if \(S\overset d=\sum_{\gamma>0}(\pi/\gamma)U_\gamma\) with independent \(U_\gamma\) uniform on \([-1,1]\), the sum over the positive ordinates \(\gamma\) of the zeros. The book notes that this is Newman’s criterion with the harmonic scaling and gives no new route to RH.

Source: ch/fm3.tex, label fm:thm:harmonic.

Möbius inversion identity for harmonic sums of independent symmetric variables

Volume I, Proposition 20.4.14, page 618 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Y\) be a symmetric random variable with finite variance, \(S_Y=\sum_{n\ge1}Y_n/n\) for independent copies \(Y_n\) of \(Y\), and let \(S^{(k)}\), \(k\ge1\), be independent copies of \(S_Y\), independent of \(Y\). With \(\mu\) the Möbius function and \(S_\pm=\sum_{\mu(k)=\pm1}S^{(k)}/k\), one has \(Y+S_-\overset d=S_+\). No hypothesis on zeros of \(\zeta\) is needed. If moreover \(Y\) has finite moments of order \(2m\), then \(\kappa_{2m}(Y)=\kappa_{2m}(S_Y)/\zeta(2m)\), with \(\kappa_{2m}\) the cumulant of order \(2m\). The identity is a convolution identity between laws, not a signed sum. For the symmetric Davenport and Heilbronn law the spin reading of \(S_{DH}\) fails, since its moment generating function vanishes at \(n(\rho-\frac12)\) for the nonreal zero \(\rho=0.8085\ldots+85.6993\ldots i\) (the location of that zero is computed).

Source: ch/fm3.tex, label fm:prop:mobius.

Weighted sums of tilted Pólya variables are a tilt of a ferromagnetic law only under RH

Volume I, Proposition 20.4.15, page 618 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(c_j>0\), \(\alpha_j\ge1\), let \(V^{(j)}_{\alpha_j}\) be independent Esscher tilts of the Pólya variable (the variable with moment generating function \(\xi(\alpha+z)/\xi(\alpha)\), \(\xi\) the completed zeta function), and \(Z=\sum_jc_jV^{(j)}_{\alpha_j}\) converging in law, with finite moment generating function \(\prod_j\xi(\alpha_j+c_jz)/\xi(\alpha_j)\). This function has no zero on the imaginary axis. \(Z\) is an Esscher tilt by a field \(\theta\) of a ferromagnetic law if and only if the Riemann Hypothesis holds and \((\alpha_j-\frac12)/c_j=\theta\) for every \(j\), and then the family is finite. So no weighting of the levels \(\alpha\ge1\) gives a ferromagnetic system up to a tilt without proving RH.

Source: ch/fm3.tex, label fm:prop:weights.

The tilted Pólya variable minus an exponential, and its Gaussian limit

Volume I, Proposition 20.4.16, page 619 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha>1\), let \(V_\alpha\) be the Esscher tilt of the Pólya variable with moment generating function \(\xi(\alpha+z)/\xi(\alpha)\) (\(\xi\) the completed zeta function), \(G\sim\mathrm{Gamma}(\frac\alpha2+1,1)\), \(N\) with \(P(N=n)=n^{-\alpha}/\zeta(\alpha)\) and \(e\sim\mathrm{Exp}(\alpha-1)\) (rate \(\alpha-1\)), all independent of one another and of \(V_\alpha\). Then \(V_\alpha-e\overset d=\frac12\log(G/\pi)-\log N\). Consequently \(\sqrt{2\alpha}\,(V_\alpha-\frac12\log\frac\alpha{2\pi})\) converges in law to the standard normal \(\mathrm{N}(0,1)\) as \(\alpha\to\infty\).

Source: ch/fm3.tex, label fm:prop:vminuse.

Cosh moment of the difference of two tilted Pólya variables, kernel with cos(theta x)

Volume I, Proposition 20.4.17, page 619 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha\in\mathbb{R}\), \(\theta=\alpha-\frac12\), \(V_\alpha\) the Esscher tilt of the Pólya variable with moment generating function \(\xi(\alpha+z)/\xi(\alpha)\) (\(\xi\) the completed zeta function), and \(D_\alpha=V_\alpha-V_\alpha'\) with an independent copy, so that \(\mathbb{E}\cosh(vD_\alpha)=\xi(\alpha+v)\xi(\alpha-v)/\xi(\alpha)^2\). Then for real \(v\), \(\log\mathbb{E}\cosh(vD_\alpha)=\int_{\mathbb{R}}2\sin^2(vx/2)\cos(\theta x)\,k(|x|)\,dx/|x|\) with \(k(x)=2\sum_{\gamma_\rho>0}e^{-g_\rho x}\) and \(g_\rho=-i(\rho-\frac12)\), the sum over zeros with positive ordinate, and no hypothesis on the zeros. Moreover the tilt exponent satisfies \(v_{1-\alpha}=-v_\alpha\) (the differentiated functional equation), and \(D_\alpha\) is not ferromagnetic for \(\alpha\ne\frac12\).

Source: ch/fm3.tex, label fm:prop:modul.

Fluctuation relation for the density of the tilted Pólya variable

Volume I, Proposition 20.4.18, page 619 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(V_\alpha\) be the Esscher tilt of the Pólya variable by the field \(\theta=\alpha-\frac12\), with moment generating function \(\xi(\alpha+z)/\xi(\alpha)\) (\(\xi\) the completed zeta function) and density \(p_\alpha\). Then \(p_\alpha(-y)=e^{-(2\alpha-1)y}p_\alpha(y)\), and at \(\alpha=1\), \(\log(p_1(y)/p_1(-y))=y\). The reading as a fluctuation relation of statistical mechanics is an analogy only.

Source: ch/fm3.tex, label fm:prop:fluct.

Zeros of the harmonic sum of uniforms counted by a divisor function

Volume I, Proposition 20.4.19, page 620 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(V_k\) be independent uniform on \([-1,1]\), \(K\ge1\) an integer, \(X_K=\frac12\sum_{k\ge K}V_k/k\), with characteristic function \(\varphi_K(t)=\prod_{k\ge K}\operatorname{sinc}(t/2k)\), \(\operatorname{sinc}(x)=\sin x/x\). The zeros of \(\varphi_K\) are real, at \(t=\pm2\pi m\) with multiplicity \(d_K(m)=\#\{k\ge K:k\mid m\}\), so with \(x=T/2\pi\) the number of zeros in \((0,T]\) is \(\sum_{m\le x}d_K(m)=x\log x+O_K(x)\), which matches the main term of the Riemann and von Mangoldt count \(x\log x-x+O(\log T)\). For \(K=1\) the coefficient of \(x\) is \(2\gamma_{\mathrm E}-1\), and for \(K=3\) it is \(2\gamma_{\mathrm E}-\frac52\), with \(\gamma_{\mathrm E}\) Euler’s constant. Moreover \(\log\varphi_K(t)=-\sum_{j\ge1}S_j^{(K)}t^{2j}/j\) with \(S_j^{(K)}=\zeta(2j)\big(\zeta(2j)-\sum_{k<K}k^{-2j}\big)/(2\pi)^{2j}\).

Source: ch/fm3.tex, label fm:prop:divisor.

No ferromagnetic deformation of the harmonic sums X1 and X2 has the Pólya law

Volume I, Proposition 20.4.20, page 620 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a_k=1/(2k)\) for \(k\ge K\), \(X_K=\sum_{k\ge K}a_kV_k\) with independent uniform \(V_k\) on \([-1,1]\), and let \(X_{K,J}\) be a ferromagnetic deformation of \(X_K\) with these weights (the \(V_k\) under the joint density proportional to \(\exp(\sum_{k<l}J_{kl}v_kv_l)\), \(J_{kl}\ge0\)), or a limit in law of such. Then \(\operatorname{Var}X_{K,J}\ge\operatorname{Var}X_K\). Consequently no ferromagnetic deformation of \(X_1\) or \(X_2\) has the Pólya law (the symmetric law with characteristic function \(\Xi(t)/\Xi(0)\), variance \(0.0462099862\)), since \(\operatorname{Var}X_1=0.137\) and \(\operatorname{Var}X_2=0.0537\). The proof uses Ginibre’s form of the second Griffiths inequality, and the variance of \(X_3\) is \(0.0329\), below that of the Pólya law, so there is no such obstruction for \(K=3\).

Source: ch/fm3.tex, label fm:prop:gks.

Zero count of the shifted harmonic sum of uniforms

Volume I, Proposition 20.4.21, page 621 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(c>0\) let \(X^{(c)}=\frac12\sum_{k\ge0}V_k/(k+c)\) with independent uniform \(V_k\) on \([-1,1]\), whose characteristic function has real zeros \(2\pi n(k+c)\), \(n\ge1\), \(k\ge0\). With \(x=T/2\pi\), the number of zeros of \(X^{(c)}\) in \((0,T]\) is \(\sum_{k\ge0}\lfloor x/(k+c)\rfloor=x\log x+(\gamma_{\mathrm E}-1-\psi(c))x+o(x)\), where \(\psi\) is the digamma function and \(\gamma_{\mathrm E}\) Euler’s constant.

Source: ch/fm3.tex, label fm:prop:shiftcount.

Principal part of the zero zeta function of the shifted harmonic sum

Volume I, Proposition 20.4.22, page 621 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\psi\) be the digamma function, \(\gamma_{\mathrm E}\) Euler’s constant, \(\zeta(s,c)\) the Hurwitz zeta function, \(Z^{(c)}(s)=\sum_{n\ge1,k\ge0}(2\pi n(k+c))^{-s}=(2\pi)^{-s}\zeta(s)\zeta(s,c)\), and \(Z_\Xi(s)=\sum_{\gamma>0}\gamma^{-s}\) summed over the ordinates \(\gamma>0\) of the zeros \(\frac12+i\gamma\) of \(\zeta\), with multiplicity. Then \(Z^{(c)}\) and \(Z_\Xi\) have the same principal part at \(s=1\) if and only if \(\psi(c)=\gamma_{\mathrm E}\), namely \(\frac1{2\pi}\big[(s-1)^{-2}-\log(2\pi)(s-1)^{-1}\big]\).

Source: ch/fm3.tex, label fm:prop:polar.

Cosine power spin with real zeros at pi (n plus b)

Volume I, Lemma 20.4.23, page 622 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(b>\frac12\) let \(W\) have density proportional to \(\cos(\pi w/2)^{2b-2}\) on \([-1,1]\). Then \(\mathbb{E}e^{itW}=\Gamma(b)^2/\big(\Gamma(b-t/\pi)\Gamma(b+t/\pi)\big)\), with zeros only at \(t=\pm\pi(n+b)\), \(n\ge0\). So \(W\) is a bounded single spin with real zeros; \(b=1\) is the uniform spin and \(b\to\frac12\) the Ising spin.

Source: ch/fm3.tex, label fm:lem:bspin.

Row bound for ferromagnetic realisations of the Pólya law with a cosine power spin tail

Volume I, Theorem 20.4.25, page 622 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(\Xi(t)=\xi(\frac12+it)\), \(E(t)=\big|\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\big|\) at \(s=\frac12+it\), so that \(\Xi(t)/E(t)=\pm Z(t)\) with \(Z\) Hardy’s function. Let \(X=Y+\sum_{k\ge L}a_kW_k\), where \(Y\) has any law, the \(W_k\) are independent of each other and of \(Y\), \(a_k=1/(2(k+c))\) with \(c>0\), and each \(W_k\) is a cosine power spin with parameter \(b>\frac12\) (density proportional to \(\cos(\pi w/2)^{2b-2}\) on \([-1,1]\)). Let \(\varphi_X\) be the characteristic function of \(X\), \(P_L(T)=\{2\pi(n+b)(k+c)\le T:n\ge0,\ k\ge L\}\), and \(\mathcal E_T(\varphi)=\sup_{0\le t\le T}|\Xi(t)-\Xi(0)\varphi(t)|/E(t)\). Then \(\mathcal E_T(\varphi_X)\ge\max_{t\in P_L(T)}|Z(t)|\). The bound holds for every ferromagnetic coupling of the block \(k<L\).

Source: ch/fm3.tex, label fm:thm:rowbound.

Powers of the exponential of the Pólya variable are not infinitely divisible for exponent below 2

Volume I, Proposition 20.5.1, page 624 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\). For every real \(\beta\) with \(0<|\beta|<2\) the random variable \(e^{\beta X}\) is not infinitely divisible. No hypothesis on the zeros of \(\zeta\) is used.

Earlier editions: Proposition 13.6.1, page 564 (edition 1).

Source: ch/fm5.tex, label fm:prop:notid.

The exponential of the Pólya variable is not a generalised gamma convolution

Volume I, Theorem 20.5.3, page 625 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), and let \(B_0=\sum_{m\ge1}\Gamma_m/(\pi m^2)\) with independent Gamma\((2,1)\) variables \(\Gamma_m\); this is the variable \(\frac\pi2S_2\) of Biane, Pitman and Yor, a generalised gamma convolution (GGC) with Laplace transform \((\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda})^2\) and \(\mathbb{E}B_0^\sigma=2\xi(2\sigma)\) for every complex \(\sigma\). For \(0<q<1\) let \(B_q\) be \(B_0\) size biased by \(B_0^q\), that is \(\mathbb{E}g(B_q)=\mathbb{E}[B_0^qg(B_0)]/\mathbb{E}B_0^q\), so that \(\mathbb{E}B_q^\sigma=\xi(2\sigma+2q)/\xi(2q)\). A GGC is a law on \([0,\infty)\) with Laplace transform \(\exp\{-b\lambda-\int\log(1+\lambda/t)\,\mu(dt)\}\) for a drift \(b\ge0\) and a positive Thorin measure \(\mu\). Then for every \(0<q<1\) the variable \(B_q\) is not a GGC. In particular \(e^{2X}\) is not a GGC, where \(X\) is the Pólya variable, the symmetric random variable with density \(\Phi/\xi(\frac12)\), \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and moment generating function \(\xi(\frac12+s)/\xi(\frac12)\), because \(e^{2X}\overset{d}{=}B_{1/4}\). No hypothesis on the zeros is used.

Earlier editions: Theorem 13.6.3, page 565 (edition 1).

Source: ch/fm5.tex, label fm:thm:notggc.

Size bias principle for sums of gamma variables with integer shapes

Volume I, Theorem 20.5.5, page 627 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Y=\sum_{n=1}^K\Gamma_n/\tau_n\) with \(0<\tau_1<\dots<\tau_K\) and independent gamma variables \(\Gamma_n\) with integer shapes \(t_n\ge1\) and unit scale, let \(0<q<1\), and let \(Y_q\) be \(Y\) size biased by \(Y^q\), that is \(\mathbb{E}g(Y_q)=\mathbb{E}[Y^qg(Y)]/\mathbb{E}Y^q\). Then \(Y_q+G\overset{d}{=}\sum_{n=1}^K\Gamma_n'/\tau_n\), where the \(\Gamma_n'\) are independent gamma variables with shapes \(t_n+q\) and unit scale, and \(G\), independent of \(Y_q\), is a generalised gamma convolution without drift whose Thorin measure has a continuous distribution function \(\theta_G\) that vanishes on \((0,\tau_1]\), increases by exactly \(q\) on each interval \([\tau_k,\tau_{k+1}]\) and is constant on \([\tau_K,\infty)\).

Earlier editions: Theorem 13.6.5, page 567 (edition 1).

Source: ch/fm5.tex, label fm:thm:sbp.

Factorisation of the exponential of the Pólya variable against a Biane Pitman Yor variable

Volume I, Theorem 20.5.6, page 628 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(B_0=\sum_{m\ge1}\Gamma_m/(\pi m^2)\) with independent Gamma\((2,1)\) variables \(\Gamma_m\) (the variable \(\frac\pi2S_2\) of Biane, Pitman and Yor, with \(\mathbb{E}B_0^\sigma=2\xi(2\sigma)\)), and for \(0<q<1\) let \(B_q\) be \(B_0\) size biased by \(B_0^q\), so that \(\mathbb{E}B_q^\sigma=\xi(2\sigma+2q)/\xi(2q)\). Let \(\frac\pi2S_{2+q}=\sum_{m\ge1}\Gamma_m'/(\pi m^2)\) with independent gamma variables \(\Gamma_m'\) of shape \(2+q\) and unit scale, with Laplace transform \((\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda})^{2+q}\). Then \(B_q+G_q\overset{d}{=}\frac\pi2S_{2+q}\), where \(G_q\) is independent of \(B_q\) and is a generalised gamma convolution (a law on \([0,\infty)\) with Laplace transform \(\exp\{-b\lambda-\int\log(1+\lambda/t)\,\mu(dt)\}\)) without drift whose Thorin measure has a continuous distribution function \(\theta_q\) with \(\theta_q=0\) on \((0,\pi]\) and \(\theta_q(\pi m^2)=(m-1)q\) for every \(m\ge1\). In particular \(e^{2X}+G_{1/4}\overset{d}{=}\frac\pi2S_{9/4}\), where \(X\) is the Pólya variable, the symmetric random variable with density \(\Phi/\xi(\frac12)\), \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and moment generating function \(\xi(\frac12+s)/\xi(\frac12)\), because \(e^{2X}\overset{d}{=}B_{1/4}\). No hypothesis on the zeros is used.

Earlier editions: Theorem 13.6.6, page 568 (edition 1).

Source: ch/fm5.tex, label fm:thm:factor.

The exponential of the Pólya variable is self decomposable with a Lévy density built from the theta function

Volume I, Theorem 20.5.7, page 629 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\omega(x)=\sum_{n\ge1}e^{-\pi n^2x}\), let \(B_q\), \(0<q<1\), be the variable \(B_0=\sum_{m\ge1}\Gamma_m/(\pi m^2)\) (independent Gamma\((2,1)\) variables \(\Gamma_m\), the variable \(\frac\pi2S_2\) of Biane, Pitman and Yor) size biased by \(B_0^q\), and let \(\theta_q\) be the distribution function of the Thorin measure of the generalised gamma convolution \(G_q\) in the identity \(B_q+G_q\overset{d}{=}\frac\pi2S_{2+q}\), where \(\frac\pi2S_{2+q}\) has Laplace transform \((\sqrt{\pi\lambda}/\sinh\sqrt{\pi\lambda})^{2+q}\); \(\theta_q\) is continuous, vanishes on \((0,\pi]\), and satisfies \(\theta_q(\pi m^2)=(m-1)q\). Then \(B_q\) is self decomposable, with no drift and Lévy density \(k_q(x)/x\), where \(k_q(x)=(2+q)\,\omega(x)-\int_0^\infty e^{-tx}\,d\theta_q(t)\) and \(2\,\omega(x)\le k_q(x)\le(2+q)\,\omega(x)\). In particular \(e^{2X}\), with \(X\) the Pólya variable (the symmetric random variable with density \(\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and with moment generating function \(\xi(\frac12+s)/\xi(\frac12)\)), which has the law of \(B_{1/4}\), is self decomposable, hence infinitely divisible and unimodal; its Lévy density lies between \(2\omega(x)/x\) and \(\frac94\omega(x)/x\), and it is not a generalised gamma convolution. No hypothesis on the zeros is used.

Earlier editions: Theorem 13.6.7, page 569 (edition 1).

Source: ch/fm5.tex, label fm:thm:sd.

Hyperbolically completely monotone densities of the powers of the exponential of the Pólya variable

Volume I, Proposition 20.5.8, page 631 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\). A density \(f\) on \((0,\infty)\) is hyperbolically completely monotone (HCM) if for every \(u>0\) the function \(f(uv)f(u/v)\) of \(w=v+1/v\) is completely monotone; an HCM law is a generalised gamma convolution (Bondesson). The variable \(e^{\beta X}\) has an HCM density, and so is a generalised gamma convolution, for \(|\beta|\ge5.5791422\), and it does not have an HCM density for \(|\beta|\le5.5791395\). The proof of the first statement is computer assisted, in ball arithmetic. No hypothesis on the zeros is used.

Earlier editions: Proposition 13.6.8, page 571 (edition 1).

Source: ch/fm5.tex, label fm:prop:hcm.

The Stern row counts c_N(m) as coprime pairs of the Calkin Wilf tree, and the limit of Knauf’s partition function, with a new proof of Knauf’s limit

Volume I, Proposition 20.6.1, page 632 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(s\) be Stern’s diatomic sequence, \(s(1)=1\), \(s(2n)=s(n)\), \(s(2n+1)=s(n)+s(n+1)\), let \(Z_N(\beta)=\sum_{k=2^N}^{2^{N+1}-1}s(k)^{-\beta}=\sum_{m\ge1}c_N(m)m^{-\beta}\) for complex \(\beta\), where \(c_N(m)\) is the number of \(k\) with \(2^N\le k<2^{N+1}\) and \(s(k)=m\), let \(\varphi\) be Euler’s function and \(F_n\) the Fibonacci numbers with \(F_1=F_2=1\). Then \(c_N(1)=1\), and for \(m\ge2\), \(c_N(m)\) is the number of coprime pairs \((p,q)\) of positive integers with \(p+q=m\) for which \(p/q\) has depth at most \(N-1\) in the Calkin Wilf tree (root \(1/1\), the children of \(p/q\) being \(p/(p+q)\) and \((p+q)/q\)). Hence \(c_N(m)\) does not decrease with \(N\), \(c_N(m)=\varphi(m)\) for \(m\le N+1\), \(c_N(m)=0\) for \(m>F_{N+2}\), and \(c_N(m)\to\varphi(m)\) as \(N\to\infty\). Moreover \(Z_N(\beta)\to\zeta(\beta-1)/\zeta(\beta)\) as \(N\to\infty\) for \(\operatorname{Re}\beta>2\), increasingly for real \(\beta>2\), and \(Z_N(-1)=\sum_m m\,c_N(m)=3^N\).

Earlier editions: Proposition 13.7.1, page 573 (edition 1).

Source: ch/fm5k.tex, label fm:prop:stern.

Perron identity for Knauf’s partition function, with the depth profile term and the remainder R_N

Volume I, Proposition 20.6.2, page 634 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(s\) be Stern’s diatomic sequence, \(s(1)=1\), \(s(2n)=s(n)\), \(s(2n+1)=s(n)+s(n+1)\), let \(c_N(m)\) be the number of \(k\) with \(2^N\le k<2^{N+1}\) and \(s(k)=m\), which vanishes for \(m>F_{N+2}\) (\(F_n\) the Fibonacci numbers, \(F_1=F_2=1\)), let \(Z_N(\beta)=\sum_{k=2^N}^{2^{N+1}-1}s(k)^{-\beta}=\sum_m c_N(m)m^{-\beta}\) be the partition function of Knauf’s spin chain, let \(\varphi\) and \(\mu\) be Euler’s and Möbius’s functions, and let \(w_N(x)=c_N(m)/\varphi(m)\) for \(m\le x<m+1\), \(m\ge1\), and \(w_N(x)=1\) for \(0<x<1\), and let \(W_N(z)=\int_0^\infty w_N(x)x^{z-1}\,dx\) for \(\operatorname{Re}z>0\), which is \(1/z\) plus an entire function. For \(\operatorname{Re}\beta>1\), \(\beta\ne2\), \(Z_N(\beta)=\zeta(\beta-1)/\zeta(\beta)+W_N(2-\beta)/\zeta(2)+R_N(\beta)\), where \(R_N(\beta)=\sum_{d\ge1}\mu(d)d^{-\beta}r_d(\beta)\), \(r_d(\beta)=\sum_{n\ge1}n^{1-\beta}w_N(dn)-\zeta(\beta-1)-d^{\beta-2}W_N(2-\beta)\), and the series converges absolutely. \(R_N\) extends analytically to \(\beta=2\), and \(C_N(\beta)=Z_N(\beta)-W_N(2-\beta)/\zeta(2)\) is meromorphic on the plane with a single pole, at \(\beta=2\), which is simple with residue \(1/\zeta(2)\).

Earlier editions: Proposition 13.7.2, page 574 (edition 1).

Source: ch/fm5k.tex, label fm:prop:perron.

Computed zeros of Knauf’s partition function and of the corrected sum near the points 1 plus rho

Volume I, Result 20.6.2, page 634 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(Z_N(\beta)=\sum_{k=2^N}^{2^{N+1}-1}s(k)^{-\beta}\) be the partition function of Knauf’s spin chain, \(s\) Stern’s diatomic sequence, let \(C_N=Z_N-W_N(2-\beta)/\zeta(2)\) be the corrected sum with the profile term removed, and let \(\rho_n\) be the nontrivial zeros of \(\zeta\) with positive ordinate, in increasing order of ordinate. For every even \(N\) from \(10\) to \(34\) and every \(n\le20\), the disc of radius \(0.3\) about \(1+\rho_n\) contains exactly one zero of \(Z_N\) and exactly one zero of \(C_N\), found by Newton’s method from \(1+\rho_n\) (winding number on the circle, sampled; a computation, not a ball proof), and the largest distance of such a zero from \(1+\rho_n\) is \(0.243\). Removing the profile term does not bring the zeros consistently closer: the mean distance is smaller for \(C_N\) only at \(N=14\) and at every computed \(N\) from \(24\) on. At \(N=34\) the mean distances are \(0.0081\) for \(Z_N\) and \(0.0047\) for \(C_N\).

Earlier editions: Result 13.7.2, page 575 (edition 1).

Source: ch/fm5k.tex, label fm:res:fisher.

Conjecture that the remainder R_N of Knauf’s partition function tends to zero on a strip around the line 3/2

Volume I, Conjecture 20.6.3, page 635 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(R_N\) be the remainder of the Perron identity for the partition function \(Z_N(\beta)=\sum_{k=2^N}^{2^{N+1}-1}s(k)^{-\beta}\) of Knauf’s spin chain, where \(s\) is Stern’s diatomic sequence, so that \(Z_N-W_N(2-\beta)/\zeta(2)=\zeta(\beta-1)/\zeta(\beta)+R_N\) with \(W_N(z)=\int_0^\infty w_N(x)x^{z-1}\,dx\), where \(w_N(x)=c_N(m)/\varphi(m)\) for \(m\le x<m+1\) and \(w_N=1\) on \((0,1)\), \(c_N(m)\) is the number of \(k\) with \(2^N\le k<2^{N+1}\) and \(s(k)=m\) (\(s\) Stern’s diatomic sequence), and \(\varphi\) is Euler’s function. Conjecture: there is an open strip \(\sigma_1<\operatorname{Re}\beta<\sigma_2\) with \(1\le\sigma_1<\frac32<\sigma_2\le2\) on which \(R_N\to0\) locally uniformly as \(N\to\infty\). If it holds, then by Hurwitz’s theorem every point \(1+\rho\) in the strip, for \(\rho\) a nontrivial zero of \(\zeta\), is the limit of as many zeros of the corrected sum \(Z_N-W_N(2-\beta)/\zeta(2)\) as its multiplicity, and no other limit points occur in the strip. The computations support it only at the first twenty points \(1+\rho_n\) on the line \(\operatorname{Re}\beta=\frac32\), and at \(\beta=\frac32+2i\) the modulus of \(R_N\) stays between \(0.021\) and \(0.051\) for \(10\le N\le34\) with no decrease. The consequence does not decide the Riemann Hypothesis, since the zeros of the corrected sum would converge to the points \(1+\rho\) whether or not \(\operatorname{Re}\rho=\frac12\).

Earlier editions: Conjecture 13.7.3, page 575 (edition 1).

Source: ch/fm5k.tex, label fm:conj:R.

Circle lemma for palindromic polynomials with small middle coefficients, and the Farey chain above beta star of N

Volume I, Lemma 20.6.5, page 637 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P(x)=1+x^N+Q(x)\) with \(Q(x)=\sum_{k=1}^{N-1}a_kx^k\) and \(a_k=a_{N-k}\ge0\). If \(Q(1)<2\), the zeros of \(P\) are simple and lie on the unit circle. Consequence: the Farey chain of Kleban and Özlük has configurations the words \(\varpi_1\cdots\varpi_N\) in \(A=\begin{pmatrix}1&0\\1&1\end{pmatrix}\) and \(B=A^{\mathsf T}\), energy \(\log\operatorname{tr}(\varpi_1\cdots\varpi_N)\), and partition function in the field \(x=e^h\) equal to \(Z^{\rm F}_N(\beta,x)=\sum(\operatorname{tr}\varpi_1\cdots\varpi_N)^{-\beta}x^{n_B}\), with \(n_B\) the number of letters \(B\). Since \(A^N\) and \(B^N\) have trace \(2\) and every other word has trace at least \(3\), \(Z^{\rm F}_N(\beta,x)=2^{-\beta}(1+x^N+Q_{N,\beta}(x))\) with \(Q_{N,\beta}(1)=\sum(2/\operatorname{tr}\varpi_1\cdots\varpi_N)^\beta\) over the other words, strictly decreasing in \(\beta\), so for every \(\beta>\beta^*(N)\), the root of \(2^\beta Z^{\rm F}_N(\beta,1)=4\), all zeros of \(x\mapsto Z^{\rm F}_N(\beta,x)\) are simple and on the unit circle (the Lee and Yang property at \((\beta,N)\)).

Earlier editions: Lemma 13.7.5, page 577 (edition 1).

Source: ch/fm5k.tex, label fm:lem:circle.

Computed Lee and Yang threshold of the Farey chain in a field for even N from 10 to 36

Volume I, Result 20.6.3, page 637 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. For the Farey chain in a field, with partition function \(Z^{\rm F}_N(\beta,x)=\sum(\operatorname{tr}\varpi_1\cdots\varpi_N)^{-\beta}x^{n_B}\), with \(n_B\) the number of letters \(B\), over the words \(\varpi_1\cdots\varpi_N\) in \(A=\begin{pmatrix}1&0\\1&1\end{pmatrix}\) and \(B=A^{\mathsf T}\), the computation finds for each even \(N\) from \(10\) to \(36\) a threshold \(\beta_{\rm LY}(N)\) such that all zeros of \(x\mapsto Z^{\rm F}_N(\beta,x)\) lie on the unit circle at every computed \(\beta\ge\beta_{\rm LY}(N)\) and some do not at every computed \(\beta\) with \(0<\beta<\beta_{\rm LY}(N)\) (grid of step \(0.1\) on \(0.2\le\beta\le4\), one transition at every \(N\), then bisection). For \(N\le32\) the bracket of width \(3\times10^{-11}\) is certified by a Sturm count in ball arithmetic at 400 bits, and for \(N=34,36\) it is a double precision bisection. At the threshold four zeros meet on the circle as two double zeros \(e^{\pm i\theta_N}\), and just below it they leave the circle as a quartet \(e^{\pm u\pm i\theta}\). The threshold rises with \(N\) from \(1.2424\) at \(N=10\) to \(1.8329\) at \(N=36\) and stays below \(\beta^*(N)\), the root of \(2^\beta Z^{\rm F}_N(\beta,1)=4\), which falls from \(2.1056\) to \(2.0525\). This is a computation for these \(N\) and says nothing about larger \(N\).

Earlier editions: Result 13.7.3, page 578 (edition 1).

Source: ch/fm5k.tex, label fm:res:ly.

Conjecture on the Lee and Yang property of the Farey chain in a field above and below beta equal to 2

Volume I, Conjecture 20.6.6, page 637 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For the Farey chain in a field, with partition function \(Z^{\rm F}_N(\beta,x)=\sum(\operatorname{tr}\varpi_1\cdots\varpi_N)^{-\beta}x^{n_B}\), with \(n_B\) the number of letters \(B\), over the words \(\varpi_1\cdots\varpi_N\) in \(A=\begin{pmatrix}1&0\\1&1\end{pmatrix}\) and \(B=A^{\mathsf T}\), say that it has the Lee and Yang property at \((\beta,N)\) when every zero of \(x\mapsto Z^{\rm F}_N(\beta,x)\) lies on the unit circle. Conjecture: for every \(\beta>2\) the property holds at \((\beta,N)\) for all large \(N\), and for every \(\beta\) with \(0<\beta<2\) it fails for all large \(N\). The value \(2\) is the inverse temperature of the phase transition of the chain, not a fit: free fits of the computed thresholds \(\beta_{\rm LY}(N)\) in powers of \(1/N\) extrapolate to values between \(2.017\) and \(2.086\), all above \(2\).

Earlier editions: Conjecture 13.7.6, page 578 (edition 1).

Source: ch/fm5k.tex, label fm:conj:ly.

Mixed shoulders and the sine transform of the slope (open problem)

Volume I, Problem 20.1, page 638 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. A real random variable is called ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function vanishes only on the imaginary axis. Let \(p\) be a symmetric probability density on \([-1,1]\), absolutely continuous and nonincreasing on \([0,1]\) with \(p(1)=0\), and put \(g=-p'\ge0\) on \((0,1)\). For a variable \(X\) with density \(p\) and characteristic function \(\varphi_X\), integration by parts gives \(z\,\varphi_X(z)=2\int_0^1g(y)\sin(zy)\,dy\), so \(X\) is ferromagnetic if and only if the sine transform of the slope \(g\) has only real zeros. The book decides the case in which \(g\) is monotone: \(X\) is ferromagnetic if \(p\) is concave on \([0,1]\), and not ferromagnetic if \(p\) is convex on \([0,1]\) and not linear on its support. Open problem: decide the slopes that first rise and then fall. One class is settled. If \(g\) is symmetric about \(\frac12\), then \(p(0)=\int_0^1g=1\), so that \(u\mapsto g(\frac12+u)\) is a probability density on \([-\frac12,\frac12]\), and \(X\overset{d}{=}\frac12U+B\) with \(U\) uniform on \([-1,1]\) and \(B\) independent with that density; since \(\sin(z/2)/(z/2)\) has only real zeros, \(X\) is then ferromagnetic exactly when \(B\) is. For the beta slopes \(g(y)\propto y^a(1-y)^b\) with \(a,b\ge0\) this decides the diagonal \(a=b\), where \(B\) is a symmetric beta law and \(X\) is ferromagnetic, and the shoulder theorem decides the edges: \(X\) is ferromagnetic for \(b=0\), where \(p\) is concave, and not for \(a=0<b\), where \(p\) is convex and not linear. The rest of the quadrant is not decided. On the half line the slope of the Pólya law, the symmetric law with density \(\Phi/\xi(\frac12)\) where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), rises and then falls, and the Riemann Hypothesis is the statement that its sine transform has only real zeros.

Earlier editions: Problem 13.1, page 579 (edition 1).

Source: ch/fm6.tex, label fm:prob:shoulder.

Which ferromagnetic laws have all nonreal zeros of the clock Mellin transform on one vertical line? (open problem)

Volume I, Problem 20.2, page 638 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a ferromagnetic law \(X\) let \(\mathcal D_X(w)=\Gamma(w)\mathbb{E}T_X^{-w}\) be the Mellin transform of the Laplace transform of its clock \(T_X\) (the sum of \(e_k/c_k^2\) over the zeros \(\pm c_k\) of the characteristic function). Open problem: for which ferromagnetic laws \(X\) does \(\mathcal D_X\) have nonreal zeros, all of them on one vertical line? The uniform pair \(\pi U+\pi U'\) (\(U,U'\) independent uniform on \([-1,1]\)) has this property if and only if the Riemann Hypothesis holds, since \(\mathcal D_{\pi U+\pi U'}(w)=2\pi^{-w}\Gamma(w)\xi(1+2w)\). The sums of one, four and six independent copies of \(\pi U\) do not have it (for four and six computed), and neither do the pairs \(X_\varepsilon+X_\varepsilon'\) with density proportional to \(1+\varepsilon t^2\) on \([-\pi,\pi]\) for small \(\varepsilon>0\) (computed), so no theorem whose hypotheses hold for every symmetric density on an interval that is nondecreasing in \(|t|\) can give it. Is the uniform pair, up to scale, the only pair \(X+X'\) with \(X\) a symmetric density on an interval nondecreasing in \(|t|\) that has it? A first step is the first order motion of the zeros along the densities \(\frac1{2\pi}+\varepsilon g\) on \([-\pi,\pi]\), which is linear in \(g\): is there an even \(g\ne0\) with \(\int g=0\) along which every zero \(\frac{\rho_k-1}2\) keeps its real part to first order?

Earlier editions: Problem 13.2, page 579 (edition 1).

Source: ch/fm6.tex, label fm:prob:mellin.

Does the Pólya law belong to the Griffiths Simon class (open problem)

Volume I, Problem 20.3, page 639 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The Griffiths Simon class consists of the weak limits of suitably scaled sums of ferromagnetically coupled Ising spins; its laws are ferromagnetic, that is symmetric with a moment generating function that vanishes only on the imaginary axis. The class of Ellis, Monroe and Newman consists of the laws with density \(e^{-V(x)}\), \(V\) even and \(V'\) convex on \([0,\infty)\), and their weak limits; by a theorem of Newman the Pólya law (the symmetric law with density \(\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and with moment generating function \(\xi(\frac12+s)/\xi(\frac12)\)) belongs to it. Open problem: does the Pólya law belong to the Griffiths Simon class? A positive answer would imply the Riemann Hypothesis. The book proves, with no hypothesis on the zeros, that finite sums of independent symmetric Ising spins, independent uniform variables on \([-1,1]\) and an independent Gaussian, with nonnegative coefficients, do not converge in law to the Pólya law, so the spins of any approximating system would have to be coupled.

Earlier editions: Problem 13.3, page 579 (edition 1).

Source: ch/fm6.tex, label fm:prob:gs.

Splitting the zeros of the Pólya characteristic function into two characteristic functions (open problem)

Volume I, Problem 20.4, page 639 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Assume the Riemann Hypothesis, and let \(X\) be the Pólya variable (the symmetric random variable with density \(\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and with moment generating function \(\xi(\frac12+s)/\xi(\frac12)\)), so that its characteristic function is \(\varphi_X(t)=\prod_\gamma(1-t^2/\gamma^2)\) over the positive ordinates \(\gamma\) of the zeros of \(\zeta\), with no Gaussian factor. Open problem: can the ordinates be split into two nonempty sets \(A\) and \(A'\) such that \(\prod_{\gamma\in A}(1-t^2/\gamma^2)\) and \(\prod_{\gamma\in A'}(1-t^2/\gamma^2)\) are both characteristic functions? This is the same as writing \(X=D+R\) with \(D\) and \(R\) independent, ferromagnetic and not \(0\), and both sets are then infinite. Under Ingham’s hypothesis, that the positive ordinates are linearly independent over the rationals, no three ordinates are in arithmetic progression, so no piece is a scaled fair sign or a scaled uniform variable, whose zeros form arithmetic progressions.

Earlier editions: Problem 13.4, page 580 (edition 1).

Source: ch/fm6.tex, label fm:prob:split.

Positivity of the zero completed laws and the threshold near K = 5000 (open problem)

Volume I, Problem 20.5, page 639 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), with characteristic function \(\varphi_X\), let \(\gamma_1\le\gamma_2\le\cdots\) be the positive ordinates of the zeros \(\frac12\pm i\gamma_k\) of \(\zeta\) on the critical line, with multiplicity, put \(s_K=\sum_{k\le K}\gamma_k^{-2}\) and \(\beta_K=\frac12\operatorname{Var}X - s_K>0\), and let \(f_K\) be the function with Fourier transform \(\varphi_K(\theta)=e^{-\beta_K\theta^2}\prod_{k\le K}(1-\theta^2/\gamma_k^2)\). For \(v>0\) the function \(e^{-v\theta^2/2}\prod_{k\le K}(1-\theta^2/\gamma_k^2)\) is the Fourier transform of \(f_{K,v}(x)=g_v(x)\sum_{j=0}^Ke_jv^{-j}\mathrm{He}_{2j}(x/\sqrt{v})\), where \(g_v\) is the \(N(0,v)\) density, \(\mathrm{He}_n(y)=\mathbb{E}(y+iZ)^n\) with \(Z\) standard normal is the Hermite polynomial and \(e_j\) is the \(j\)th elementary symmetric function of \(\gamma_1^{-2},\dots,\gamma_K^{-2}\); there is a threshold \(v^*(K)\) with \(f_{K,v}\ge0\) if and only if \(v\ge v^*(K)\), and \(f_K=f_{K,2\beta_K}\). The book proves that \(f_K\) is a ferromagnetic probability density for \(K\le16\) by the Maxwell construction, and for every \(K\le200\) and for \(K=300\), \(400\), \(600\) and \(800\) by a computer assisted proof in ball arithmetic; the ratio \(v^*(K)/2\beta_K\) increases slowly, to \(0.733\) at \(K=800\). Open problem: compute the threshold \(v^*(K)\) near \(K=5000\), where a straight line in \(\log K\) through the last computed points, which reaches \(1\) near \(K=1.8\times10^4\), and the power law \(1-v^*(K)/2\beta_K\approx1.65K^{-0.27}\), which never does, part ways; the cancellation there calls for a working precision of tens of thousands of bits. Is \(v^*(K)<2\beta_K\) for every \(K\)? If so, every \(f_K\) is a ferromagnetic probability density, and with no hypothesis their limit in law, with characteristic function \(e^{-\beta_\infty\theta^2}\prod_k(1-\theta^2/\gamma_k^2)\) over the zeros on the critical line, where \(\beta_\infty=\lim_K\beta_K\) is \(0\) under the Riemann Hypothesis, is a ferromagnetic law; it is the Pólya law exactly when the Riemann Hypothesis holds.

Earlier editions: Problem 13.5, page 580 (edition 1).

Source: ch/fm6.tex, label fm:prob:zc.

The cosine constant Omega, whether the good tilts form an interval, and improved bounds (open problem)

Volume I, Problem 20.6, page 639 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(p\) be the Pólya density \(\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and let \(\Omega\) be the infimum of the \(a_0\ge0\) such that the law with density proportional to \(\cosh(at)\,p(t)\) is ferromagnetic (symmetric with a moment generating function vanishing only on the imaginary axis) for every \(a\ge a_0\). By de Bruijn’s theorem, with no hypothesis, \(\Omega\le\frac12\); \(\Omega\le\frac38\) if the half plane \(\operatorname{Re}s>\frac78\) is free of zeros of \(\zeta\), as announced by OpenAI in October 2026 and not refereed; and the Riemann Hypothesis holds if and only if \(\Omega=0\). Open problem: is the set of \(a\ge0\) for which \(\cosh(at)\,p\) is ferromagnetic an interval? Improve these bounds by computation, as computations with the zeros have improved de Bruijn’s bound \(\frac12\) for the de Bruijn Newman constant \(\Lambda\) to \(\Lambda\le0.2\) (Platt and Trudgian). A first case is \(a=\sqrt{0.1}\), below both bounds: the law \(\cosh(\sqrt{0.1}\,t)\,p\) is at total variation distance \(1.094\times10^{-3}\) from \(p\), as close as the de Bruijn Newman flow at time \(\lambda=0.2\), the law with density proportional to \(e^{\lambda t^2/4}p(t)\) (distance \(1.097\times10^{-3}\)), which is ferromagnetic, and deciding it is a computation with the zeros of \(\frac12(\xi(s+a)+\xi(s - a))\), of the kind that gave the bound \(0.2\) for the flow.

Earlier editions: Problem 13.6, page 580 (edition 1).

Source: ch/fm6.tex, label fm:prob:omega.

The size bias principle for real shapes (open problem)

Volume I, Problem 20.7, page 640 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(Y=\sum_{n=1}^K\Gamma_n/\tau_n\) with \(0<\tau_1<\dots<\tau_K\) and independent gamma variables \(\Gamma_n\) of shapes \(t_n>0\) and unit scale, let \(0<q<1\), and let \(Y_q\) be \(Y\) size biased by \(Y^q\), that is \(\mathbb{E}g(Y_q)=\mathbb{E}[Y^qg(Y)]/\mathbb{E}Y^q\). The size bias principle proved in the book states that for integer shapes \(t_n\ge1\), \(Y_q+G\overset{d}{=}\sum_n\Gamma_n'/\tau_n\), where the \(\Gamma_n'\) are independent gamma variables of shapes \(t_n+q\) and unit scale and \(G\), independent of \(Y_q\), is a generalised gamma convolution without drift whose Thorin measure has a continuous distribution function that vanishes on \((0,\tau_1]\), increases by exactly \(q\) on each interval \([\tau_k,\tau_{k+1}]\) and is constant on \([\tau_K,\infty)\). Open problem: does it hold for all real shapes \(t_n>0\)? For \(K=1\) it is the gamma identity, with \(G=0\). Integer shapes enter the proof twice, through the divided difference formula for the density of \(Y\) and through a sign lemma, and together they show that on each gap \((\tau_k,\tau_{k+1})\) the phase, a continuous determination of \(-\frac1\pi\) times the argument of the boundary value at \(-r+i0\) of the Laplace transform of \(Y_q\), is nonincreasing and falls by exactly \(q\). Is this still true for real shapes? With it, the rest of the proof goes through, provided the estimates of the transform near the points \(-\tau_n\) and at infinity can be checked as well.

Earlier editions: Problem 13.7, page 580 (edition 1).

Source: ch/fm6.tex, label fm:prob:sbp.

Infinite divisibility of powers of the exponential of the Pólya variable below the HCM threshold (open problem)

Volume I, Problem 20.8, page 640 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\). The variable \(e^{\beta X}\) is not infinitely divisible for \(0<|\beta|<2\), is self decomposable and not a generalised gamma convolution (GGC) at \(|\beta|=2\), and has a hyperbolically completely monotone (HCM) density, so is a GGC, for \(|\beta|\ge5.5791422\); it has no HCM density for \(|\beta|\le5.5791395\). The HCM threshold, the least \(|\beta|\) for which \(e^{\beta X}\) has an HCM density, lies between \(5.5791395\) and \(\pi/0.563096=5.57914219\ldots\). Open problem: for \(2<|\beta|\) below the HCM threshold, is \(e^{\beta X}\) infinitely divisible, self decomposable or a GGC? Below the threshold it has no HCM density, which leaves the GGC property open, and self decomposability is established only at \(|\beta|=2\). Is the HCM threshold exactly \(\pi/v_m=5.5791396\), where \(v_m=0.5630962646\) is the point at which \(F(v)=\Phi(iv)\) has its local maximum on the imaginary axis? That requires \(h\ge0\) on the open strip \(0<v<v_m\), where \(h(x,v)=\operatorname{Im}\big(-\Phi'/\Phi\big)(x+iv)\) and \(h(0,v)\to0\) as \(v\uparrow v_m\), so that a certificate must replace the balls with positive lower bounds near \(iv_m\) by a local argument.

Earlier editions: Problem 13.8, page 580 (edition 1).

Source: ch/fm6.tex, label fm:prob:powers.

The depth profile of Knauf’s chain and the sums of partial quotients (open problem)

Volume I, Problem 20.9, page 640 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(Z_N(\beta)=\sum_{k=2^N}^{2^{N+1}-1}s(k)^{-\beta}\) for Stern’s diatomic sequence \(s\), and let \(R_N\) be the remainder in the Perron identity \(Z_N(\beta)=\zeta(\beta-1)/\zeta(\beta)+W_N(2-\beta)/\zeta(2)+R_N(\beta)\), with \(W_N(z)=\int_0^\infty w_N(x)x^{z-1}\,dx\), where \(w_N(x)=c_N(m)/\varphi(m)\) for \(m\le x<m+1\) and \(w_N=1\) on \((0,1)\), \(c_N(m)\) is the number of \(k\) with \(2^N\le k<2^{N+1}\) and \(s(k)=m\) (\(s\) Stern’s diatomic sequence), and \(\varphi\) is Euler’s function. Open problem: prove that \(R_N\to0\) locally uniformly on some open strip \(\sigma_1<\operatorname{Re}\beta<\sigma_2\) with \(1\le\sigma_1<\frac32<\sigma_2\le2\), or find the strips on which it holds. For \(m\ge2\), \(c_N(m)\) is the number of integers \(a\) prime to \(m\), \(1\le a<m\), for which \(a/m=[0;b_1,\dots,b_k]\) has \(b_1+\dots+b_k\le N+1\). The mean of \(b_1+\dots+b_k\) over \(1\le a\le m\) is asymptotic to \(\frac6{\pi^2}\log^2m\) (Yao and Knuth 1975), and over the \(a\) prime to \(m\) a quadratic fit in \(\log m\) to the computed means at \(m=10^5,\dots,10^8\) has leading coefficient \(0.6073\) (a computation). The consequence of the conjecture does not decide the Riemann Hypothesis.

Earlier editions: Problem 13.9, page 581 (edition 1).

Source: ch/fm6.tex, label fm:prob:profile.

The Lee and Yang threshold of the Farey chain (open problem)

Volume I, Problem 20.10, page 640 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For the Farey chain in a field, with partition function \(Z^{\rm F}_N(\beta,x)=\sum(\operatorname{tr}\varpi_1\cdots\varpi_N)^{-\beta}x^{n_B}\), with \(n_B\) the number of letters \(B\), over the words \(\varpi_1\cdots\varpi_N\) in \(A=\begin{pmatrix}1&0\\1&1\end{pmatrix}\) and \(B=A^{\mathsf T}\), open problem: prove that for every \(\beta>2\) all zeros of \(x\mapsto Z^{\rm F}_N(\beta,x)\) lie on the unit circle for all large \(N\), and that for every \(\beta\) with \(0<\beta<2\) some zeros leave the circle for all large \(N\). The first half follows if for every \(\beta>2\) the sum \(\sum(2/\operatorname{tr}\varpi_1\cdots\varpi_N)^\beta\) over the words other than \(A^N\) and \(B^N\) is below \(2\) for all large \(N\). The computed root \(\beta^*(N)\) of \(2^\beta Z^{\rm F}_N(\beta,1)=4\) falls from \(2.1056\) to \(2.0525\) for \(10\le N\le36\), and for \(2<\beta\le\beta^*(N)\) the circle lemma does not decide. At the computed thresholds the zeros that leave the circle first are at an angle \(\theta_N\) with \(N\theta_N\) between \(8.61\) and \(9.95\).

Earlier editions: Problem 13.10, page 581 (edition 1).

Source: ch/fm6.tex, label fm:prob:fareyly.

Volume I, Chapter 21. Cauchy mixtures, the reverse sine squared lemma and a two spin Lee and Yang form of RH

The Thorin density of a law of Xi type is a signed Cauchy mixture

Volume I, Theorem 21.1.2, page 644 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be a symmetric random variable with \(\mathbb E e^{aX^2}<\infty\) for some \(a>0\) whose moment generating function is \(L(s)=\mathbb E e^{sX}=e^{bs^2}\prod_r(1-s^2/r^2)\), with \(b\ge0\) and \(\sum_r|r|^{-2}<\infty\) (a law of Xi type), and write its zeros as \(r=\delta+i\gamma\), both members of each pair \(\pm r\) included. Let \(u_y(v)=\pi^{-1}\operatorname{Re}(L'/L)(y+iv)\) and \(P_a(x)=a/(\pi(a^2+x^2))\). For \(y>0\) with \(y\ne\delta\) for every zero, [u_y(v)=,y+r(y-),P{|y-|}(v-),] the sum taken over the pairs \(\{r,-\bar r\}\), absolutely and locally uniformly convergent in \(v\).

Source: ch/nc_cauchy.tex, label cm:thm:signed.

Positive Cauchy mixtures at every level are exactly the ferromagnetic laws

Volume I, Theorem 21.2.1, page 645 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(X\) of Xi type (symmetric, \(\mathbb E e^{aX^2}<\infty\) for some \(a>0\), with \(L(s)=\mathbb E e^{sX}=e^{bs^2}\prod_r(1-s^2/r^2)\) as above) the following are equivalent. (i) \(u_y(v)\ge0\) for every \(y>0\) and every \(v\) at which it is defined. (ii) There are \(\eta\ge0\) and a positive measure \(\nu\) with \(\int(1+x^2)^{-1}\nu(dx)<\infty\) such that \(u_y=\eta y+P_y*\nu\) for every \(y>0\). (iii) \(X\) is ferromagnetic, that is, \(L\) vanishes only on the imaginary axis. (iv) \(u\mapsto1/L(\sqrt u)\) is the Laplace transform of a generalised gamma convolution with drift \(b\) and Thorin measure \(\sum_km_k\delta_{c_k^2}\), the \(c_k\) being the zeros of the characteristic function with multiplicities \(m_k\). Then \(\eta=2b/\pi\) and \(\nu=\sum_km_k(\delta_{c_k}+\delta_{-c_k})\), both unique, and \(b=0\) if \(X\) has Gaussian moments of all orders.

Source: ch/nc_cauchy.tex, label cm:thm:ferro.

Strip classes of laws of Xi type and the Cauchy defect

Volume I, Proposition 21.2.4, page 646 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(\delta\ge0\) let \(\mathcal C_\delta\) be the set of laws of Xi type whose characteristic function has all its zeros in \(|\operatorname{Im}\theta|\le\delta\). For \(X\) of Xi type, \(u_y\ge0\) for every \(y>\delta\) if and only if \(X\in\mathcal C_\delta\); then for \(y>\delta\), \(u_y\) is a positive Cauchy mixture with widths in \([y-\delta,y+\delta]\), plus the drift. The classes are closed under independent sums (the Thorin densities add) and under the tilts \(m(x)=\mathbb E e^{xY}\), \(Y=\sqrt{2\mu}Z+\sum_j\lambda_j\varepsilon_j\) (\(Z\) standard Gaussian, \(\varepsilon_j\) signs), which map \(\mathcal C_\delta\) into \(\mathcal C_{\delta'}\) with \(\delta'=(\max(\delta^2-\operatorname{Var}Y,0))^{1/2}\) when the density decays faster than \(e^{-|x|^\kappa}\) for some \(\kappa>2\). They are not closed under mixtures.

Source: ch/nc_cauchy.tex, label cm:prop:strip.

The Ellis, Monroe and Newman class is not inside the ferromagnetic class

Volume I, Corollary 21.2.5, page 646 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Given \(\Lambda\ge0\) (Rodgers and Tao), the class of laws with density \(e^{-V}\), \(V\) even and \(V'\) convex on \([0,\infty)\) (and their weak limits), the class of Ellis, Monroe and Newman, is not contained in the class of ferromagnetic laws.

Source: ch/nc_cauchy.tex, label cm:cor:emn.

The reverse sine squared kernel lemma

Volume I, Lemma 21.3.1, page 646 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(a>0\) and real \(\gamma,\theta\), \(\int_{\mathbb R}e^{-a|x|}\cos(\gamma x)\cos(\theta x)\,dx=\pi\big[P_a(\theta-\gamma)+P_a(\theta+\gamma)\big]\) with \(P_a(x)=a/(\pi(a^2+x^2))\). For a zero \(\rho=\beta+i\gamma\) of \(\xi\) with \(\beta<\alpha\), \(\log(1+s/(\alpha-\rho))=\int_0^\infty(1-e^{-sx})e^{-(\alpha-\rho)x}\,dx/x\), and the canonical density of \(\rho\) and \(\bar\rho\) together is \(2e^{-(\alpha-\beta)x}\cos\gamma x\).

Source: ch/nc_cauchy.tex, label cm:lem:kernel.

The reverse sine squared lemma, from the zeros to the canonical measure

Volume I, Theorem 21.3.2, page 647 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha\ge1\), or more generally let \(\alpha>\frac12\) with every zero of \(\xi\) having \(\beta<\alpha\). The series \(K_\alpha(x)=\sum_{\gamma>0}e^{-(\alpha-\beta)|x|}\cos\gamma x\) converges in \(\mathcal S'(\mathbb R)\); its Fourier cosine transform is \(v_\alpha\); on \((0,\infty)\) it equals \(-\frac12xL_\alpha\), where \(L_\alpha\) is the Lévy measure of \(\log\xi(\alpha+s)/\xi(\alpha)=cs+\int(e^{-sx}+sx-1)L_\alpha(dx)\), with \(c=b_\alpha=\sum_\rho(\alpha-\rho)^{-1}\); and it is positive definite, with \(v_\alpha>0\).

Source: ch/nc_cauchy.tex, label cm:thm:reverse.

A single width Cauchy mixture at one basepoint at least one implies RH

Volume I, Theorem 21.3.3, page 647 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha\ge1\). If \(v_\alpha/\pi=P_{\alpha-1/2}*\nu\) for a positive measure \(\nu\), so that the Thorin density of \(1/\xi\) at the single basepoint \(\alpha\) is a positive Cauchy mixture of the single width \(\alpha-\frac12\), then RH holds, the Pólya law is ferromagnetic, and \(\nu=\sum_\gamma\delta_\gamma\). The converse holds at every \(\alpha>\frac12\).

Source: ch/nc_cauchy.tex, label cm:thm:single.

The undamped reverse sine squared series is Weil’s functional

Volume I, Theorem 21.3.4, page 647 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(h\) be even and real with \(e^{|x|/2}(|h|+|h''|)\) integrable, and \(\alpha\ge1\), \(w=\alpha-\frac12\). Group the zeros in pairs \(\{\rho,1-\bar\rho\}\) with real parts \(\frac12\pm\delta\), a zero on the line (\(\delta=0\)) forming a pair by itself, with the single term \(e^{(\frac12-\alpha)|x|}\) in the sum below. Then [e^{w|x|}K_,h:={}(e^{(+-)|x|}+e{(--)|x|})e^{w|x|}(x)h(x),dx=h(t_),] with \(t_\rho=(\rho-\frac12)/i\) and \(\widehat h(t)=\int h(x)e^{-itx}dx\); the series converges absolutely and does not depend on \(\alpha\). It is Weil’s functional; under RH it equals \(\frac12\sum_\gamma|\widehat g(\gamma)|^2\ge0\) for \(h=g*\tilde g\), and Weil’s criterion gives the converse.

Source: ch/nc_cauchy.tex, label cm:thm:weil.

Cosine tilted laws are ferromagnetic when the tilt exceeds the strip

Volume I, Theorem 21.4.1, page 648 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(X\) of Xi type and \(w>0\), the law \(\cosh(wx)\,P(dx)/\mathbb E\cosh(wX)\) is ferromagnetic whenever every zero of \(L\) has \(|\operatorname{Re}r|<w\). In particular the cosine tilt \(p_w=\cosh(wx)\,p/\mathbb E\cosh(wX)\) of the Pólya law \(p\) is ferromagnetic for every \(w\ge\frac12\), with no hypothesis.

Source: ch/nc_cauchy.tex, label cm:thm:tilt.

The two spin Lee and Yang property is a zero free half plane

Volume I, Theorem 21.5.1, page 650 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be of Xi type and \(w>0\), let \(\varepsilon\) be a fair sign independent of \(X\), and let \(Z_w(h_1,h_2)=\mathbb E\sum_{\varepsilon=\pm1}e^{wX\varepsilon+h_1X+h_2\varepsilon}=e^{h_2}L(h_1+w)+e^{-h_2}L(h_1-w)\). The pair has the Lee and Yang property if \(Z_w\ne0\) whenever \(\operatorname{Re}h_1>0\) and \(\operatorname{Re}h_2>0\). (a) If no zero of \(L\) has \(\operatorname{Re}r>w\), the pair has the Lee and Yang property. (b) If the pair has the Lee and Yang property, no zero of \(L\) has \(\operatorname{Re}r>w\); so the pair has the Lee and Yang property at the coupling \(w\) exactly when \(X\in\mathcal C_w\). (c) For \(w_0\ge0\), the pair has the Lee and Yang property for every \(w>w_0\) if and only if no zero has \(\operatorname{Re}r>w_0\).

Source: ch/nc_cauchy.tex, label cm:thm:pair.

The Pólya spin coupled to an Ising spin, unconditional at one half and equivalent to RH

Volume I, Corollary 21.5.2, page 650 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. The Pólya spin coupled to one Ising spin with coupling \(w\) is a Lee and Yang ferromagnet for every \(w\ge\frac12\), with no hypothesis. For \(w_0\in[0,\frac12)\) it is one for every \(w>w_0\) if and only if \(\zeta\) has no zero with \(\operatorname{Re}s>\frac12+w_0\). RH holds if and only if it is one for every \(w>0\).

Source: ch/nc_cauchy.tex, label cm:cor:pair.

Truncated Euler products sit exactly at the coupling one half

Volume I, Proposition 21.5.3, page 650 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(N\ge2\) let \(E_N(s)=\prod_{p\le N}(1-p^{-1/2-s})(1-p^{-1/2+s})\), the truncated Euler product of \(\zeta(\frac12+s)^{-1}\zeta(\frac12-s)^{-1}\). Its zeros are \(\pm\frac12+2\pi ik/\log p\), and the two spin system built from \(E_N\) in place of \(L\) has the Lee and Yang property exactly for \(w\ge\frac12\).

Source: ch/nc_cauchy.tex, label cm:prop:euler.

The sum of two three point spins, an exact law whose tilt threshold lies inside the strip

Volume I, Proposition 21.6.1, page 651 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0<q<\frac14\), \(\cosh a=\frac{1-2q}{2q}\) (so \(a>0\)), and \(X=Y_1+Y_2\) with \(Y_1,Y_2\) independent and \(P(Y_i=\pm1)=q\), \(P(Y_i=0)=1-2q\). Then \(X\) is of Xi type, and its cosine tilt is ferromagnetic exactly for \(w\ge w^*\), where \(\cosh2w^*=\cosh^2a\). Moreover \(\sinh w^*=\sinh a/\sqrt2\), \(a/\sqrt2<w^*<a\), and \(w^*/a\to\cos\frac\pi4\) as \(q\uparrow\frac14\). Since \(L\) has its zeros at real parts \(\pm a\), \(X\) lies in \(\mathcal C_a\) and not in \(\mathcal C_w\) for \(w<a\). So for every \(w\in[w^*,a)\) the cosine tilt of \(X\) at \(w\) is ferromagnetic while \(X\notin\mathcal C_w\). For \(w>0\) and \(w<w^*\) the tilt has no zero on the imaginary axis at all.

Source: ch/nc_cauchy.tex, label cm:prop:twospin.

A pair planted in Xi has slice threshold 0.2951 below its width 0.3

Volume I, Proposition 21.6.2, page 652 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(F(z)=\Xi(z)\,\dfrac{(z^2-c^2)(z^2-\bar c^2)}{(z^2-\gamma_k^2)(z^2-\gamma_{k+1}^2)}\) with \(c=\frac12(\gamma_k+\gamma_{k+1})+i\eta\), an even entire function, real on the real axis, with strip half width \(\eta\). For \(k=1\) and \(\eta=0.3\), \(F(z+iw)+F(z-iw)\) has only real zeros in \([1,45.3]\times[-2.2,2.2]\) at \(w=0.299<\eta\). The nonreal zero born from \(c\) reaches the real axis at \(w^*\approx0.2951\), its imaginary part behaving like \((w^*-w)^{1/2}\). The inverse Fourier transform of \(F\) is positive at \(x=0,0.25,\dots,1.25\), with ratio to the Pólya density between \(0.989\) and \(1.111\).

Source: ch/nc_cauchy.tex, label cm:prop:counter.

The phase criterion for the real zeros of the cosine tilt

Volume I, Proposition 21.6.3, page 652 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be real entire and even, and let \(\Theta_w(x)\) be a continuous argument of \(F(x+iw)\) on an interval \(I\) where \(F(x+iw)\ne0\). The real zeros of the tilt \(F(z+iw)+F(z-iw)\) in \(I\) are the points where \(\Theta_w\in\frac\pi2+\pi\mathbb Z\). A real zero \(a\) of \(F\) contributes \(\arg(x-a+iw)\) to \(\Theta_w\), which decreases by \(\pi\) across \(a\); a zero \(c=m+i\eta\) contributes \(\arg(x-m+i(w-\eta))\), which increases by \(\pi\) across \(m\) when \(w<\eta\) and decreases when \(w>\eta\), while \(\bar c\) contributes a decrease. So for \(w<\eta\) the planted pair produces a local rise of \(\Theta_w\) near \(m\), of height tending to \(\pi\) as \(w\uparrow\eta\), and two nonreal zeros of the tilt become real exactly when that rise first reaches a level \(\frac\pi2+k\pi\).

Source: ch/nc_cauchy.tex, label cm:prop:phase.

The gap law eta minus w* equals phi squared eta cubed over two for a simple pair

Volume I, Theorem 21.7.1, page 654 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be real entire and even with a planted pair \(c=m+i\eta\), \(\bar c\), and write \(G=F/((z-c)(z-\bar c))\), real on the real axis with \(G(m)\ne0\). Let \(r>0\) be such that \(G\) has no zero in the disc \(|z-m|\le2r\). For \(\eta\) small enough the tilt \(F(z+iw)+F(z-iw)\) has exactly two zeros in \(|z-m|<r\) for every \(w\in(0,\eta)\), and they are real if and only if \(\Phi_w(u)=\arctan\dfrac{2wu}{u^2+\eta^2-w^2}+\psi_w(m+u)\), with \(\psi_w(x)=\arg G(x+iw)\) a continuous branch near \(m\), takes a value in \(\frac\pi2+\pi\mathbb Z\) for some \(u\in(-r,r)\). If \(\phi=(G'/G)(m)\ne0\), the slice threshold \(w^*\) satisfies [-w^*=,^2,3(1+O())(),] and if \(\phi=0\), then \(\eta-w^*=O(\eta^7)\).

Source: ch/nc_cauchy.tex, label cm:thm:exact.

Slice thresholds for several pairs and pairs of higher multiplicity

Volume I, Theorem 21.8.1, page 658 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F=G_0\prod_{j=1}^K\big((z^2-c_j^2)(z^2-\bar c_j^2)\big)^{n_j}\) with distinct heights \(m_j>0\), multiplicities \(n_j\ge1\), \(c_j=m_j+i\eta_j\), \(\eta_j=t_j\eta\), \(t_j\in(0,1]\), and \(G_0\) real entire, real on the real axis, with \(G_0(m_j)\ne0\). Let \(G_j=F/((z-c_j)(z-\bar c_j))^{n_j}\) and let \(G_j^0\) be its limit as \(\eta\to0\) near \(m_j\), with field \(\phi_j=\dfrac{(G_j^0)'}{G_j^0}(m_j)=\dfrac{G_0'}{G_0}(m_j)+\sum_{l\ne j}\dfrac{2n_l}{m_j-m_l}+\sum_{l}\dfrac{2n_l}{m_j+m_l}\). The local slice threshold \(w_j^*\) is the least \(w\) for which all \(2n_j\) zeros of the tilt born from \(c_j\) and \(\bar c_j\) are real. Let \(r>0\) be such that \(G_j^0\) has no zero in \(|z-m_j|\le2r\) for every \(j\). For \(\eta\) small enough the tilt \(F(z+iw)+F(z-iw)\) has exactly \(2n_j\) zeros in \(|z-m_j|<r\) for every \(w\in[0,\eta]\); they are all real for \(w\ge\eta_j\), and (i) for \(n_j\) odd, \(\eta_j-w_j^*=\dfrac{\phi_j^2}{2n_j^2}\,\eta_j^3\big(1+O(\eta)\big)\) when \(\phi_j\ne0\), and \(O(\eta^7)\) when \(\phi_j=0\); (ii) for \(n_j\) even, \(w_j^*=\eta_j\cos\dfrac{\pi}{2n_j}+O(\eta^2)\), and more precisely \(w_j^*=\eta_j\sin\Big(\dfrac{(n_j-1)\pi}{2n_j}+\dfrac{|\psi_j|}{n_j}\Big)+O(\eta^3)\) with \(\psi_j=\operatorname{dist}(\arg G_j(m_j+iw_j^*),\pi\mathbb Z)=O(\eta)\). The configuration is seen by the one variable test, in the sense that every zero born from a planted zero is real, exactly for \(w\ge\max_jw_j^*\).

Source: ch/nc_cauchy.tex, label cm:thm:several.

Ferromagnetic coupling never widens the strip, for spins of exponential type

Volume I, Theorem 21.9.1, page 660 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Rests on Proposition 2.2 of Lieb and Sokal (1981), as restated by Kozitsky, not read in the original.

Statement. Let \(\chi_1,\dots,\chi_N\) be symmetric probability measures on \(\mathbb R\) whose Laplace transforms \(\varphi_i(x)=\int e^{x\sigma}\chi_i(d\sigma)\) are entire of exponential type, for instance measures of bounded support, and suppose \(\varphi_i(x)\ne0\) whenever \(\operatorname{Re}x>\delta_i\). Let \(J_{ij}\ge0\). Then the partition function \(Z(h_1,\dots,h_N)=\int\exp\big(\tfrac12\sum_{i,j}J_{ij}\sigma_i\sigma_j+\sum_ih_i\sigma_i\big)\prod_i\chi_i(d\sigma_i)\) does not vanish when \(\operatorname{Re}h_i>\delta_i\) for every \(i\). It does not vanish when \(\operatorname{Re}h_i<-\delta_i\) for every \(i\) either, and in a common field all its zeros lie in \(|\operatorname{Re}h|\le\max_i\delta_i\).

Source: ch/nc_cauchy.tex, label cm:thm:stripLY.

Ferromagnetic coupling never widens the strip, for laws of Xi type with Gaussian moments

Volume I, Theorem 21.9.2, page 661 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Rests on Proposition 2.2 of Lieb and Sokal (1981), as restated by Kozitsky, not read in the original.

Statement. Let \(\chi_1,\dots,\chi_N\) be laws of Xi type with Gaussian moments of all orders, whose Laplace transforms satisfy \(\varphi_i(x)\ne0\) for \(\operatorname{Re}x>\delta_i\), and let \(J_{ij}\ge0\). Then \(Z(h_1,\dots,h_N)=\int\exp\big(\tfrac12\sum_{i,j}J_{ij}\sigma_i\sigma_j+\sum_ih_i\sigma_i\big)\prod_i\chi_i(d\sigma_i)\ne0\) whenever \(\operatorname{Re}h_i>\delta_i\) for every \(i\). In particular \(N\) Pólya spins in any ferromagnetic coupling have all the zeros of their partition function in a common field in the strip \(|\operatorname{Re}h|\le\sup_\rho|\beta-\frac12|\), with no hypothesis.

Source: ch/nc_cauchy.tex, label cm:thm:stripXi.

The bare Pólya Ising pair has its field zeros inside the strip

Volume I, Proposition 21.9.3, page 662 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be of Xi type in \(\mathcal C_\delta\) and couple it to a fair Ising spin with strength \(w\ge0\). In a common field \(h\) the partition function \(e^{h}L(h+w)+e^{-h}L(h-w)\) has no zero with \(|\operatorname{Re}h|>\delta\), and none off the imaginary axis when every zero of \(L\) has \(|\operatorname{Re}r|<w\).

Source: ch/nc_cauchy.tex, label cm:prop:strippair.

A field strip criterion for zeros of the Laplace transform of an Ising magnetisation

Volume I, Proposition 21.9.4, page 663 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\sigma_1,\dots,\sigma_n\in\{-1,1\}\) be Ising spins with couplings \(J_{kl}\ge0\), real fields \(b_k\) and weights \(a_k>0\), and let \(M=\sum_ka_k\sigma_k\) under the probability measure proportional to \(\exp\big(\sum_{k<l}J_{kl}\sigma_k\sigma_l+\sum_kb_k\sigma_k\big)\). Then every zero \(w\in\mathbb{C}\) of \(w\mapsto\mathbb{E}e^{wM}\) satisfies \(\min_k(-b_k/a_k)\le\operatorname{Re}w\le\max_k(-b_k/a_k)\), in particular \(|\operatorname{Re}w|\le\max_k|b_k|/a_k\). Consequently, if magnetisations \(M_N\) of such systems with \(\max_k|b_k|/a_k\le\delta\) have characteristic functions \(t\mapsto\mathbb{E}e^{itM_N}\) converging locally uniformly on \(\mathbb{C}\) to \(\Xi(t)/\Xi(0)\) (for instance if \(M_N\) converges in law and \(\sup_N\mathbb{E}e^{R|M_N|}<\infty\) for every \(R>0\)), where \(\Xi(t)=\xi(\frac12+it)\) and \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) is the completed Riemann zeta function, then every zero of \(\xi\) has real part at most \(\frac12+\delta\).

Source: ch/nc_cauchy.tex, label cm:prop:fieldstrip.

The two spin Lee and Yang property at every coupling is monotonicity of the modulus of xi

Volume I, Theorem 21.11.1, page 665 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. The Pólya spin coupled to one Ising spin is a Lee and Yang ferromagnet for every coupling \(w>0\) if and only if \(\sigma\mapsto|\xi(\sigma+it)|\) is nondecreasing on \([\frac12,\infty)\) for every \(t\), if and only if RH holds.

Source: ch/nc_cauchy.tex, label cm:thm:mono.

A Frobenius spin is ferromagnetic exactly when the Hasse bound holds

Volume I, Proposition 21.11.2, page 665 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For an elliptic curve over \(\mathbb F_q\) with \(\#E(\mathbb F_q)=q+1-a\), put \(L_E(x)=q^x\big(1-aq^{-1/2-x}+q^{-2x}\big)=2\cosh(x\log q)-a/\sqrt q\), where \(x=s-\frac12\). For \(a\le0\), \(L_E\) is the moment generating function of the three point measure with masses \(1\), \(|a|/\sqrt q\), \(1\) at \(-\log q\), \(0\), \(\log q\); for \(a>0\) the imaginary field shift \(x\mapsto x+i\pi/\log q\) turns it into minus such a function and changes no real part. Its zeros are imaginary if and only if \(|a|\le2\sqrt q\), and otherwise lie at real parts \(\pm\delta\), \(\cosh(\delta\log q)=|a|/(2\sqrt q)\). Hence the Hasse bound holds exactly when this Frobenius spin is ferromagnetic, when its Thorin density is a single width Cauchy mixture at every level, and when the pair formed with an Ising spin is a Lee and Yang ferromagnet at every coupling.

Source: ch/nc_cauchy.tex, label cm:prop:frob.

Explicit formula with a finite prime side for the Cauchy times Fejér window

Volume I, Theorem 21.12.1, page 667 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(\delta>0\) and \(L>0\) let \(G(x)=e^{-\delta|x|}(1-|x|/L)_+\) with Fourier transform \(H\), and for a centre \(w\) let \(g_w(x)=G(x)\cos wx\), with transform \(h_w(r)=\frac12[H(r-w)+H(r+w)]\). The flow \(O_{\delta,L}(w)\) is the right side of the explicit formula for \(g_w\) (the archimedean and pole terms minus \(2\sum_{n\le e^L}\Lambda(n)n^{-1/2-\delta}(1-\log n/L)\cos(w\log n)\), with \(\Lambda\) the von Mangoldt function). Then, for every \(\delta>0\) and \(L>0\), unconditionally, \(O_{\delta,L}(w)=\sum_\rho h_w(\gamma_\rho)\) over the nontrivial zeros \(\rho\) of \(\zeta\) with ordinates \(\gamma_\rho\), the series converging absolutely, and the prime side contains only the prime powers \(n\le e^L\).

Source: ch/nc_cauchy.tex, label cm:thm:flow.

Height test at level one excluding off line and multiple zeros in an ordinate window

Volume I, Theorem 21.12.3, page 668 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(G\) be even, \(G(0)=1\), \(G\ge0\), supported in \([-L,L]\), with \(\widehat{g_w}\) in the class of test functions for which the explicit formula of the book holds, where \(g_w(x)=G(x)\cos wx\), and such that \(G(x)\cosh(\eta x)\) is positive definite for every \(\eta\in[0,\frac12)\). Examples are \(e^{-|x|/2}(1-|x|/L)_+\) and \(\operatorname{sech}(x/2)(1-|x|/L)_+\). Let \(O(w)\) be the flow of \(G\) (the right side of the explicit formula for \(g_w\), containing the prime powers \(n\le e^L\)) and \(H_0(0)=\int G\). If \(\sup_{w\in I}O(w)<2H_0(0)\) on an interval \(I\), then every zero of \(\zeta\) with ordinate in \(I\) is simple and on the critical line. The book’s computations with this test (maximum of \(O/H_0(0)\) equal to \(1.5085\) over the ordinates in \([5,160]\) for \(G=e^{-|x|/2}(1-|x|/14.51)_+\), in double precision on a grid of step \(0.01\)) are not part of this theorem.

Source: ch/nc_cauchy.tex, label cm:thm:height.

Ceiling of the level one height test

Volume I, Theorem 21.12.4, page 669 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For any window \(G\) as in the level one height test (even, \(G(0)=1\), \(G\ge0\), compactly supported, with \(G(x)\cosh\eta x\) positive definite for every \(\eta\in[0,\frac12)\)) one has \(G\le\operatorname{sech}(x/2)\) and \(H_0(0)=\int G\le2\pi\), and the hypothesis \(\sup_IO<2H_0(0)\) on a window \(I=[T,T+W]\) with \(W\ge W_0(G)\) can hold only if \(\log(T/2\pi)<4\pi\), that is \(T<2\pi e^{4\pi}\approx1.8\times10^6\). If it is known a priori that all zeros satisfy \(|\beta-\frac12|\le\eta_0\), the same argument with \(G\cosh\eta_0x\) gives \(H_0(0)\le\pi/\eta_0\) and the necessary condition \(\eta_0<2\pi/\log(T/2\pi)\).

Source: ch/nc_cauchy.tex, label cm:thm:ceiling.

Interference of an off line pair below level one is exponentially large in L

Volume I, Proposition 21.12.5, page 669 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\delta<\frac12\) and consider the Cauchy times Fejér window with damping \(\delta\) and length \(L\). A pair of zeros with \(\eta=|\beta-\frac12|>\delta\) at distance \(d\) from the centre contributes about \(2e^{(\eta-\delta)L}/(L((\eta-\delta)^2+d^2))\) to the flow, with a sign that depends on \(dL\). The counting function \(N(T)\) does not see \(\eta\), and without the Riemann Hypothesis the only a priori limit is \(\eta<\frac12\), so a bound on the interference from \(N(T)\) alone is at least of order \(e^{(1/2-\delta)L}/(Ld^2)\), against \(H(0)\le L\). This is a size statement; the Davenport and Heilbronn function, which has a functional equation of the same shape, realises it (computed: for the pair at \(240.40\), at distance \(5.03\) and \(\delta=0.05\), the contributions are \(-0.02\), \(+0.035\), \(-0.155\) and \(-1.93\) in units of \(H(0)\) for \(L=10,14.5,20,30\)).

Source: ch/nc_cauchy.tex, label cm:prop:nt.

Open problem on the one variable tilt, intervals and the constant kappa

Volume I, Problem 21.1, page 671 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For the one variable tilt, is the set of \(w\) for which \(\cosh(wx)\,P(dx)\) is ferromagnetic always an interval? Extend the gap law to clusters whose width is comparable to the spacing of the neighbouring zeros. Is there a constant \(\kappa>0\) such that ferromagnetism of \(\cosh(wx)\,P(dx)\) for every \(w>w_0\) implies that no zero has \(|\operatorname{Re}r|>w_0/\kappa\)? The sum of two three point spins gives \(\kappa\le\cos\frac\pi4\).

Source: ch/nc_cauchy.tex, label cm:prob:interval.

Open problem, a Lee and Yang system built from the Euler product

Volume I, Problem 21.2, page 671 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Find a coupled system of Lee and Yang spins built from the Euler product, not from \(\xi\), with coupling below \(\frac12\) and a partition function that contains \(L(h_1\pm w)\); its Lee and Yang property would be a zero free half plane.

Source: ch/nc_cauchy.tex, label cm:prob:euler.

Open problem, strict shrinking of the strip under ferromagnetic coupling

Volume I, Problem 21.3, page 671 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Ferromagnetic couplings never widen the strip of a law of Xi type with Gaussian moments of all orders. Do they shrink it strictly, by at least an amount comparable to the coupling?

Source: ch/nc_cauchy.tex, label cm:prob:strip.

Volume I, Chapter 22. Pólya and the zeros of entire functions

Five equivalent forms of the Riemann Hypothesis in the Thorin variable: Laguerre Pólya class, Pólya Schur multipliers, Jensen polynomials and GGC

Volume I, Proposition 22.4.1, page 678 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, \(F(s)=\xi(\frac12+\sqrt s)/\xi(\frac12)=\sum_{k\ge0}a_ks^k\) (an entire function of order one half), \(g_k=k!\,a_k\), and \(J_d(x)=\sum_{k=0}^d\binom dk g_kx^k\) the Jensen polynomial of degree \(d\). The following are equivalent. (i) The Riemann Hypothesis holds. (ii) \(F\) belongs to the Laguerre Pólya class of type I, that is, \(F\) is a locally uniform limit of polynomials with only negative zeros. (iii) \((g_k)\) is a Pólya Schur multiplier sequence, that is, \(\sum_kg_kc_kx^k\) has only real zeros whenever the polynomial \(\sum_kc_kx^k\) has only real zeros. (iv) For every \(d\) the Jensen polynomial \(J_d\) has only real, negative zeros. (v) The reciprocal Xi law, the probability law on \([0,\infty)\) with Laplace transform \(1/F(s)\), is a generalized gamma convolution (GGC). The equivalence of (ii), (iii) and (iv) is the theorem of Pólya and Schur (1914) applied to \(F(s)=\sum_kg_ks^k/k!\).

Earlier editions: Proposition 16.4.1, page 644 (edition 1).

Source: ch/polya3.tex, label pol:prop:lp.

Open problem: an exact arithmetic proof that the reciprocal Xi density is log concave, and then Pólya frequency of every order n

Volume I, Problem 22.1, page 680 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function, \(Y\) have the reciprocal Xi law with Laplace transform \(\xi(\frac12)/\xi(\frac12+\sqrt s)\), \(Z\) be a standard normal variable independent of \(Y\), and \(f_X\) the density of \(X=Z\sqrt{2Y}\). Prove unconditionally, in exact or interval arithmetic, that \(f_X\) is log concave, that is, a Pólya frequency function of order two (PF\(_2\)); a floating point certificate is given in the book. Then prove that \(f_X\) is PF\(_n\) for every \(n\ge3\), that is, \(\det[f_X(x_i-y_j)]_{i,j\le n}\ge0\) whenever \(x_1<\dots<x_n\) and \(y_1<\dots<y_n\).

Earlier editions: Problem 16.1, page 646 (edition 1).

Source: ch/polya3.tex, label pol:op:pf2.

Even part basepoint laws: GGC laws whose Thorin atoms are the zeros of the real part of xi on a vertical line

Volume I, Corollary 22.6.2, page 681 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function and, for \(\alpha>\frac12\), \(A_\alpha(w)=\frac12(\xi(\alpha+w)+\xi(\alpha-w))\), the even part in \(w\) of \(w\mapsto\xi(\alpha+w)\), so that \(A_\alpha(it)=\operatorname{Re}\xi(\alpha+it)\) for real \(t\) and \(A_\alpha(0)=\xi(\alpha)>0\). Whenever \(A_\alpha\) has only purely imaginary zeros \(\pm i\tau_k\) (\(\tau_k>0\), with multiplicity), the function \(\varphi^P_\alpha(s)=\xi(\alpha)/A_\alpha(\sqrt s)\) is the Laplace transform of a generalized gamma convolution \(\mu^P_\alpha\) with Thorin measure \(\sum_k\delta_{\tau_k^2}\), so that \(\mu^P_\alpha\) is the law of \(\sum_kE_k/\tau_k^2\) with independent standard exponential variables \(E_k\). By the theorems of Pólya, de Bruijn and Lagarias, this hypothesis holds for every \(\alpha\ge1\) unconditionally and for every \(\alpha>\frac12\) if the Riemann Hypothesis holds. Conversely, if it holds along a sequence \(\alpha_n\downarrow\frac12\), then the Riemann Hypothesis holds.

Earlier editions: Corollary 16.6.2, page 647 (edition 1).

Source: ch/polya4.tex, label pol:cor:evenpart.

Half integer crossings of the clock: zeros of the real part of xi on a vertical line and the even part law as a midpoint rule

Volume I, Proposition 22.6.3, page 681 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, and let \(\alpha>\frac12\) be such that \(\xi\) has no zero in \(\operatorname{Re}s\ge\alpha\) (this holds for every \(\alpha\) larger than \(\sigma_{\max}=\sup\operatorname{Re}\rho\) over the zeros \(\rho\) of \(\xi\), and unconditionally for every \(\alpha\ge1\)). Let \(U_\alpha(\tau)=\frac1\pi\Delta_0^\tau\arg\xi(\alpha+it)\) be the clock, the variation of the argument of \(\xi\) along the segment from \(\alpha\) to \(\alpha+i\tau\) divided by \(\pi\), so that \(U_\alpha'(\tau)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\tau)\), and let \(\tau_\alpha\) be its inverse. Then \(U_\alpha\) is continuous and strictly increasing with \(U_\alpha(0)=0\), and the positive zeros of \(t\mapsto\operatorname{Re}\xi(\alpha+it)\) are exactly \(\tau_k=\tau_\alpha(k-\frac12)\), \(k=1,2,\dots\), all simple. Consequently the function \(\xi(\alpha)/A_\alpha(\sqrt s)\), with \(A_\alpha(w)=\frac12(\xi(\alpha+w)+\xi(\alpha-w))\), is the Laplace transform of a generalized gamma convolution \(\mu^P_\alpha\), the even part law, whose Thorin distribution is the rounding of the Thorin distribution \(U_\alpha\) of the basepoint law \(\mu^{(\alpha)}\) (the law with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt s)\)): \(U^P_\alpha((0,T^2])=\lfloor U_\alpha(T)+\frac12\rfloor\) for \(T>0\). In law, \(\mu^P_\alpha=\mathcal{L}\big(\sum_{k\ge1}E_k/\tau_\alpha(k-\frac12)^2\big)\) with independent standard exponential variables \(E_k\), while \(\mu^{(\alpha)}=\mathcal{L}\big(\int_0^\infty dg_v/\tau_\alpha(v)^2\big)\) with \(g\) a standard gamma subordinator: the even part law is the midpoint rule for the gamma clock integral.

Earlier editions: Proposition 16.6.3, page 647 (edition 1).

Source: ch/polya4.tex, label pol:prop:round.

Bounded counting error for the zeros of the real part of xi on a line to the right of one (Polson note)

Volume I, Proposition 22.6.5, page 682 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(\alpha>1\), let \(0<\tau_1<\tau_2<\cdots\) be the positive zeros of \(t\mapsto\operatorname{Re}\xi(\alpha+it)\) (all real, simple, and the half integer crossings of the clock \(U_\alpha(t)=\pi^{-1}\arg\xi(\alpha+it)\)), and \(N_\alpha(T)=\#\{k:\tau_k\le T\}\). Let \(\theta_\infty(T)\) be the continuous argument of \(\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\) along \(s=\alpha+it\), \(0\le t\le T\), starting from \(0\). Then \(-1-\log\zeta(\alpha)/\pi<N_\alpha(T)-\theta_\infty(T)/\pi-\frac12\le\log\zeta(\alpha)/\pi\), so the counting error is bounded by \(1+\log\zeta(\alpha)/\pi\). At \(\alpha=\frac32\) the bound is \(1.30566\), and for \(T\le400\) the computed supremum of the deviation is \(1.1845\).

Source: ch/polya4.tex, label pol:prop:count.

Gap sandwich for the zeros of the real part of xi on a line to the right of one (Polson note)

Volume I, Theorem 22.6.6, page 683 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(\alpha>1\), and let \(\tau_k<\tau_{k+1}\) be consecutive positive zeros of \(t\mapsto\operatorname{Re}\xi(\alpha+it)\). Let \(G_\alpha(y)=\operatorname{Re}\frac1{\alpha-1+iy}-\frac12\log\pi+\frac12\operatorname{Re}\psi(1+\frac{\alpha+iy}2)\) (the archimedean part of \(\operatorname{Re}\xi'/\xi(\alpha+iy)\), with \(\psi=\Gamma'/\Gamma\)), and let \(G^+\), \(G^-\) be the maximum and the minimum of \(G_\alpha\) over \([\tau_k,\tau_{k+1}]\). Then \(\pi/(G^+-\frac{\zeta'}\zeta(\alpha))\le\tau_{k+1}-\tau_k\), and if \(G^-+\frac{\zeta'}\zeta(\alpha)>0\) then also \(\tau_{k+1}-\tau_k\le\pi/(G^-+\frac{\zeta'}\zeta(\alpha))\). Consequently the normalised gaps \(\delta_k=(\tau_{k+1}-\tau_k)\frac1{2\pi}\log\frac{\tau_k}{2\pi}\) satisfy \(|\delta_k-1|\le\frac{2|\zeta'/\zeta(\alpha)|}{\log(\tau_k/2\pi)}(1+o(1))\). At \(\alpha=\frac32\) (\(\zeta'/\zeta(\frac32)=-1.50524\)) the lower bound holds for all \(201\) gaps below height \(400\) (computed).

Source: ch/polya4.tex, label pol:thm:gaps.

Is the set of h with real zeros of Re xi(1/2+h+it) an interval (open problem)

Volume I, Problem 22.2, page 684 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\). Is the set of \(h\ge0\) for which all zeros of \(t\mapsto\operatorname{Re}\xi(\frac12+h+it)\) are real an interval? With \(\Delta=\sup|\beta-\frac12|\le\frac12\) over the zeros \(\beta+i\gamma\) of \(\zeta\), by the strip theorem of de Bruijn the set contains \([\Delta,\infty)\), and the Riemann Hypothesis is equivalent to its being \([0,\infty)\). Whether nonreal zeros persist for every \(h<\Delta\) when \(\Delta>0\) is open: the strip bound only locates zeros and does not produce nonreal ones.

Source: ch/polya4.tex, label pol:prob:mono.

Zeros of Pólya’s Xi star at half Gram points

Volume I, Proposition 22.7.2, page 685 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_\nu\) be the modified Bessel function of the second kind and \(\Xi^*(t)=4\pi^2\big[K_{\frac94+\frac{it}2}(2\pi)+K_{\frac94-\frac{it}2}(2\pi)\big]=8\pi^2\operatorname{Re}K_{\frac94+\frac{it}2}(2\pi)\), Pólya’s approximation of Riemann’s Xi function (it has only real zeros), let \(t_k^*\) be its positive zeros in increasing order, and let \(\vartheta(t)=\operatorname{Im}\log\Gamma(\frac14+\frac{it}2)-\frac t2\log\pi\) be the Riemann Siegel theta function. Then \(\vartheta(t_k^*)/\pi=k-\frac32-\frac{2\pi^2-3}{\pi\,t_k^*}+O\big((t_k^*)^{-2}\big)\) as \(k\to\infty\). Hence, with \(N^*(T)=\#\{k:t_k^*\le T\}\), one has \(N^*(T)=\lfloor\vartheta(T)/\pi+\frac32\rfloor\) for all large \(T\) outside the intervals \([t_k^*,t_k^*+O(1/(t_k^*\log t_k^*)))\), on which \(N^*(T)\) is one larger; in particular \(N^*(T)-\vartheta(T)/\pi\) is bounded, whereas the number \(N(T)\) of zeros of the zeta function with ordinate in \((0,T]\) satisfies \(N(T)=\vartheta(T)/\pi+1+S(T)\) with \(S(T)=\pi^{-1}\arg\zeta(\frac12+iT)\).

Earlier editions: Proposition 16.7.2, page 649 (edition 1).

Source: ch/polya4.tex, label pol:prop:gram.

Calibration of Thorin recovery on the line to the right of one (open problem)

Volume I, Problem 22.3, page 685 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(h>\frac12\) the atoms \(\tau_k^2\) of the Thorin measure of the law with Laplace transform \(\xi(\frac12+h)/A(\sqrt s)\), \(A(w)=\frac12(\xi(\frac12+h+w)+\xi(\frac12+h-w))\), with \(\tau_k\) the positive zeros of \(t\mapsto\operatorname{Re}\xi(\frac12+h+it)\), are known from the phase, so this family is a test case on which a method that recovers a Thorin measure from its transform at a fixed level \(\eta_0\) has a known answer. Problem: run deconvolution methods from \(\eta_0\) (say \(\frac12\)) down to \(\eta=0.05\) at \(h=1\) and report the total variation error as a function of \(\eta_0\) and \(h\).

Source: ch/polya4.tex, label pol:prob:calib.

Exact Gram law for the zeros of the two term kernel

Volume I, Theorem 22.7.3, page 686 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(s=\frac12+it\), \(G(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\), so that \(\arg G(\frac12+it)=\vartheta(t)+\pi\) with \(\vartheta\) the Riemann Siegel theta function, \(K_\nu\) the modified Bessel function of the second kind, and \(F(t)=2\pi^2K_{\frac94+\frac{it}2}(2\pi)-3\pi K_{\frac54+\frac{it}2}(2\pi)\). Then \(\frac12\operatorname{Re}F(t)\) is the characteristic function, at \(2t\), of the density proportional to \((2\pi^2\cosh9u-3\pi\cosh5u)e^{-2\pi\cosh4u}\). Let \(g_k\) be the Gram points, \(\vartheta(g_k)=k\pi\), so \(g_{-1}=9.667\) and \(g_0=17.846\). Then \(F(t)=\frac12G(s)(1+E(t))\) with \(E(t)=\frac{2\pi^2i}t(1+o(1))\), and: (i) for every \(k\ge0\) the interval \([g_k,g_{k+1})\) contains exactly one zero of \(\operatorname{Re}F\), and it is simple; there is exactly one zero in \((0,g_{-1})\), at \(t=7.604\), and none in \([g_{-1},g_0)\); (ii) with \(N_*(T)\) the number of positive zeros of \(\operatorname{Re}F\) up to \(T\), for \(T\ge34\), \(N_*(T)=\lfloor(\vartheta(T)+\delta(T))/\pi+\frac12\rfloor+1\) with \(\delta=\arg(1+E)\) and \(|\delta(T)|\le\arcsin|E(T)|<\frac\pi2\), so that \(|N_*(T)-\vartheta(T)/\pi-1|\le\frac12+|\delta(T)|/\pi\). The proof is analytic for \(t\ge34\) and in ball arithmetic on \([0,g_5]\). For the base kernel and for the kernel of \(\Xi^*\) the one zero per Gram interval law is only computed, between \(g_0\) and \(700\).

Source: ch/polya4.tex, label pol:thm:gramexact.

The zeros of the even part at basepoint one quarter sit at half Gram points, displaced from the zeta zeros by the argument S

Volume I, Proposition 22.7.4, page 688 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(K_\nu(a)=\int_0^\infty e^{-a\cosh u}\cosh(\nu u)\,du\) be the Macdonald function, \(\Xi^\circ(t)=\operatorname{Re}K_{\frac14+\frac{it}2}(2\pi)\) and \(\psi(t)=\arg K_{\frac14+\frac{it}2}(2\pi)\) with \(\psi(0)=0\) and continuous argument (up to a constant \(\Xi^\circ\) is the transform of the law proportional to \(\cosh(\frac x2)e^{-2\pi\cosh2x}\)). Let \(\vartheta\) be the Riemann Siegel theta function and \(S(T)=\frac1\pi\arg\zeta(\frac12+iT)\) (the mean of the one sided limits at an ordinate of a zero). (i) The zeros of \(\Xi^\circ\) are real and simple, \(\psi\) is strictly increasing on \([0,\infty)\), and the positive zeros \(t^\circ_1<t^\circ_2<\cdots\) are the solutions of \(\psi(t^\circ_n)=(n-\frac12)\pi\); so \(N^\circ(T)=\#\{n:t^\circ_n\le T\}=\lfloor\psi(T)/\pi+\frac12\rfloor\). (ii) \(\psi(t)=\vartheta(t)+O(t^{-1/2})\); hence \(\vartheta(t^\circ_n)/\pi=n-\frac12+O((t^\circ_n)^{-1/2})\), and \(t^\circ_n-g_n=O\big((t^\circ_n)^{-1/2}/\log t^\circ_n\big)\) for the half Gram points \(g_n\), defined by \(\vartheta(g_n)=(n-\frac12)\pi\). (iii) Let \(0<\gamma_1\le\gamma_2\le\cdots\) be the ordinates of all zeros of the Riemann xi function \(\xi\) in the upper half plane, with multiplicity and whatever their real parts. Suppose that exactly one zero of \(\xi\), counted with multiplicity, has ordinate \(\gamma_{n+1}\). Then \(\vartheta(\gamma_{n+1})=(n-\frac12)\pi-\pi S(\gamma_{n+1})\), and \(t^\circ_n-\gamma_{n+1}=\frac{\pi S(\gamma_{n+1})}{\vartheta'(\gamma_{n+1})}+O\big(\frac{\gamma_{n+1}^{-1/2}}{\log\gamma_{n+1}}\big)\) with an absolute implied constant.

Earlier editions: Proposition 16.7.3, page 650 (edition 1).

Source: ch/polya4.tex, label pol:prop:skeleton.

The base spin in the Griffiths Simon class implies Pólya’s spin is (implication)

Volume I, Corollary 22.7.6, page 690 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_{2\pi}\) be the base spin density and \(\rho_P(u)\propto\rho_{2\pi}(u)\cosh9u\) Pólya’s spin, where \(\cosh9u=\mathbb{E}e^{uY}\) with \(Y=\pm9\) a single Ising spin. If \(\rho_{2\pi}\) is in the Griffiths Simon class (the weak limits of suitably scaled sums of ferromagnetically coupled Ising spins), then so is \(\rho_P\). The book proves the implication, from a closure lemma under multiplication by \(\mathbb{E}e^{xY}\) for \(Y\) in the class with \(4b_Xb_Y<1\) (here \(b_Y=0\)), and does not prove that \(\rho_{2\pi}\) is in the class.

Source: ch/polya4.tex, label pol:cor:polya.

Polynomial rungs of the cosh ladder are Lee Yang and GHS spins

Volume I, Theorem 22.7.7, page 690 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_K(x)=\int_0^xt\prod_{k\le K}(1+t^2/k^2\pi^2)\,dt\), an even polynomial of degree \(2K+2\) with positive coefficients, and \(\rho_{a,K}(u)\propto e^{-af_K(4u)}\). Then: (i) each \(\rho_{a,K}\) is a Lee Yang spin (its Fourier transform has only real zeros, by de Bruijn’s theorem) and a GHS spin (by the criterion of Ellis, Monroe and Newman); (ii) \(f_K\uparrow\cosh-1\) locally uniformly, so \(\rho_{a,K}\) converges weakly to the base spin \(\rho_a\propto e^{-a(\cosh 4u-1)}\); (iii) \(\rho_{a,1}\), with \(f_1(x)=\frac{x^2}2+\frac{x^4}{4\pi^2}\), is a \(\varphi^4\) spin with positive mass and lies in the Griffiths Simon class (Simon and Griffiths 1973). Consequently the base spin \(\rho_a\), and through the implication for Pólya’s spin, Pólya’s spin are in the class if infinitely many of the rungs \(\rho_{a,K}\) are. Whether any rung with \(K\ge2\) is in the Griffiths Simon class is open.

Source: ch/polya4.tex, label pol:thm:ladder.

Open problem: a sign condition on the primal measure of the 2 sin squared lemma that holds for the Bessel model, survives the Euler product and fails for Davenport Heilbronn

Volume I, Problem 22.4, page 693 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For an entire function \(f\) that is zero free on \(\operatorname{Re}z\ge\alpha\), call \(\nu\) the primal measure of \(f\) at the basepoint \(\alpha\) when \(\frac{f'}f(\alpha+w)-\frac{f'}f(\alpha)=\int(1-e^{-wx})\,\nu(dx)\) for \(\operatorname{Re}w>0\). The \(2\sin^2\) lemma states that when \(\nu\ge0\) the compensated reciprocal \(e^{wf'(\alpha)/f(\alpha)}f(\alpha)/f(\alpha+w)\) is, in the variable \(u=w^2/2\), the Laplace transform of a generalized gamma convolution with Thorin density \(\frac1{\pi r}\int2\sin^2(rx/2)\,\nu(dx)\) at \(z>0\), where \(r=\sqrt{2z}\). For Riemann’s xi function and \(\alpha>1\) the prime powers contribute the part \(\sum_{n\ge2}\Lambda(n)n^{-\alpha}\delta_{\log n}\) of \(\nu\) and the gamma factor an archimedean part. For the Bessel model \(z\mapsto K_z(2\pi)\), with \(K_z\) the modified Bessel function of the second kind of index \(z\), which is even in \(z\), of order one, and has only purely imaginary zeros (its analogue of the Riemann Hypothesis is a theorem of Pólya), every basepoint law is a generalized gamma convolution although the primal measure is signed at every basepoint tested. Find a sign condition on the primal measure that holds for the Bessel model at every basepoint \(\alpha>0\), is preserved by the Euler product, that is, by adding the primal measures of the Euler factors, and fails for the Davenport Heilbronn function (a linear combination of two Dirichlet \(L\) functions with a functional equation of Riemann type, no Euler product, and zeros off the critical line).

Earlier editions: Problem 16.2, page 654 (edition 1).

Source: ch/polya5.tex, label pol:op:sign.

Open problem: only imaginary zeros of the even part of xi at one basepoint inside the critical strip, without a quasi Riemann hypothesis

Volume I, Problem 22.5, page 694 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and, for \(\alpha>\frac12\), \(A_\alpha(w)=\frac12(\xi(\alpha+w)+\xi(\alpha-w))\) and \(B_\alpha(w)=\frac1{2i}(\xi(\alpha+w)-\xi(\alpha-w))\) the even and odd parts of \(w\mapsto\xi(\alpha+w)\). Prove, for one fixed \(\alpha\in(\frac12,1)\), that \(A_\alpha\) has only purely imaginary zeros (equivalently, that the entire function \(t\mapsto A_\alpha(it)\), which equals \(\operatorname{Re}\xi(\alpha+it)\) for real \(t\), has only real zeros), without proving the quasi Riemann hypothesis at \(\alpha\) (that \(\zeta\) has no zero with real part greater than \(\alpha\)), by a Pólya type argument that uses the Euler product. The odd part \(B_\alpha\) need not have only imaginary zeros.

Earlier editions: Problem 16.3, page 654 (edition 1).

Source: ch/polya5.tex, label pol:op:even.

Open problem: are the derivative laws of xi, or of F in the Thorin variable, generalized gamma convolutions for some n

Volume I, Problem 22.6, page 694 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and \(F(s)=\xi(\frac12+\sqrt s)/\xi(\frac12)\). Determine whether, for some \(n\ge1\), the function \(\xi^{(2n)}(\frac12)/\xi^{(2n)}(\frac12+\sqrt s)\) is the Laplace transform of a generalized gamma convolution (GGC), and whether, for some \(n\ge1\), the function \(F^{(n)}(0)/F^{(n)}(s)\), with the derivatives of \(F\) taken in the variable \(s\), is. The case \(n=0\) is the reciprocal Xi law, a GGC if and only if the Riemann Hypothesis holds. Partial results are known for large \(n\); for the second family, by Griffin, Ono, Rolen and Zagier, for every degree \(d\) there is \(N(d)\) such that the shifted Jensen polynomials \(\sum_{j=0}^d\binom dj g_{n+j}x^j\), with \(g_k=k!\,a_k\) and \(F(s)=\sum_ka_ks^k\), have only real zeros for all \(n\ge N(d)\). For neither family is a global statement known for any \(n\).

Earlier editions: Problem 16.4, page 654 (edition 1).

Source: ch/polya5.tex, label pol:op:deriv.

Exact ratio of the even part and basepoint Laplace transforms, and the exact defect of the midpoint rule

Volume I, Theorem 22.11.1, page 694 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function. For \(\alpha>\frac12\) and real \(s\ge0\) put \(R_\alpha(s)=\xi(\alpha-\sqrt s)/\xi(\alpha+\sqrt s)\), \(\varphi^{(\alpha)}(s)=\xi(\alpha)/\xi(\alpha+\sqrt s)\), \(\varphi^P_\alpha(s)=\xi(\alpha)/A_\alpha(\sqrt s)\) with \(A_\alpha(w)=\frac12(\xi(\alpha+w)+\xi(\alpha-w))\), \(\psi_\alpha=-\log\varphi^{(\alpha)}\) and \(\psi^P_\alpha=-\log\varphi^P_\alpha\). Then, unconditionally, \(R_\alpha(0)=1\), \(0<R_\alpha(s)<1\) for \(s>0\), \(\varphi^P_\alpha(s)/\varphi^{(\alpha)}(s)=2/(1+R_\alpha(s))\) and \(\psi^P_\alpha(s)-\psi_\alpha(s)=\log\frac{1+R_\alpha(s)}2\), which lies in \((-\log2,0)\) for \(s>0\). Hence \(\varphi^{(\alpha)}\le\varphi^P_\alpha\le2\varphi^{(\alpha)}\) on \([0,\infty)\), and as \(\alpha=\frac12+a\downarrow\frac12\), \(\psi^P_\alpha(s)-\psi_\alpha(s)=-a\frac{\xi'}\xi(\frac12+\sqrt s)+O(a^2)\) locally uniformly in \(s\). In particular \(\varphi^P_\alpha\) and \(\varphi^{(\alpha)}\) both converge pointwise to \(\xi(\frac12)/\xi(\frac12+\sqrt s)\), so whenever the two families are Laplace transforms of probability laws they converge weakly to the reciprocal Xi law. If moreover \(\xi\) has no zero in \(\operatorname{Re}s\ge\alpha\) and \(\tau_\alpha\) is the inverse of the clock \(U_\alpha(\tau)=\frac1\pi\Delta_0^\tau\arg\xi(\alpha+it)\), then for every \(s\ge0\), \(\sum_{k\ge1}\log\big(1+s/\tau_\alpha(k-\frac12)^2\big)=\int_0^\infty\log\big(1+s/\tau_\alpha(v)^2\big)\,dv+\log\frac{1+R_\alpha(s)}2\).

Earlier editions: Theorem 16.11.1, page 655 (edition 1).

Source: ch/polya6.tex, label pol:thm:exact.

Log concavity of the two sided reciprocal Xi density: a floating point certificate

Volume I, Theorem 22.11.5, page 696 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\xi\) be Riemann’s xi function, \(\varphi(t^2)=\xi(\frac12)/\xi(\frac12+t)\), \(\Xi(z)=\xi(\frac12+iz)\) with positive zeros \(\gamma_1<\gamma_2<\cdots\), and \(f_X\) the density of \(X=Z\sqrt{2Y}\), where \(Y\) has Laplace transform \(\varphi(s)\) and the standard normal variable \(Z\) is independent of \(Y\). Put \(N=f_Xf_X''-(f_X')^2\). The computation shows that \(N<0\) on \(\mathbb{R}\), so that \(f_X\) is log concave, in three parts for \(x\ge0\) (the density is even). (A) On \([0,0.95]\), direct quadrature of \(f^{(j)}(x)=\frac1\pi\int_0^\infty\varphi(t^2)\frac{d^j}{dx^j}\cos(tx)\,dt\) at the left end \(x_i\) of each of \(770\) cells, with the supremum of \(|N'|\) on a cell of length \(h\) bounded through \(|f^{(j)}(y)|\le|f^{(j)}(x_i)|+h\,m_{j+1}\) and \(m_j=\frac1\pi\int_0^\infty t^j\varphi(t^2)\,dt\), proves \(N<0\) on each cell. (B) On \([0.95,2.1]\), the residue expansion \(f_X(x)=\sum_{\gamma_k<T}\eta_ke^{-\gamma_kx}+E_T(x)\), \(\eta_k=-\Xi(0)/\Xi'(\gamma_k)\), \(|E_T^{(j)}(x)|\le C_j(T)e^{-Tx}\), with the first twelve zeros and \(T=57.90\), gives \(Ne^{(\gamma_1+\gamma_2)x}\) as an explicit exponential sum plus a remainder with a decreasing majorant, and \(26\) equal cells certify negativity. (C) For \(x\ge2.1\) the two term form with \(T=23\) gives \(Ne^{(\gamma_1+\gamma_2)x}=\eta_1\eta_2(\gamma_2-\gamma_1)^2+\varepsilon(x)\) with \(\eta_1\eta_2(\gamma_2-\gamma_1)^2=-4.776\times10^8\) and \(|\varepsilon(x)|\le4.687\times10^8\), a margin of \(1.9\%\) at \(x=2.1\) that a relative error of \(2\%\) in the constants \(C_j(23)\) would break. The quadratures are floating point (mpmath, \(80\) digits) with explicit analytic bounds, not interval arithmetic, and the only arithmetic input is that the first thirteen zeros of \(\xi\) are simple and on the critical line.

Earlier editions: Theorem 16.11.5, page 657 (edition 1).

Source: ch/polya6.tex, label pol:thm:pf2.

The two sided reciprocal Xi density is uniformly close to a Pólya frequency density of every order

Volume I, Theorem 22.11.6, page 697 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(H=3\times10^{12}\) the height up to which all zeros of \(\xi\) are verified to be simple and on the critical line, \(\gamma_1<\gamma_2<\cdots\) the positive ordinates of the zeros on the critical line, \(Y\) the random variable with Laplace transform \(\xi(\frac12)/\xi(\frac12+\sqrt s)\), \(Z\) a standard normal variable independent of \(Y\), and \(f_X\) the density of \(X=Z\sqrt{2Y}\). Let \(X_{\mathrm{on}}=\sum_\gamma L_\gamma/\gamma\), the sum over the zeros on the critical line with independent standard Laplace variables \(L_\gamma\). Unconditionally \(X_{\mathrm{on}}\) has a density \(f_{\mathrm{on}}\) that is a Pólya frequency function of every order, and for \(j=0,1,2\), \(\sup_x|f_X^{(j)}(x)-f_{\mathrm{on}}^{(j)}(x)|\le\varepsilon_j:=\frac1\pi\int_0^\infty t^j\,\Pi_K(t)\big(2St^2+S^2t^4\big)\,dt\), where \(\Pi_K(t)=\prod_{k\le K}(1+t^2/\gamma_k^2)^{-1}\) for any \(K\) with \(\gamma_K<H\) and \(S=\sum_{\gamma>H}\gamma^{-2}\le(\log\frac H{2\pi}+2)/(2\pi H)=1.533\times10^{-12}\), the sum over the ordinates of all zeros above \(H\). With \(K=300\) the integrals evaluate, in floating point, to \(\varepsilon_0\le1.57\times10^{-10}\), \(\varepsilon_1\le1.3\times10^{-9}\) and \(\varepsilon_2\le1.3\times10^{-8}\). Consequently every minor \(\det[f_X(x_i-y_j)]_{i,j\le n}\) with \(x_1<\dots<x_n\) and \(y_1<\dots<y_n\) is at least \(-n\cdot n!\,(2\varepsilon_0+f_{\mathrm{on}}(0))^{n-1}\varepsilon_0\), which is \(-1.1\times10^{-8}\) for \(n=3\) and \(-1.2\times10^{-7}\) for \(n=4\), since \(f_{\mathrm{on}}(0)\le\frac1\pi\int_0^\infty\Pi_K\,dt=1.976\).

Earlier editions: Theorem 16.11.6, page 658 (edition 1).

Source: ch/polya6.tex, label pol:thm:approxpf.

Exact form of the 2 sin squared lemma: the compensated reciprocal is a GGC if and only if the clock runs at least as fast at every height as at height zero

Volume I, Theorem 22.11.7, page 698 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be a real entire function with no zeros in the half plane \(\operatorname{Re}z>\alpha-\delta\) for some \(\delta>0\), so that \(f(\alpha)\) is real and nonzero, and with \(\frac{f'}f(z)=O(\log|z|)\) uniformly on \(\operatorname{Re}z\ge\alpha\). (This holds for Riemann’s xi function with \(\alpha>\sigma_{\max}=\sup\operatorname{Re}\rho\) over its zeros \(\rho\) and \(\delta=\alpha-\sigma_{\max}\), and for \(\nu\mapsto K_\nu(2\pi)\), with \(K_\nu\) the modified Bessel function of the second kind, with \(\alpha>0\) and \(\delta=\alpha\).) Let \(\psi(u)=\log f(\alpha+w)-\log f(\alpha)-w\frac{f'}f(\alpha)\) with \(w=\sqrt{2u}\) the principal root. Then \(\psi'\) is holomorphic on \(\mathbb{C}\setminus(-\infty,0]\), \(\psi'(u)=O(|u|^{-1/2}\log|u|)\), and \(\psi'(u)=\int_0^\infty\frac{\omega_\alpha(z)}{z+u}\,dz\) with \(\omega_\alpha(z)=\frac1{\pi\sqrt{2z}}\Big[\operatorname{Re}\frac{f'}f\big(\alpha+i\sqrt{2z}\big)-\frac{f'}f(\alpha)\Big]\). Hence the compensated reciprocal \(e^{wf'(\alpha)/f(\alpha)}f(\alpha)/f(\alpha+w)\) is the Laplace transform in \(u=w^2/2\) of a generalized gamma convolution if and only if \(\operatorname{Re}\frac{f'}f(\alpha+ir)\ge\frac{f'}f(\alpha)\) for all \(r>0\), that is, if and only if the clock at the basepoint \(\alpha\) runs at least as fast at every height as at height zero.

Earlier editions: Theorem 16.11.7, page 659 (edition 1).

Source: ch/polya7.tex, label pol:thm:exactlemma.

Primes and the compensated clock: the prime sum, the archimedean bracket and the threshold 7.78918

Volume I, Proposition 22.11.8, page 700 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Lambda\) the von Mangoldt function and \(\mathcal{A}(s)=\frac1s+\frac1{s-1}-\frac12\log\pi+\frac12\frac{\Gamma'}\Gamma(\frac s2)\), so that \(\frac{\xi'}\xi=\mathcal{A}+\frac{\zeta'}\zeta\). For \(\alpha>1\) and real \(r\) put \(D_\alpha(r)=\operatorname{Re}\frac{\xi'}\xi(\alpha+ir)-\frac{\xi'}\xi(\alpha)\). Then \(D_\alpha(r)=\big[\operatorname{Re}\mathcal{A}(\alpha+ir)-\mathcal{A}(\alpha)\big]+\sum_{n\ge2}\frac{\Lambda(n)}{n^\alpha}\,2\sin^2\!\big(\frac{r\log n}2\big)\). The prime sum is nonnegative. The archimedean bracket equals \(\sum_{k\ge1}q_{\alpha+2k}(r)-q_{\alpha-1}(r)\) with \(q_c(r)=r^2/(c(c^2+r^2))\), so that only the pole term is negative, and it is nonnegative for every real \(r\) if and only if \(\zeta(3,1+\frac\alpha2)\ge8(\alpha-1)^{-3}\) (the Hurwitz zeta function), that is, if and only if \(\alpha\ge\alpha_1=7.78918\ldots\). For \(\alpha\ge\alpha_1\) the inequality \(D_\alpha(r)\ge0\) therefore follows from the identity alone. For \(1<\alpha<\alpha_1\) the bracket takes negative values (numerically, its minimum is \(-4.83\) at \(\alpha=1.2\), \(r=1.57\); \(-0.74\) at \(\alpha=2\); \(-0.011\) at \(\alpha=6\)).

Earlier editions: Proposition 16.11.8, page 660 (edition 1).

Source: ch/polya7.tex, label pol:prop:primes.

The pole term needs the gamma factor, the compensated reciprocal of (s minus 1) zeta is not a GGC

Volume I, Remark 22.11.9, page 700 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f(s)=(s-1)\zeta(s)\), a real entire function with no zeros in \(\operatorname{Re}s\ge1\). For every \(\alpha>1\) the compensated reciprocal \(e^{w\frac{f'}f(\alpha)}f(\alpha)/f(\alpha+w)\) is not the Laplace transform in \(u=w^2/2\) of a generalized gamma convolution. The proof shows that \(D^f_\alpha(r)=\operatorname{Re}\frac{f'}f(\alpha+ir)-\frac{f'}f(\alpha)\) would have to be nonnegative for every real \(r\), whereas its mean over \([0,R]\) tends to \(-\frac{f'}f(\alpha)<0\) as \(R\to\infty\), because \(f'(\alpha)\ge\frac12\) and \(f(\alpha)>0\) give \(\frac{f'}f(\alpha)>0\). Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Remark 16.11.9, page 661 (edition 1).

Source: ch/polya7.tex, label pol:rem:pole.

The compensated clock condition on every line to the right of alpha zero implies a zero free half plane

Volume I, Proposition 22.11.10, page 701 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(b_\alpha=\frac{\xi'}\xi(\alpha)\) and \(\alpha_0\ge\frac12\). If \(\operatorname{Re}\frac{\xi'}\xi(\alpha+ir)\ge b_\alpha\) for all real \(r\) and every \(\alpha>\alpha_0\), then no zero of \(\zeta\) has real part greater than \(\alpha_0\). For \(\alpha_0=\frac12\) the hypothesis implies the Riemann Hypothesis.

Earlier editions: Proposition 16.11.10, page 662 (edition 1).

Source: ch/polya7.tex, label pol:prop:comp.

Compensated clock condition at almost all heights in the critical strip, unconditionally, with the exceptional set bounded by zero density

Volume I, Theorem 22.11.14, page 703 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(b_\alpha=\frac{\xi'}\xi(\alpha)\), \(D_\alpha(r)=\operatorname{Re}\frac{\xi'}\xi(\alpha+ir)-\frac{\xi'}\xi(\alpha)\), and let \(N(\sigma,T)\) be the number of zeros \(\beta+i\gamma\) of \(\xi\) with \(\beta>\sigma\) and \(0<\gamma\le T\). Fix \(\frac12<\alpha\le1\) and put \(\sigma=\frac12(\alpha+\frac12)\). There is \(T_0\) such that for \(T\ge T_0\) the exceptional set \(E_\alpha(T)=\{r\in[T,2T]:D_\alpha(r)\le0\}\) has measure \(|E_\alpha(T)|\le C\big(1+\frac1{\alpha-1/2}\big)\big(N(\sigma,3T)+1\big)\) with an absolute constant \(C\). By Ingham’s density theorem, \(|E_\alpha(T)|\ll_\alpha T^{3(1-\sigma)/(2-\sigma)}(\log T)^5=o(T)\), and under the density hypothesis, \(N(\sigma,T)\ll T^{2(1-\sigma)+\epsilon}\) for every \(\epsilon>0\), one has \(|E_\alpha(T)|\ll T^{2(1-\sigma)+\epsilon}\). In particular, for every such \(\alpha\) the compensated clock condition \(D_\alpha(r)>0\) holds on a set of heights \(r\) of full density.

Earlier editions: Theorem 16.11.14, page 664 (edition 1).

Source: ch/polya8.tex, label pol:thm:almostall.

Hankel determinants of the two sided reciprocal Xi density have the Pólya frequency sign far in the tail

Volume I, Proposition 22.11.15, page 705 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(z)=\xi(\frac12+iz)\) with positive zeros \(\gamma_1<\gamma_2<\cdots\), \(H=3\times10^{12}\) the height up to which the zeros of \(\xi\) are verified to be simple and on the critical line, and \(f_X\) the density of \(X=Z\sqrt{2Y}\), where \(Y\) has Laplace transform \(\xi(\frac12)/\xi(\frac12+\sqrt s)\) and the standard normal variable \(Z\) is independent of \(Y\). Put \(\eta_k=-\Xi(0)/\Xi'(\gamma_k)\) (the signs alternate, \(\eta_1>0>\eta_2\), and so on) and \(\mathcal{H}_n(x)=\det[f_X^{(i+j)}(x)]_{0\le i,j<n}\). For an exponential sum \(f=\sum_k\eta_ke^{-\gamma_kx}\) one has \(\det[f^{(i+j)}(x)]_{0\le i,j<n}=\sum_{|S|=n}\big(\prod_{k\in S}\eta_k\big)\prod_{k<l\in S}(\gamma_l-\gamma_k)^2\,e^{-x\sum_{k\in S}\gamma_k}\), the sum over \(n\) element sets \(S\) of indices. Unconditionally, for every \(n\) with \(\gamma_n<H\), and for \(x>0\) with the residue expansion \(f_X(x)=\sum_{\gamma_k<T}\eta_ke^{-\gamma_kx}+E_T(x)\), \(|E_T^{(j)}(x)|\le C_j(T)e^{-Tx}\), for some \(T\in(\gamma_n,\min(\gamma_{n+1},H))\), the term of \(S=\{1,\dots,n\}\) dominates: \(\mathcal{H}_n(x)=\eta_1\cdots\eta_n\prod_{1\le k<l\le n}(\gamma_l-\gamma_k)^2\,e^{-x(\gamma_1+\cdots+\gamma_n)}\big(1+O(e^{-(T-\gamma_n)x})\big)\) as \(x\to\infty\). Its sign is \((-1)^{\lfloor n/2\rfloor}=(-1)^{n(n-1)/2}\), so \((-1)^{n(n-1)/2}\mathcal{H}_n(x)>0\) for all large \(x\), which is the Pólya frequency condition of order \(n\) at coalescing points.

Earlier editions: Proposition 16.11.15, page 665 (edition 1).

Source: ch/polya8.tex, label pol:prop:hankel.

Zero accounting for the even and odd parts of xi: almost all zeros are purely imaginary

Volume I, Theorem 22.11.16, page 705 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\frac12<\alpha\le1\) such that no zero of \(\xi\) lies on the line \(\operatorname{Re}s=\alpha\) (this excludes at most countably many \(\alpha\)). Let \(U_\alpha(T)=\frac1\pi\Delta_0^T\arg\xi(\alpha+it)\) be the clock, \(N(\alpha,T)\) the number of zeros \(\beta+i\gamma\) of \(\xi\) with \(\beta>\alpha\) and \(0<\gamma\le T\), and \(A_\alpha(w)=\frac12(\xi(\alpha+w)+\xi(\alpha-w))\), \(B_\alpha(w)=\frac1{2i}(\xi(\alpha+w)-\xi(\alpha-w))\) the even and odd parts of \(w\mapsto\xi(\alpha+w)\), so that \(A_\alpha(it)=\operatorname{Re}\xi(\alpha+it)\) and \(B_\alpha(it)=\operatorname{Im}\xi(\alpha+it)\). Let \(N_A(T)\) and \(N_B(T)\) count the zeros \(w\) of \(A_\alpha\) and \(B_\alpha\) with \(0<\operatorname{Im}w\le T\), and \(N_A^0(T)\), \(N_B^0(T)\) those among them on the imaginary axis. Then, with no other hypothesis, as \(T\to\infty\) through values that are not ordinates of zeros of \(\xi\), \(A_\alpha\) or \(B_\alpha\), \(N_A(T)=U_\alpha(T)+2N(\alpha,T)+O(\log T)\) and \(N_A^0(T)\ge\lfloor U_\alpha(T)+\frac12\rfloor\), and the same holds for \(B_\alpha\) with \(\lfloor U_\alpha(T)\rfloor\) in place of \(\lfloor U_\alpha(T)+\frac12\rfloor\). Consequently \(N_A(T)-N_A^0(T)\le2N(\alpha,T)+O(\log T)\) and \(N_B(T)-N_B^0(T)\le2N(\alpha,T)+O(\log T)\), so by Ingham’s density theorem all but \(O(T^{3(1-\alpha)/(2-\alpha)}\log^5T)\) of the zeros of \(A_\alpha\) and of \(B_\alpha\) up to height \(T\) are purely imaginary, a proportion tending to one.

Earlier editions: Theorem 16.11.16, page 666 (edition 1).

Source: ch/polya8.tex, label pol:thm:accounting.

Volume I, Chapter 23. Haseo Ki, derivatives and the Keiper Li coefficients

Eventually Thorin: after any positive time the reciprocal of the flowed Xi function is a GGC transform times a finite rational factor

Volume I, Proposition 23.2.1, page 710 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For real \(t\) let \(\Xi_t(\theta)=\int_{\mathbb{R}}\Phi(u)e^{tu^2/4}\cos(\theta u)\,du\) be the de Bruijn Newman flow of Riemann’s kernel \(\Phi\) (so \(\Xi_0=\Xi\), Riemann’s Xi function). For every \(t>0\) there are a finite, conjugation invariant multiset \(E_t\subset\mathbb{C}\setminus(0,\infty)\) and a generalised gamma convolution (GGC) Laplace transform \(L_t\), with discrete Thorin measure on \((0,\infty)\) whose atoms are simple beyond a finite level, such that \(\Xi_t(0)/\Xi_t(i\sqrt s)=L_t(s)\prod_{\eta\in E_t}(1+s/\eta)^{-1}\). The function \(\Xi_t\) has only real zeros exactly when \(E_t=\emptyset\). (This is the theorem of Ki, Kim and Lee on the zeros of \(\Xi_t\), written through the Thorin dictionary.)

Earlier editions: Proposition 23.2.1, page 842 (edition 1).

Source: ch/ki1.tex, label ki:prop:eventual.

Shrinking parabolas: Thorin atoms of the flowed Xi function lie to the right of a parabola that collapses at t = 1/2

Volume I, Corollary 23.2.2, page 710 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For real \(t\) let \(\Xi_t(\theta)=\int_{\mathbb{R}}\Phi(u)e^{tu^2/4}\cos(\theta u)\,du\) be the de Bruijn Newman flow of Riemann’s kernel \(\Phi\) (so \(\Xi_0=\Xi\), Riemann’s Xi function). For \(0<t<\frac12\) put \(\Delta(t)^2=\frac14-\frac t2\). Every Thorin atom of \(\Xi_t\), that is every square of a zero of \(\Xi_t\) and in particular every \(\eta\in E_t\) of the eventually Thorin factorisation, lies in the closed region \(\{x+iy:\ x\ge y^2/(4\Delta(t)^2)-\Delta(t)^2\}\) to the right of the parabola with focus \(0\) and vertex \(-\Delta(t)^2\). The parabolas are nested and collapse onto \([0,\infty)\) at \(t=\frac12\), where de Bruijn’s theorem takes over.

Earlier editions: Corollary 23.2.2, page 843 (edition 1).

Source: ch/ki1.tex, label ki:cor:parab.

No slack in the Gaussian direction: no class of kernels closed under Gaussian damping and containing Phi has only real zeros

Volume I, Proposition 23.2.3, page 711 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. Let \(\mathcal K\) be a class of positive even kernels closed under \(K\mapsto Ke^{-bu^2}\) for every \(b>0\). If every \(K\in\mathcal K\) has a Fourier transform with only real zeros, then \(\Phi\notin\mathcal K\). (The proof uses the theorem of Rodgers and Tao, \(\Lambda\ge0\), for the de Bruijn Newman constant \(\Lambda\).)

Earlier editions: Proposition 23.2.3, page 843 (edition 1).

Source: ch/ki1.tex, label ki:prop:obstruction.

Brownian exit time limit: the reciprocals of the even derivatives of Xi, rescaled, converge to 1 over cosh of the square root of 2s

Volume I, Proposition 23.3.1, page 712 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. Let \(X_n\) be the symmetric random variable with density proportional to \(u^{2n}\Phi(u)\) and \(m_n=\mathbb{E}|X_n|\), so that \(\Xi^{(2n)}(\theta)/\Xi^{(2n)}(0)=\mathbb{E}\cos(\theta X_n)\) for the even derivatives of \(\Xi\), and put \(\psi_n(s)=\Xi^{(2n)}(0)/\Xi^{(2n)}(i\sqrt{2s}/m_n)=1/\mathbb{E}[\cosh(\sqrt{2s}X_n/m_n)]\). Then \(\psi_n(s)\to1/\cosh\sqrt{2s}\) locally uniformly off the poles of the limit. The limit is \(\mathbb{E}[e^{-sT}]\) for \(T\) the exit time of standard Brownian motion from \((-1,1)\), a generalised gamma convolution (GGC) with Thorin measure \(\sum_{k\ge1}\delta_{c_k^2/2}\), \(c_k=(2k-1)\pi/2\). For each \(K\) there is \(n_K\) such that for \(n\ge n_K\) the \(K\) smallest Thorin atoms of \(\psi_n\) are positive and simple and converge to \(c_k^2/2\). If the Riemann Hypothesis holds, each \(\psi_n\) is the Laplace transform of a GGC \(T_n\) and \(T_n\to T\) in law.

Earlier editions: Proposition 23.3.1, page 844 (edition 1).

Source: ch/ki2.tex, label ki:prop:bm.

Conjecture: the Edgeworth rate of convergence of the zeros and clocks of the derivatives of Xi

Volume I, Remark 23.3.3, page 713 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. With \(X_n\) the symmetric variable of density proportional to \(u^{2n}\Phi(u)\), \(m_n=\mathbb{E}|X_n|\), \(V_n=|X_n|-m_n\), \(\sigma_n^2=\mathbb{E}V_n^2\), \(\kappa_n=\mathbb{E}V_n^3\), \(c_k=(2k-1)\pi/2\) and \(\psi_n(s)=1/\mathbb{E}[\cosh(\sqrt{2s}X_n/m_n)]\), conjecturally, as \(n\to\infty\) and for each fixed \(k\), the \(k\)th positive zero \(z_{k,n}\) of \(\Xi^{(2n)}\) satisfies \(z_{k,n}m_n=c_k+\frac{\kappa_nc_k^3}{6m_n^3}(1+o(1))\), and for every \(S>0\), \(\sup_{0\le s\le S}|\psi_n(s)-1/\cosh\sqrt{2s}|=\big(\sup_{x\le\sqrt{2S}}\frac{x^2}{2\cosh x}\big)\frac{\sigma_n^2}{m_n^2}(1+o(1))\). For \(S\ge2.133\) the constant is \(\max_xx^2/(2\cosh x)=0.53222\), attained at \(x=2.06534\), the root of \(\tanh x=2/x\) (computed).

Earlier editions: Remark 23.3.3, page 845 (edition 1).

Source: ch/ki2.tex, label ki:conj:rate.

Prove the Edgeworth rate for the zeros of derivatives of Xi (open problem)

Volume I, Problem 23.1, page 717 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi\) be the positive Pólya kernel with \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\), let \(X_n\) be the random variable with density proportional to \(u^{2n}\Phi(u)\), \(m_n=\mathbb{E}|X_n|\), \(V_n=|X_n|-m_n\), \(\sigma_n^2=\mathbb{E}V_n^2\) and \(\kappa_n=\mathbb{E}V_n^3\). A real \(\theta\) is a zero of \(\Xi^{(2n)}\) if and only if \(\cos(\theta m_n)\mathbb{E}\cos(\theta V_n)=\sin(\theta m_n)\mathbb{E}\sin(\theta V_n)\). The book conjectures, supported by numerics, that for fixed \(k\) the \(k\)th positive zero \(z_{k,n}\) of \(\Xi^{(2n)}\) satisfies \(z_{k,n}m_n=c_k+\kappa_nc_k^3/(6m_n^3)(1+o(1))\) with \(c_k=(k-\frac12)\pi\). Problem: prove this with explicit error terms by a saddle point analysis of the logarithm of the transform of the tilted law, and determine how large the number \(K_n\) of atoms known to be real may be unconditionally. The local theorem of Ki and Kim gives \(K_n\to\infty\), and the expansion is governed by \(c_{K_n}^3|\kappa_n|/m_n^3\to0\), that is \(K_n=o((n^2\log n)^{1/3})\); reality of the atoms is a separate question.

Source: ch/ki3.tex, label ki:prob:rate.

Effective reality of zeros of high derivatives of Xi (open problem)

Volume I, Problem 23.2, page 717 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Xi\) be Riemann’s Xi function in the variable \(t\) with \(\Xi(t)=\xi(\frac12+it)\). Find an explicit \(n_0\) and heights \(T(n)\to\infty\) such that all zeros of \(\Xi^{(2n)}\) below \(T(n)\) are real for \(n\ge n_0\). This is the effective form of the local theorem of Ki and Kim, by which for every \(R\) the zeros of \(\Xi^{(2n)}\) in \(|\theta|<R\) are real for large \(n\). It would give a sequence of certified positive Thorin measures computed from the moments of the Pólya kernel alone, and a derivative analogue of the barrier of the de Bruijn Newman computations.

Source: ch/ki3.tex, label ki:prob:ladder.

A property of the Pólya kernel that breaks the Gaussian tilt obstruction (open problem)

Volume I, Problem 23.3, page 717 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi\) be the Pólya kernel of Riemann’s Xi function. Multiplying by \(e^{-bu^2}\), \(b>0\), preserves every condition that bounds \((\log\Phi)''\) from above, so no such condition can imply the Riemann Hypothesis. Problem: identify a property of \(\Phi\) used by the sine squared lemma that is destroyed by \(\Phi\mapsto\Phi e^{-bu^2}\) for every \(b>0\) and that fails for the Pólya kernels of Epstein zeta functions of class number greater than one.

Source: ch/ki3.tex, label ki:prob:property.

Thorin measures of reciprocal Weng Xi functions of rank two and three (open problem)

Volume I, Problem 23.4, page 717 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Compute the Thorin measures of the reciprocals of the Weng Xi functions of rank two and three through the sine squared lemma, and compare their margin of positivity with that of the Thorin measure \(\sum_{\gamma>0}\delta_{\gamma^2}\) of the reciprocal Xi function of zeta under the Riemann Hypothesis, where \(\gamma\) runs over the positive ordinates of the nontrivial zeros.

Source: ch/ki3.tex, label ki:prob:weng.

Eventual Thorin property of the reciprocal friable approximant (open problem)

Volume I, Problem 23.5, page 717 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Adapt Ki’s argument principle and sign change method to prove that the reciprocal of the friable approximant of Xi of the book’s chapter on friable approximants is eventually Thorin, that is, it equals the Laplace transform of a generalised gamma convolution times a finite rational factor \(\prod_{\eta\in E}(1+s/\eta)^{-1}\) over a finite conjugation invariant multiset \(E\subset\mathbb{C}\setminus(0,\infty)\) (the exceptional set), with an explicit bound on \(E\).

Source: ch/ki3.tex, label ki:prob:friable.

Keiper Li coefficients as values of the Thorin functional

Volume I, Proposition 23.6.1, page 719 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s completed zeta function and, for \(n\ge1\), let \(\lambda_n=\sum_\rho\big[1-(1-1/\rho)^n\big]\) be the Keiper Li coefficient, the sum over the zeros \(\rho\) of \(\xi\) taken in conjugate pairs. Let \(U\) be the Thorin functional of the zeros, which gives a unit point mass at \(-(\rho-\tfrac12)^2\) to each pair \(\{\rho,1-\rho\}\) of zeros of \(\xi\), on the critical line or off it, so that \(\langle U,\varphi\rangle=\sum_{\{\rho,1-\rho\}}\varphi\big(-(\rho-\tfrac12)^2\big)\). With no hypothesis on the zeros, \(\lambda_n=\langle U,\varphi_n\rangle\), where \(\varphi_n(z)=2-\Big(\frac{2\sqrt z+i}{2\sqrt z-i}\Big)^n-\Big(\frac{2\sqrt z-i}{2\sqrt z+i}\Big)^n\) is a rational function of \(z\) with a pole of order \(n\) at \(z=-\tfrac14\) and \(\varphi_n(z)\sim n^2/z\) as \(z\to\infty\); for example \(\varphi_1(z)=(z+\tfrac14)^{-1}\) and \(\varphi_2(z)=4z(z+\tfrac14)^{-2}\). For \(z>0\), \(\varphi_n(z)=4\sin^2\big(n\arctan\tfrac1{2\sqrt z}\big)\ge0\). Consequently, by Li’s criterion, the Riemann Hypothesis holds if and only if \(\langle U,\varphi_n\rangle\ge0\) for every \(n\ge1\).

Earlier editions: Proposition 8.3.1, page 226 (edition 1).

Source: ch/ch20.tex, label prop:ch20:thorin.

Li coefficients from the heat trace and from the cumulants of the clock tilted at one quarter

Volume I, Proposition 23.6.2, page 720 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s completed zeta function and \(\lambda_n=\sum_\rho\bigl(1-(1-1/\rho)^n\bigr)\) the Keiper Li coefficients, the sum over the zeros \(\rho\) of \(\xi\) taken in pairs \(\{\rho,1-\rho\}\) as \(\sum_{\{\rho,1-\rho\}}\bigl(2-(1-1/\rho)^n-(1-1/(1-\rho))^n\bigr)\). Let \(W(t)=\sum_{\{\rho,1-\rho\}}e^{(\rho-1/2)^2t}\) be the heat trace of \(\xi\), let \(\ell_n\) be the polynomials defined by \(\sum_{n\ge1}\ell_n(t)y^{n-1}=\frac{1+y}{(1-y)^3}\exp\bigl(-\frac{ty}{(1-y)^2}\bigr)\) (so \(\ell_1=1\), \(\ell_2=4-t\), \(\ell_3=9-6t+\frac12t^2\)), and let \(\widetilde T\) be the Esscher tilt at \(u=\frac14\) of the clock variable \(T\) (the nonnegative random variable whose Laplace transform is \(\xi(\frac12)/\xi(\frac12+\sqrt u)\)), the law \(e^{-x/4}P(T\in dx)/\mathbb{E}e^{-T/4}\), with cumulants \(\kappa_j(\widetilde T)\). Without any hypothesis on the zeros: (i) \(\lambda_n=\int_0^\infty W(t)e^{-t/4}\ell_n(t)\,dt\), because the test function \(\varphi_n(z)=2-r^n-r^{-n}\), with \(r=(2\sqrt z+i)/(2\sqrt z-i)\), equals \(\int_0^\infty e^{-(z+1/4)t}\ell_n(t)\,dt\) for \(\operatorname{Re}z>-\frac14\); (ii) for \(j\ge1\), \(\kappa_j(\widetilde T)=\int_0^\infty t^{j-1}e^{-t/4}W(t)\,dt=(j-1)!\sum_{\{\rho,1-\rho\}}(\rho(1-\rho))^{-j}\), and for \(n\ge1\), \(\lambda_n=\sum_{j=1}^n(-1)^{j+1}\frac n{j!}\binom{n+j-1}{2j-1}\kappa_j(\widetilde T)\). The relation (ii) between \(\lambda_n\) and the sums \(\sum(\rho(1-\rho))^{-j}\) is that of Voros (2010); what is added is that \((j-1)!\sum(\rho(1-\rho))^{-j}\) are the cumulants of a probability law. The relation is triangular with diagonal \((-1)^{n+1}/(n-1)!\), so the \(\lambda_n\) and the cumulants of \(\widetilde T\) determine each other, with \(\mathbb{E}\widetilde T=\lambda_1\).

Earlier editions: Proposition 8.3.2, page 227 (edition 1).

Source: ch/ch20.tex, label prop:ch20:heat.

Volume I, Chapter 24. A route to the Riemann Hypothesis

The Riemann Hypothesis for a curve over a finite field is equivalent to positive semidefinite Toeplitz matrices

Volume I, Proposition 24.4.1, page 732 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(C\) be a smooth, projective, geometrically irreducible curve of genus \(g\) over \(\mathbb F_q\) with \(N_n\) points over \(\mathbb F_{q^n}\), and put \(s_0=2g\) and \(s_n=s_{-n}=(q^n+1-N_n)/q^{n/2}=\sum_{j=1}^{2g}\alpha_j^n/q^{n/2}\) for \(n\ge1\), the \(\alpha_j\) being the \(2g\) Frobenius eigenvalues. The Riemann Hypothesis for \(C\), \(|\alpha_j|=\sqrt q\) for every \(j\), holds if and only if the Toeplitz matrices \((s_{k-l})_{0\le k,l<n}\) are positive semidefinite for every \(n\).

Source: ch/nc_route.tex, label route:prop:toeplitz.

A Curie Weiss coupling of strength a shrinks the zero strip of Xi type laws from delta squared to delta squared minus a

Volume I, Proposition 24.5.1, page 734 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_1,\dots,\chi_N\) be laws of Xi type in \(\mathcal C_\delta\) whose densities are \(O(e^{-|x|^\kappa})\) for some \(\kappa>2\), and couple them with the Curie Weiss weight \(e^{\frac a2S^2}\), \(S=\sum_i\sigma_i\), \(a\ge0\). In a common field the partition function \(\mathbb E e^{\frac a2S^2+hS}\) has all its zeros in \(|\operatorname{Re}h|\le\big(\max(\delta^2-a,0)\big)^{1/2}\).

Source: ch/nc_route.tex, label route:prop:cw.

Problem: construct from the primes a positive semidefinite form for Weil’s criterion

Volume I, Problem 24.1, page 741 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Construct, from the primes, a positive semidefinite Hermitian form \(\mathcal Q\) on a space of test functions dense enough for Weil’s criterion, such that \(\mathcal Q(g)=\sum_\rho\widehat g(t_\rho)\overline{\widehat g(\bar t_\rho)}\) for every \(g\) in it. Equivalently: prove the monotonicity of \(|\xi(\sigma+it)|\) in \(\sigma\ge\tfrac12\); or prove that the coupling of the Pólya spin to an Ising spin can be lowered from \(\tfrac12\) to \(0\) without losing the Lee and Yang property.

Source: ch/nc_route.tex, label route:prob:missing.

Problem: lower bounds for the margins that the runs only observe

Volume I, Problem 24.2, page 741 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Prove, at fixed windows \(X\) and bounded heights, a lower bound for the margin that the runs only observe: the least eigenvalue of Weil’s form on \(N\) modes at the window \(X\) (exponential in \(N\) to \(N=40\), then saturating at small \(X\)), the Thorin margin linear in \(\alpha-\tfrac12\) with a constant that grows like \((\log T)^2\), and the curve margin \(\varepsilon_N\approx(6.8\,x^{-4})^N\) with its threshold at \(\tau=2g\). A bound uniform in \(X\) or in the height would imply RH, so only the restricted form is a problem of this plan. Computationally open: the levels \(0.51\) to \(0.55\) above \(10^4\), the Weil grid beyond \(N=600\), and the Hankel orders beyond \(k=128\).

Source: ch/nc_route.tex, label route:prob:margin.

Problem: strip shrinking for pair couplings and the multiplicity rate

Volume I, Problem 24.3, page 741 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Ferromagnetic couplings never widen the strip, and the Curie Weiss proposition gives the shrinking \(\delta^2\mapsto\delta^2-a\) for a Curie Weiss coupling of strength \(a\). Prove a shrinking bound for pair couplings that do not act through the magnetisation, such as nearest neighbour chains, and prove the multiplicity rate \(\delta^2\mapsto\delta^2-Na\), which is expected only for \(N\) identical copies and to first order.

Source: ch/nc_route.tex, label route:prob:shrink.

Problem: the gap law for clusters as wide as the spacing of their neighbours

Volume I, Problem 24.4, page 741 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For one pair off the line, the exact criterion of the book gives the law \(\eta-w^*=\tfrac12\phi^2\eta^3(1+O(\eta))\), so the one variable test, ferromagnetism of \(\cosh(wx)\,p\), measures the distance of an isolated pair from the line to third order. For clusters that are small compared with their distance from the other zeros each cluster is seen through its own field, among simple pairs the widest decide the threshold, and a double pair is missed by the fixed fraction \(1-\cos\tfrac\pi4\) of its distance rather than to third order. What remains is clusters as wide as the spacing of their neighbours, so that a certificate for the tilt in a window says exactly how much of the strip it excludes; such a certificate can exclude at most the zeros with \(\operatorname{Re}s>\tfrac12+w/\cos\tfrac\pi4\) unless simplicity of the zeros is also certified.

Source: ch/nc_route.tex, label route:prob:gap.

Problem: a family along the Euler product with propagating Lee and Yang property

Volume I, Problem 24.5, page 742 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a curve the point counts over all extensions \(\mathbb F_{q^n}\) are governed by the same \(2g\) eigenvalues, and positivity of the Toeplitz form propagates along \(n\). Find a family of laws or spins indexed by an integer, built from the Euler product of \(\zeta\), whose Lee and Yang property, or Toeplitz positivity, can be propagated from one member to the next, and whose limit is the Pólya spin. Finite truncations of the Euler product are not such a family.

Source: ch/nc_route.tex, label route:prob:family.

Problem: the two spin Lee and Yang property for every w above some w_0 below one half

Volume I, Problem 24.6, page 742 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Prove the two spin Lee and Yang property for every \(w>w_0\), for one \(w_0<\tfrac12\). By the book’s pair corollary the two spin form at every \(w>w_0\) is the zero free half plane \(\operatorname{Re}s>\tfrac12+w_0\), the quasi Riemann hypothesis. A proof by the methods of this book would be a proof of positivity from the primes at a level below one.

Source: ch/nc_route.tex, label route:prob:qrh.

Volume I, Chapter 25. The Berry Keating operator and the Thorin programme

Spectral measure of Pólya’s kernel for the Berry Keating dilation generator

Volume I, Proposition 25.3.1, page 744 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H=-i(x\,d/dx+\frac12)\) be the generator of the unitary dilations \(f\mapsto e^{-t/2}f(e^{-t}x)\) on \(L_2(\mathbb{R})\) (the symmetrised quantisation of \(xp\)), and let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\). Let \(\Phi\) be the even positive Pólya kernel with \(\xi(\frac12+iE)=\int_{\mathbb{R}}\Phi(u)e^{iEu}\,du\). On even functions define \((\mathcal M^+f)(E)=\frac1{2\sqrt\pi}\int_{\mathbb{R}}|x|^{-1/2-iE}f(x)\,dx\), which turns \(H\) into multiplication by \(E\). Put \(g(x)=\frac12|x|^{-1/2}\Phi(\log|x|)\). Then \(g\in L_2(\mathbb{R})\) is even, \(\mathcal M^+\) is unitary from the even subspace onto \(L_2(\mathbb{R})\), and \((\mathcal M^+g)(E)=\xi(\frac12+iE)/(2\sqrt\pi)\). Hence the spectral measure of \(g\) for \(H\) is \(|\xi(\frac12+iE)|^2\,dE/(4\pi)\), of total mass \(\|g\|^2=\frac12\int_{\mathbb{R}}\Phi^2\,du=0.159875906\) (the number computed, both sides of Parseval agreeing). The ordinates of the zeros on the critical line are the points where this density vanishes; \(H\) has no eigenvalue there. The statement says nothing about the location of the zeros.

Source: ch/nc_bk.tex, label bk:prop:mellin.

Finite rank changes cannot give the zeros as spectrum of a Gram regular operator

Volume I, Proposition 25.4.1, page 745 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A selfadjoint operator \(A\), bounded below with compact resolvent, is Gram regular if the number of its eigenvalues \(\mu\le t^2/4\) is \(\vartheta(t)/\pi+O(1)\), where \(\vartheta\) is the Riemann Siegel phase; the target is the multiset \(\{\gamma^2/4\}\) over the ordinates \(\gamma>0\) of the nontrivial zeros of \(\zeta\), with multiplicity. Let \(A\) be Gram regular. Then no selfadjoint operator whose resolvent differs from that of \(A\) by an operator of finite rank, in particular no other selfadjoint realisation obtained by changing a finite number of boundary conditions, has spectrum \(\{\gamma^2/4\}\) with multiplicity, even after a finite set is discarded. Likewise, if \(K\) is a positive first order operator (a confinement of \(xp\)) with \(\vartheta(t)/\pi+O(1)\) levels up to \(t\), no selfadjoint \(K'\) whose resolvent differs from that of \(K\) by finite rank has as spectrum the positive ordinates with multiplicity, even after a finite set is discarded. The same holds if the phase of \(A\) carries a finite number of prime terms. The proof is that a finite rank change moves the counting function by \(O(1)\), whereas \(S(T)=\pi^{-1}\arg\zeta(\frac12+iT)\) is unbounded (Selberg). The statement is unconditional and offers no candidate operator.

Source: ch/nc_bk.tex, label bk:prop:nogo.

Cauchy form of the Thorin density and its comb on the prime side (conditional on RH)

Volume I, Proposition 25.5.1, page 746 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis.

Statement. Assume the Riemann Hypothesis. Let \(\gamma\) run over the ordinates, positive and negative, of the nontrivial zeros \(\rho=\frac12+i\gamma\) of \(\zeta\) with multiplicity, let \(\mu_0=\sum_\gamma\delta_\gamma\), \(P_a(x)=a/(\pi(a^2+x^2))\), \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(u_\sigma(t)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\sigma+it)\). For \(\sigma>\frac12\) and \(t\in\mathbb{R}\), \(u_\sigma=P_{\sigma-1/2}*\mu_0\). With \(\widehat{\mu_0}(\omega)=\sum_\gamma e^{-i\gamma\omega}\), the explicit formula gives, on compactly supported test functions in \(\omega\), \(\widehat{\mu_0}(\omega)=2\cosh\frac\omega2+\Gamma(\omega)-\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\big(\delta_{\log n}+\delta_{-\log n}\big)\), where \(\Lambda\) is the von Mangoldt function and \(\Gamma(\omega)\) is the contribution of the Gamma factor, smooth off \(\omega=0\), and \(\widehat{u_\sigma}=e^{-(\sigma-1/2)|\omega|}\widehat{\mu_0}\) for \(\omega\ne0\). The deconvolution from level one to the critical line is the multiplier \(e^{|\omega|/2}\), which on the comb takes the weight \(\Lambda(n)/n\) to \(\Lambda(n)/\sqrt n\). Without RH the measure \(\mu_0\) and its transform do not exist as stated. The form uses the functional equation and the Hadamard product and not the Euler product, so it does not separate \(\zeta\) from the Davenport and Heilbronn function.

Source: ch/nc_bk.tex, label bk:prop:cauchy.

Krein string with spectral measure the squared ordinates of the zeros (conditional on RH)

Volume I, Theorem 25.6.1, page 747 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis.

Statement. Assume the Riemann Hypothesis and let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\). Let \(\gamma_1\le\gamma_2\le\cdots\) be the positive ordinates of the nontrivial zeros with multiplicity, and \(\tau=\sum_n\delta_{\gamma_n^2}\), so that \(\tau\) has weight \(m_j\) at the distinct value \(\lambda_j\) of multiplicity \(m_j\). Krein’s correspondence is a bijection between strings with a reflecting end at \(x=0\) and a Dirichlet end at \(x=L\) and pairs \((c,\tau)\), with \(c\ge0\) the length of an initial massless gap and \(\tau\) a measure on \([0,\infty)\) with \(\int d\tau/(1+\lambda)<\infty\). (a) There is a unique string with \(c=0\) and spectral measure \(\tau\), and its Weyl function uses no zero: \(h(-s)=\sum_n(\gamma_n^2+s)^{-1}=(\xi'/\xi)(\frac12+\sqrt s)/(2\sqrt s)\) for \(s>0\). (b) Its length is \(L=h(0+)=\sum_n\gamma_n^{-2}=\xi''(\frac12)/(2\xi(\frac12))=0.02310499311541897\) (the number computed). (c) Its mass function \(M\) satisfies \(M(L-)=\infty\) and has no jump at \(x=0\). (d) If the zeros are simple, the Dirichlet eigenfunction at \(L\) is \(\varphi(L,-s)=\xi(\frac12+\sqrt s)/\xi(\frac12)\), the hitting time \(T\) of the end \(x=L\) has \(\mathbb{E}e^{-sT}=\xi(\frac12)/\xi(\frac12+\sqrt s)\) (the clock of the book) and \(\mathbb{E}T=L\). Part (c) uses that a positive proportion of the zeros are simple (Alpöge and Furman) and \(N(T)\sim(T/2\pi)\log T\). Simplicity is needed only in (d). The theorem is RH by inclusion: existence of the string uses no property of \(\zeta\), and the same holds for every even real entire function of order below two, so it is a restatement of the book’s equivalence for such functions and not a route to RH.

Source: ch/nc_bk.tex, label bk:thm:string.

Heat trace of an operator bounded below is completely monotone

Volume I, Lemma 25.8.1, page 750 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H\) be self adjoint with \(H\ge c\). For every vector \(f\) the function \(x\mapsto\langle f,e^{-xH}f\rangle\) is completely monotone on \((0,\infty)\), and so is \(x\mapsto\operatorname{tr}e^{-xH}\) when \(e^{-xH}\) is trace class.

Source: ch/nc_bk.tex, label bk:lem:spectral.

Feynman Kac trace formula for a Schrodinger operator via the Brownian bridge

Volume I, Theorem 25.8.2, page 750 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H=-\frac{d^2}{dy^2}+V\) on \(L^2(\mathbb R)\) with \(V\) continuous and bounded below, \(V(y)\to\infty\) as \(|y|\to\infty\), and \(\int_{\mathbb R}e^{-xV}dy<\infty\) for every \(x>0\). Let \(b\) be a standard Brownian bridge on \([0,1]\). Then \(e^{-xH}\) is trace class and \(\operatorname{tr}e^{-xH}=\frac1{\sqrt{4\pi x}}\int_{\mathbb R}\mathbb{E}\big[\exp\big(-x\int_0^1V(y+\sqrt{2x}\,b_u)\,du\big)\big]dy\). A hard wall, \(V=+\infty\) outside an interval, is included by reading the exponential as the indicator that the loop stays inside. The theorem is cited from Kolmogorov and Simon style sources.

Source: ch/nc_bk.tex, label bk:thm:fktrace.

Particle in a box, heat trace through Kuiper and kernel at centre through Kolmogorov

Volume I, Theorem 25.8.3, page 750 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H_L=-\frac{d^2}{dy^2}\) on \([0,L]\) with Dirichlet conditions, with eigenvalues \(\pi^2n^2/L^2\), and let \(V\) and \(K\) be Kuiper’s and Kolmogorov’s variables (the range and the maximal absolute value of a standard Brownian bridge). For every \(x>0\), \(\operatorname{tr}e^{-xH_L}=\sum_{n\ge1}e^{-\pi^2n^2x/L^2}=\frac1{\sqrt{4\pi x}}\,\mathbb{E}[(L-\sqrt{2x}\,V)_+]\) and \(e^{-xH_L}(\frac L2,\frac L2)=\frac1{\sqrt{4\pi x}}\,P\big(K\le\frac L{2\sqrt{2x}}\big)\).

Source: ch/nc_bk.tex, label bk:thm:box.

The Riemann well, semiclassical count and the heat trace constant seven eighths

Volume I, Theorem 25.8.4, page 751 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(V_0(y)=16\pi^2e^{8|y|}\) and \(H_0=-\frac{d^2}{dy^2}+V_0\), and let \(W(x)=\sum e^{-x\gamma^2}\) be the heat trace of the zeros. (i) The semiclassical count \(N_0(T)=\frac1\pi\int_{\mathbb R}\sqrt{(T^2-V_0)_+}\,dy\) equals \(\frac T{2\pi}(\operatorname{artanh}s_T-s_T)\) with \(s_T=\sqrt{1-16\pi^2/T^2}\) for \(T>4\pi\), and \(N_0(T)=\frac T{2\pi}\log\frac T{2\pi e}+O(T^{-1})\), the main term of the Riemann von Mangoldt count. (ii) The semiclassical heat trace is \(\frac1{\sqrt{4\pi x}}\int e^{-xV_0}dy=E_1(16\pi^2x)/(4\sqrt{4\pi x})\), where \(E_1\) is the exponential integral, and \(\lim_{x\to0}\big(W(x)-\frac{E_1(16\pi^2x)}{4\sqrt{4\pi x}}\big)=\frac78\).

Source: ch/nc_bk.tex, label bk:thm:well.

Born potential, first order prime perturbation of the Riemann well

Volume I, Proposition 25.8.5, page 752 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the semiclassical diagonal approximation of the heat kernel of the Riemann well.

Statement. For \(N\ge2\) let \(\delta V_N(y)=4V_0(y)\sum_{n\le N}\Lambda(n)n^{-1/2}J_0\big(4\pi e^{4|y|}\log n\big)\) with \(V_0(y)=16\pi^2e^{8|y|}\) and \(J_0\) the Bessel function. In the semiclassical diagonal approximation \(e^{-xH_0}(y,y)\approx(4\pi x)^{-1/2}e^{-xV_0(y)}\), which is heuristic, the first order change of the heat trace is \(-\sum_{n\le N}\Lambda(n)n^{-1/2}p_x(\log n)\) with \(p_x(u)=e^{-u^2/4x}/(2\sqrt{\pi x})\), up to \(\sqrt x\) times an entire function of \(x\), and the first order change of the WKB count is the truncated fluctuation \(-\frac1\pi\sum_{n\le N}\frac{\Lambda(n)}{\sqrt n\log n}\sin(T\log n)\), up to a term \(O(T^{-1})\). Given the approximation both statements are proved.

Source: ch/nc_bk.tex, label bk:prop:born.

Volume I, Chapter 26. Conclusions and open problems of the theory

Jensen polynomials of the Pólya moments as finite free products (open problem)

Volume I, Problem 14.1, page 352 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\); its moment generating function is \(\mathbb{E}e^{sX}=\xi(\frac12+s)/\xi(\frac12)\). Put \(b_j=\mathbb{E}X^{2j}/(2j)!\), so that \(\Xi(t)/\Xi(0)=\sum_{j\ge0}(-1)^jb_jt^{2j}\) with \(\Xi(t)=\xi(\frac12+it)\), and for \(d,n\ge0\) let \(J_{d,n}(x)=\sum_{i=0}^d\binom di(n+i)!\,b_{n+i}x^i\) be the Jensen polynomial of degree \(d\) and shift \(n\) of the sequence \(j!\,b_j\). Pólya proved that the Riemann Hypothesis holds if and only if every \(J_{d,n}\) has only real zeros. The finite free multiplicative convolution \(\boxtimes_d\) of polynomials of degree \(d\) multiplies their normalised coefficients (the coefficient of \(x^{d - i}\) divided by \(\binom di\), up to the sign \((-1)^i\)) and keeps roots real and of one sign when both factors have real roots of one sign. The normalised coefficients of the reversed Jensen polynomial \(x^dJ_{d,n}(1/x)\) are, up to the signs \((-1)^i\), the numbers \((n+i)!\,b_{n+i}\) for \(0\le i\le d\). A factorisation \((n+i)!\,b_{n+i}=\alpha_i\beta_i\) into the normalised coefficients of two polynomials with only real roots of one sign would therefore give real zeros of \(J_{d,n}\). Open problem: find such factorisations for every shift \(n\) at a fixed degree \(d\), or show that none exists with factors of a given kind. The obvious split, a Gaussian multiplier sequence \(q^{i^2}\) with \(0<q\le1\) times a residual close to the coefficients of \((1+x)^d\), fails, because a root of multiplicity \(d\) splits into nonreal roots under a generic perturbation. Dividing the normalised coefficients of a polynomial with real roots of one sign by \((a)_i=a(a+1)\cdots(a+i-1)\) keeps its roots real, but multiplying them by \((a)_i\) does not (it turns \((1+x)^2\) into \(1+2ax+a(a+1)x^2\), whose discriminant is \(-4a<0\)). A factorisation has to put real structure into both factors.

Earlier editions: Problem 12.1, page 524 (edition 1).

Source: ch/wigfree.tex, label prob:wigfree:jensen.

Laguerre and Meixner Pollaczek expansions of the Pólya density (open problem)

Volume I, Problem 14.2, page 353 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and let \(\Xi(t)=\xi(\frac12+it)\). The book proves that for no \(a>0\) is \(|X|\) or \(X^2\) a scale mixture of Gamma\((a)\) laws, and that, under the Riemann Hypothesis, removing a gamma layer keeps zeros real while adding one does not: the function \(G_a(u)=\sum_kb_ku^k/(a)_k\), with \(b_k=\mathbb{E}X^{2k}/(2k)!\) and \((a)_k=a(a+1)\cdots(a+k-1)\), lies in the Laguerre Pólya class (the entire functions that are locally uniform limits of real rooted polynomials) for every \(a>0\), whereas multiplying the coefficients by \((a)_k\) instead does not preserve real zeros. Open problem: find a variable in which the Pólya density, or \(\Xi\) itself, has an expansion in Laguerre or Meixner Pollaczek polynomials with positive coefficients and truncations with only real zeros. The symmetric Meixner Pollaczek expansions treated in the book converge slowly, and a real rooted construction from their coefficients is a separate open problem of the chapter on the reciprocal zeta function.

Earlier editions: Problem 12.2, page 524 (edition 1).

Source: ch/wigfree.tex, label prob:wigfree:laguerre.

Jacobi parameters of the zero measure and the local semicircle model (open problem)

Volume I, Problem 14.3, page 353 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\gamma_1\le\gamma_2\le\cdots\) be the positive ordinates of the zeros \(\frac12\pm i\gamma_j\) of \(\zeta\), put \(a_j=\gamma_j^2\) and let \(\nu=\sum_ja_j^{-1}\delta_{1/a_j}\), a measure whose moments are computed from the Taylor coefficients at the centre of the Xi function \(\Xi(t)=\xi(\frac12+it)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) is Riemann’s xi function. Under the Riemann Hypothesis \(\nu\) is a positive measure and all its Jacobi parameters exist: write \(\alpha_n\) for the diagonal entries and \(\beta_n>0\) for the off diagonal entries of its Jacobi matrix, the recurrence coefficients of its orthogonal polynomials. A semicircle law is a local model for them: constant parameters \(\alpha_n=\alpha\), \(\beta_n=\beta\) give the semicircle law on \([\alpha-2\beta,\alpha+2\beta]\), and a local model with right end \(L_n\) of the interval for row \(n\) has \(\alpha_n=\frac12L_n\) and \(\beta_n=\frac14L_n\). Let \(N\) be the smooth Riemann von Mangoldt count of the ordinates, \(n(L)=\int_{(0,L^{-1/2})}dN(T)/\sqrt{1-LT^2}\), and define \(L_n\) by \(n(L_n)=n\). Open problem: under the Riemann Hypothesis, prove that \(\alpha_n\sim2\beta_n\sim\frac12L_n\) as \(n\to\infty\), a spectral form of the Riemann von Mangoldt formula. A table in the book supports it to a few per cent for \(n\le49\), but the local model is a heuristic, outside the scaling of Kuijlaars and Van Assche (1999). Also decide between the rules \(n(L_n)=n\) and \(n(L_n')=n-\frac12\), which the data do not separate, and explain the scatter of the ratios, about six per cent, through the individual gaps between the zeros.

Earlier editions: Problem 12.3, page 525 (edition 1).

Source: ch/wigfree.tex, label prob:wigfree:jacobi.

Order at which a zero collision is registered by the Hankel test (open problem)

Volume I, Problem 14.4, page 353 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, \(\Xi(t)=\xi(\frac12+it)\) the Xi function, and \(H_t\) the de Bruijn Newman heat flow in the normalisation of Rodgers and Tao, which the book uses: \(8H_t(2\theta)=\int_{\mathbb R}\Phi(v)e^{tv^2/4}\cos(\theta v)\,dv\), where \(\Phi(v)=\sum_{n\ge1}(4\pi^2n^4e^{9v/2}-6\pi n^2e^{5v/2})\exp(-\pi n^2e^{2v})\), so that \(8H_0(2\theta)=\Xi(\theta)\). Write \(\pm\theta_j(t)\) for the zeros of \(\theta\mapsto H_t(2\theta)\), so that \(\theta_j(0)=\gamma_j\), where \(\frac12\pm i\gamma_j\) are the nontrivial zeros of \(\zeta\), numbered by height. For every \(t<0\) the function \(H_t\) has nonreal zeros, by the theorem of Rodgers and Tao, and as \(t\) decreases from \(0\) pairs of consecutive real zeros \(\theta_k(t),\theta_{k+1}(t)\), which start at \((\gamma_k,\gamma_{k+1})\), collide and leave the real axis. Let \(\mathcal H_N(t)=[m_{i+j}(t)]_{i,j<N}\) be the Hankel matrix of order \(N\) of the moments \(m_n(t)=\sum_j\theta_j(t)^{-2n-2}\) (at \(t=0\) these are \(m_n(0)=\sum_k\gamma_k^{-2n-2}\)). Under the Riemann Hypothesis \(\mathcal H_N(0)\) is positive definite. Let \(n_{\rm fail}(t)\) be the least \(N\) for which \(\mathcal H_N(t)\) is not positive definite and \(t^*_N=\sup\{t<0:n_{\rm fail}(t)\le N\}\), the time, going back from \(t=0\), at which the Hankel test of order \(N\) first fails. Under the Riemann Hypothesis every \(t^*_N\) is negative, \(t^*_N\le t^*_{N+1}\) and \(t^*_N\to0\). The book computes \(t^*_N\) for \(N\le50\) and explains the steps of the staircase by collisions. The rule suggested in an unpublished note of N. G. Polson, that the collision of \((\gamma_k,\gamma_{k+1})\) is registered from order about \(2k+2\) on, holds for \(N=30\) and \(40\) and fails at \(N=50\): it gives the collision of \((\gamma_{24},\gamma_{25})\) at \(t=-1.144\), while the matrix of order \(50\) registers \((\gamma_{27},\gamma_{28})\), for which \(2k+2=56\), at \(t=-1.0325\), before \((\gamma_{24},\gamma_{25})\) has collided. Open problem: find the rule that gives the order at which a pair is registered, as a function of the index \(k\) of the pair and of its distance from the axis, and follow the staircase of \(t^*_N\) to orders of a few hundred, towards the limit \(0\) that it has under the Riemann Hypothesis.

Earlier editions: Problem 12.4, page 525 (edition 1).

Source: ch/wigfree.tex, label prob:wigfree:window.

Unconditional brackets for the collision times of zeros along the de Bruijn Newman flow (open problem)

Volume I, Problem 14.5, page 353 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, \(\Xi(t)=\xi(\frac12+it)\) the Xi function, and \(H_t\) the de Bruijn Newman heat flow in the normalisation of Rodgers and Tao, which the book uses: \(8H_t(2\theta)=\int_{\mathbb R}\Phi(v)e^{tv^2/4}\cos(\theta v)\,dv\), where \(\Phi(v)=\sum_{n\ge1}(4\pi^2n^4e^{9v/2}-6\pi n^2e^{5v/2})\exp(-\pi n^2e^{2v})\), so that \(8H_0(2\theta)=\Xi(\theta)\); as \(t\) decreases below \(0\), pairs of adjacent real zeros of \(H_t\) collide and leave the real axis, and \(H_t\) has nonreal zeros for every \(t<0\) by the theorem of Rodgers and Tao. The book brackets the collision times of such pairs by quantities computed along the flow and through the nonreal zeros of \(H_t\), and checks the hypotheses of the brackets on a grid of times for the \(58\) collisions of a census computed in the book. Open problem: find a priori control of the nonreal zeros of \(H_t\) for \(t<0\), of how far they travel and how close to a gap between real zeros they come, which would make the brackets statements about \(H_t\) computed from \(H_0\) alone.

Earlier editions: Problem 12.5, page 525 (edition 1).

Source: ch/wigfree.tex, label prob:wigfree:collide.

Extending the derivative sign ladder of the heat trace without new information on the zeros (open problem)

Volume I, Problem 14.6, page 353 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function, \(\gamma_k\) the ordinates of its zeros \(\frac12\pm i\gamma_k\) in the upper half plane, \(a_k=\gamma_k^2\) and \(W(t)=\sum_ke^{-ta_k}\) the heat trace of the zeros. Let \(T\) be the clock, the positive random variable with Laplace transform \(1/G(u)\), \(G(u)=\xi(\frac12+\sqrt{u})/\xi(\frac12)\), which has Lévy measure \(W(x)\,dx/x\). The Upsilon transform sends an infinitely divisible law \(\mu\) with Lévy process \((X_t)\), \(X_1\sim\mu\), to the law of \(\int_0^1\log(1/t)\,dX_t\), and the class BO is the class of infinitely divisible laws on \([0,\infty)\) whose Lévy measure has a completely monotone density. The book proves the derivative signs \((-1)^kW^{(k)}>0\) on ranges that grow with the height to which the Riemann Hypothesis (RH) is verified, and complete monotonicity of \(W\) on \((0,\infty)\) is RH. Open problem: find a mechanism that extends these ranges with no new information on the zeros, for instance a functional equation for the law of \(T\) under the Upsilon transform, as suggested in an unpublished note of N. G. Polson. The book restricts the form such an equation can take: with no hypothesis \(T\) is not the Upsilon image of a law in BO, and under RH \(T\) is the Upsilon image of the law \(\mu_0\), the self decomposable law with no drift and Lévy measure \(N(x^{-1/2})\,dx/x\), where \(N(T)\) is the number of ordinates \(\gamma_k\le T\), which is neither a generalised gamma convolution (GGC) nor in BO.

Earlier editions: Problem 12.6, page 525 (edition 1).

Source: ch/wigfree.tex, label prob:wigfree:ladder.

Volume II, Chapter 1. What computation can and cannot decide

Four off line zeros that pass the finite heat trace tests

Volume II, Theorem 1.4.1, page 8 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(W(t)=\sum_je^{-a_jt}\) its heat trace (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\), counted with multiplicity); the clock is \(\xi(\tfrac12)/\xi(\tfrac12+\sqrt u)\). (a) There are ordinates \(1000<A<B\le2000\) of zeros on the critical line, with \(B-A\le2\), such that, with \(g=(A+B)/2\) and \(w=\tfrac14+ig\), \(F(s)=\frac{\xi(\frac12+s)}{\xi(\frac12)}\,\frac{(1-s^2/w^2)(1-s^2/\bar w^2)}{(1+s^2/A^2)(1+s^2/B^2)}\) is an even entire function with the order, type and reflection symmetry of \(\xi(\tfrac12+s)\) and with four zeros off the critical line, and the following hold. The reciprocal clock \(1/F(\sqrt u)\) is the Laplace transform of an infinitely divisible, self decomposable law, and its heat trace \(W_F(t)=W(t)+2\operatorname{Re}e^{-at}-e^{-A^2t}-e^{-B^2t}\), \(a=-w^2=g^2-\tfrac1{16}-\tfrac{ig}2\), which is \(t\) times its Lévy density, is strictly log convex. \(W_F\) has the same small time expansion as \(W\) through the constant term, and \(1/F\) has the same nearest poles as the reciprocal Xi function. (b) Let \(N\ge1\) and \(R\ge T_N\), where \(T_N=20000N\sqrt{1+\log N}\), and suppose that every zero of \(\xi\) with ordinate in \((0,R+50]\) lies on the critical line. Then the pair can instead be chosen with \(R<A\le B\le R+50\) and \(B-A<2\) (if \(A=B\), a multiple zero supplies both), so that \(F\) keeps every property listed in (a) and the leading \(N\times N\) Hankel blocks \([c_{i+j}]_{i,j<N}\) and \([c_{i+j+1}]_{i,j<N}\) of \(W_F\), with \(c_k=(-1)^kW_F^{(k)}\), remain positive definite at every \(t>0\). If moreover \(R\ge10^{12}\), the derivative signs are also preserved: \((-1)^kW_F^{(k)}(t)>0\) for every \(t>0\) and every integer \(0\le k\le10^{17}\). For every \(N\le3\times10^7\) the verification of RH to height \(3\times10^{12}\) (Platt and Trudgian 2021) establishes the hypothesis on the zeros for \(R=\max(T_N,10^{12})\), so that such an \(F\) with both properties exists unconditionally.

Earlier editions: Theorem 5.2.1, page 117 (edition 1).

Source: ch/ch13.tex, label thm:ch13:perturb.

A pair of zeros off the line at the height of a line zero: its contribution to the density at resolution epsilon is positive exactly when epsilon is below d over sqrt 3 or above d

Volume II, Proposition 1.4.2, page 13 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(d>0\), \(\varepsilon>0\), \(\varepsilon\ne d\) and \(u=(\theta-\gamma)^2\). The zeros \(\frac12+i\gamma\) and \(\frac12\pm d+i\gamma\) contribute to \(\pi\rho_\varepsilon(\theta)\) the quantity \(\varepsilon(3u^2+6\varepsilon^2u+(3\varepsilon^2-d^2)(\varepsilon^2-d^2))/((\varepsilon^2+u)((d-\varepsilon)^2+u)((d+\varepsilon)^2+u))\), which is positive at every \(\theta\) if and only if \(\varepsilon<d/\sqrt3\) or \(\varepsilon>d\). Hence, if every zero off the line of an entire function with the order, symmetry and reality of \(\xi\) shares its ordinate with a zero on the line of at least its multiplicity, and the distances \(d\) have a positive infimum \(d_0\), its density is positive at every height on every line \(\operatorname{Re}s=\frac12+\varepsilon\) with \(0<\varepsilon<d_0/\sqrt3\), although zeros lie to the right of these lines. To first order in \(\varepsilon\) the contribution is \(\varepsilon(3u^2+d^4)/(u(u+d^2)^2)>0\) for every \(d\), as \(P(x)=x(x^2+d^2)\) satisfies \(P'^2-PP''=3x^4+d^4>0\) on the real line.

Earlier editions: Proposition 5.2.2, page 122 (edition 1).

Source: ch/ch13.tex, label tx4:prop:triple.

Certified zero of a Mellin transform closes the Bernstein Pick route to RH

Volume II, Theorem 1.4.3, page 14 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2017, version 24 of 2026), On Hilbert’s 8th problem, arXiv:1708.02653.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\Xi(z)=\xi(\tfrac12+iz)\), \(\Phi(u)=2\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\bigr)\exp(-\pi n^2e^{2u})\) the Pólya kernel, so that \(\Xi(\theta)=\int_{\mathbb{R}}\Phi(u)\cos(\theta u)\,du\), and let \(X\) be a random variable with the Pólya density \(\Phi/\Xi(0)\). Put \(M(s)=\mathbb{E}|X|^{2s}\), analytic for \(\operatorname{Re}s>-\tfrac12\) and continued meromorphically to \(\operatorname{Re}s<0\) by \(M(s)=\frac{2\,\Gamma(2s+1)\sin(-\pi s)}{\pi\,\Xi(0)}\int_0^\infty\Xi(\theta)\,\theta^{-2s-1}\,d\theta\) for \(-\tfrac12<\operatorname{Re}s<0\), and let \(\varphi_I(s)=4(s-\tfrac12)M(s-1)/M(s)\), the canonical interpolant of the moment ratios. Then \(M\) has exactly one zero in the disc \(|s-s_1|<10^{-8}\) with centre \(s_1=-8.494182256992559+2.937767068567584i\); the zero is simple, and \(M(s-1)\) does not vanish in the disc. Hence \(\varphi_I\) has a nonreal pole, and every Bernstein interpolant of the moment ratios (a Bernstein function that takes the values \(\varphi_I(n)\) at the integers \(n\ge1\)) agrees with \(\varphi_I\) on \((1,\infty)\) and cannot be continued through that pole, so none is a complete Bernstein function. For every \(0<q<1\) the sequence \(n\varphi_I(n)/(n-q)\) has no Bernstein interpolant either.

Earlier editions: Theorem 5.2.3, page 123 (edition 1).

Source: ch/ch13.tex, label thm:ch13:mellin.

Local model for the first failing derivative order of the heat trace near an off line quadruple

Volume II, Proposition 1.4.4, page 15 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function. Let \(F\) be obtained from \(\xi\) by replacing two consecutive zeros of \(\xi\) on the critical line by the quadruple of zeros \(\tfrac12\pm\delta\pm ig\), where \(g\) is the midpoint of the two ordinates and \(\delta>0\), the other zeros being unchanged, and let \(W_F(t)=\sum_je^{-a_jt}\) be the heat trace of \(F\) (\(a_j=-(\rho_j-\tfrac12)^2\), one zero \(\rho_j\) from each pair \(\{\rho,1-\rho\}\) of zeros of \(F\), counted with multiplicity). For an order \(k>0\), not necessarily an integer, let \(c_k^F(t)=\sum_ja_j^ke^{-a_jt}\) with \(a_j^k\) the principal power (for an integer \(k\) this is \((-1)^kW_F^{(k)}(t)\)). Let \(g-\Delta_-\) and \(g+\Delta_+\) be the ordinates of the nearest zeros of \(F\) on the critical line below and above \(g\), and put \(\Delta=\max(\Delta_-,\Delta_+)\). Assume \(\delta\ll\Delta\ll g\) and write \(\omega=2k\delta\Delta/g^2\) and \(\kappa=2k\Delta^2/g^2\). Restricted to the quadruple and these two zeros, \(c_k^F(t)<0\) for some \(t>0\) if and only if, to leading order, \(\min_{0\le\eta\le1}\bigl[2\cos(\omega\eta)+e^{-\kappa(1-\eta)}\bigr]<0\) (the neglected terms are relative errors \(O(\delta/\Delta)\) in the weight of the quadruple, \(O(\delta^2/(g\Delta))\) in its phase and \(O(\kappa\Delta/g)\) in the exponent of the neighbouring zero). The first failing order, the least \(k\) for which this holds, is \(k^*=\frac{\omega^*g^2}{2\delta\Delta}\), where \(\omega^*=\frac\pi2+\frac\delta\Delta\bigl(1+\log\frac\Delta{2\delta}\bigr)+O\bigl(\frac{\delta^2}{\Delta^2}+\frac\Delta g\bigr)\), so that \(k^*=\frac\pi4\,\frac{g^2}{\delta\Delta}\,(1+o(1))\) as \(\delta/\Delta\to0\) and \(\Delta/g\to0\). The failure is at the time \(t^*=\frac{k^*}{g^2}\bigl(1+O(\Delta/g)\bigr)=\frac{\omega^*}{2\delta\Delta}\bigl(1+O(\Delta/g)\bigr)\), which involves \(\delta\) and \(\Delta\) only. The book compares the time \(t^*_{\mathrm{loc}}=\omega^*/(2\delta\Delta)\), without the remainder, with the certified failure times \(t^*\) of its table of first failing orders, seven rows with \(g\) from \(185.24\) to \(1977.22\), \(\delta=\frac14\) and \(\delta=0.1\), and \(\Delta\) read from the zeros: the difference \(100\,(t^*_{\mathrm{loc}}/t^*-1)\) is between \(0.20\) and \(2.76\) per cent. This comparison is a computation, not part of the proposition.

Earlier editions: Proposition 5.2.4, page 123 (edition 1).

Source: ch/ch13.tex, label xg:prop:local.

Volume II, Chapter 2. Evaluating the Xi function in ball arithmetic

Tilted Pólya representation of xi and its derivative

Volume II, Proposition 2.1.1, page 21 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge 1}\bigl(2\pi^2n^4e^{9u/2} - 3\pi n^2e^{5u/2}\bigr)\exp(-\pi n^2e^{2u})\) be the Pólya kernel, an even function that is analytic in the strip \(|\operatorname{Im}u|<\pi/4\), and let \(\xi\) be the completed zeta function, so that \(\xi(\frac12+s)=\int_{\mathbb{R}}\Phi(u)e^{su}\,du\). Let \(0<\delta<\pi/4\) and \(\eta=\pi/4 - \delta\). Then for every complex \(s\), \(\xi(\frac12+s)=e^{i\eta s}\int_{\mathbb{R}}\Phi(x+i\eta)\,e^{sx}\,dx\) and \(\xi'(\frac12+s)=e^{i\eta s}\int_{\mathbb{R}}(x+i\eta)\,\Phi(x+i\eta)\,e^{sx}\,dx\). For real \(x\), \(|\Phi(x+i\eta)|\le C_\delta\) and \(|(x+i\eta)\Phi(x+i\eta)|\le C_\delta\), with \(C_\delta=O(\delta^{-5/2})\) as \(\delta\to 0\).

Earlier editions: Proposition 2.1.1, page 29 (edition 1).

Source: ch/ch04.tex, label prop:tilt.

Volume II, Chapter 3. The Thorin measure at finite resolution

Mass defect of the Thorin measure on a window

Volume II, Proposition 3.1.2, page 36 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function, \(\Xi(\theta)=\xi(\frac12+i\theta)\), and for \(\varepsilon>0\) let \(\mu_\varepsilon\) be the signed measure on \((0,\infty)\) whose density in the variable \(\theta=\sqrt{z}\) is the Thorin density \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\frac12+\varepsilon+i\theta)\), so that \(\mu_\varepsilon((0,\Theta^2])=\int_0^\Theta\rho_\varepsilon(\theta)\,d\theta\). Let \(N(\Theta)\) be the number of zeros \(\rho\) of \(\xi\) with \(0<\operatorname{Im}\rho\le\Theta\), counted with multiplicity. Suppose that \(\Theta\) is not the ordinate of a zero and that every zero with \(0<\operatorname{Im}\rho\le\Theta\) lies on the line \(\operatorname{Re}s=\frac12\). Then \(\mu_\varepsilon((0,\Theta^2])=N(\Theta) - \frac{\varepsilon}{\pi}\,\frac{\Xi'(\Theta)}{\Xi(\Theta)}+O(\varepsilon^2)\) as \(\varepsilon\downarrow 0\). The book also shows that the remainder is in fact \(O(\varepsilon^3)\).

Earlier editions: Proposition 3.2.2, page 48 (edition 1).

Source: ch/ch07.tex, label prop:mass.

Signatures of an off line zero in the Thorin density

Volume II, Proposition 3.2.1, page 39 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=a/(\pi(a^2+x^2))\) for \(a>0\) be the Poisson (Cauchy) kernel, \(P_{-a}=-P_a\), and \(P_0\) the unit point mass at the origin, and write \(\|\cdot\|_{\mathrm{TV}}\) for the total variation norm on \(\mathbb{R}\). For \(\varepsilon>0\) the Thorin density \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\frac12+\varepsilon+i\theta)\) of the completed zeta function \(\xi\) is a sum over the zeros \(\rho=\beta+i\gamma\) with \(\gamma>0\) of kernels \(P_{1/2+\varepsilon - \beta}(\theta - \gamma)\) together with their reflections at \(-\gamma\): a zero on the line \(\operatorname{Re}s=\frac12\) with ordinate \(\gamma\) contributes \(P_\varepsilon\) centred at \(\gamma\), and a pair of off line zeros \(\frac12\pm\delta+i\gamma\) with \(0<\delta<\frac12\) contributes \(P_{\varepsilon+\delta}+P_{\varepsilon - \delta}\) centred at \(\gamma\). Let \(\varepsilon>0\) and \(0<\delta<\frac12\). (i) If two line zeros at height \(\gamma\) are replaced by the quadruple \(\frac12\pm\delta\pm i\gamma\), the density near \(\gamma\) changes by \(2g\), where \(g=\frac12(P_{\varepsilon+\delta}+P_{\varepsilon - \delta}) - P_\varepsilon\) is the change per displaced zero, and for \(\delta<\varepsilon\), \(\|g\|_{\mathrm{TV}}=\frac{3\sqrt{3}}{4\pi}(\delta/\varepsilon)^2+O(\delta^4/\varepsilon^4)\). (ii) If the pair \(\frac12\pm\delta+i\gamma\) exists but a list of line zeros omits it, the difference is \(P_{\varepsilon+\delta}+P_{\varepsilon - \delta}\), and its total variation norm is \(2\) for \(\delta\le\varepsilon\) and \(\frac8\pi\arctan\sqrt{(\delta+\varepsilon)/(\delta - \varepsilon)} - 2=\frac4\pi\arcsin(\varepsilon/\delta)\) for \(\delta>\varepsilon\). (iii) For \(\delta>\varepsilon\) the negative part of \(\frac12(P_{\varepsilon+\delta}+P_{\varepsilon - \delta})\), the density of one displaced zero, has mass \(\frac2\pi\arctan\sqrt{(\delta+\varepsilon)/(\delta - \varepsilon)} - \frac12=\frac1\pi\arcsin(\varepsilon/\delta)=\frac{\varepsilon}{\pi\delta}+O(\varepsilon^3/\delta^3)\). The expansion \(g=\frac{\delta^2}{2}\,\partial_a^2P_a|_{a=\varepsilon}+O(\delta^4)\) behind (i), by which the signature grows like the square of the displacement, is Proposition 19.7 of Polson (2026).

Earlier editions: Proposition 3.3.1, page 50 (edition 1).

Source: ch/ch08.tex, label prop:signature.

Littlewood identity for the cumulative Thorin mass

Volume II, Proposition 3.3.1, page 42 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function and, for \(\varepsilon>0\) and \(T>0\), let \(M_\varepsilon(T)=\pi^{-1}\int_0^T\operatorname{Re}(\xi'/\xi)(\frac12+\varepsilon+it)\,dt\) be the cumulative Thorin mass at resolution \(\varepsilon\) up to height \(T\); it equals \(\pi^{-1}\arg\xi(\frac12+\varepsilon+iT)\), with the argument continued along the vertical line from the real axis, where \(\xi>0\). Let \(T>0\). For every \(\varepsilon>0\) such that \(\xi\) has no zero on the segment from \(\frac12+\varepsilon\) to \(\frac12+\varepsilon+iT\), \(\pi M_\varepsilon(T)=\frac{\partial}{\partial\varepsilon}\int_0^T\log\bigl|\xi(\frac12+\varepsilon+it)\bigr|\,dt\), and for every \(\varepsilon>0\), \(\int_0^T\log\bigl|\xi(\frac12+\varepsilon+it)\bigr|\,dt=\int_0^T\log\bigl|\xi(\frac12+it)\bigr|\,dt+\pi\int_0^\varepsilon M_{\varepsilon'}(T)\,d\varepsilon'\), where \(M_{\varepsilon'}(T)\) is defined except for the finitely many \(\varepsilon'\) at which the segment meets a zero.

Earlier editions: Proposition 3.4.1, page 54 (edition 1).

Source: ch/ch09.tex, label prop:littlewood.

Zero density identity from the Thorin mass

Volume II, Proposition 3.3.2, page 43 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function. For \(\varepsilon>0\) and \(T>0\) let \(M_\varepsilon(T)=\pi^{-1}\arg\xi(\frac12+\varepsilon+iT)\) be the cumulative Thorin mass, the argument being continued along the vertical line from the real axis, where \(\xi>0\), and let \(\tau_\varepsilon(T)\) be \(\pi^{-1}\) times the change of \(\arg\xi\) along the horizontal segment from \(\frac12+\varepsilon+iT\) to \(\frac12+iT\). Let \(N(T)\) be the number of zeros \(\rho\) with \(0<\operatorname{Im}\rho\le T\), and \(\nu(\sigma,T)\) the number of those with \(\operatorname{Re}\rho>\sigma\), both counted with multiplicity. Let \(\varepsilon>0\) and \(T>0\), with \(T\) not the ordinate of a zero and no zero of \(\xi\) on the segment from \(\frac12+\varepsilon\) to \(\frac12+\varepsilon+iT\). Then \(M_\varepsilon(T)+\tau_\varepsilon(T)=N(T) - 2\,\nu(\frac12+\varepsilon,T)\). The segment condition holds at \(\varepsilon=\frac12\), and \(\nu(1,T)=0\), which gives the unconditional count \(N(T)=M_{1/2}(T)+\tau_{1/2}(T)\), and \(\nu(\frac12+\varepsilon,T)=\frac12\,[N(T) - M_\varepsilon(T) - \tau_\varepsilon(T)]\) for every \(\varepsilon\) satisfying the hypotheses.

Earlier editions: Proposition 3.4.2, page 54 (edition 1).

Source: ch/ch09.tex, label prop:zerodensity.

Volume II, Chapter 4. The de Bruijn Newman flow computed from the Pólya density

Velocity of a zero of the de Bruijn Newman function at time zero through Hardy’s function

Volume II, Proposition 4.1.1, page 48 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\bigr)\exp(-\pi n^2e^{4u})\) and \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\), so that \(H_0(z)=\frac18\Xi(z/2)\) with \(\Xi(\theta)=\xi(\frac12+i\theta)\) for Riemann’s completed zeta function \(\xi\), and let \(Z\) be Hardy’s function, \(Z(\theta)=e^{i\vartheta(\theta)}\zeta(\frac12+i\theta)\), with \(\vartheta\) the Riemann Siegel theta function. Let \(\gamma_n\) be the \(n\)th positive zero of \(Z\) and suppose it is simple, so that \(z_n=2\gamma_n\) is a simple zero of \(H_0\) and moves smoothly with \(t\) near \(t=0\) under the backward heat flow \(\partial_tH_t=-\partial_z^2H_t\). Then \(\dot z_n(0)=\frac12\Bigl[\frac{Z''(\gamma_n)}{Z'(\gamma_n)}+\frac{4\gamma_n}{\gamma_n^2+1/4}-\operatorname{Im}\psi\bigl(\tfrac14+\tfrac{i\gamma_n}2\bigr)\Bigr]\), where \(\psi\) is the digamma function, and the last two terms tend to \(-\frac\pi2\) as \(\gamma_n\to\infty\), a drift of \(-\frac\pi4\) in \(\dot z_n\). The first \(34\) zeros move down at \(t=0\) and the \(35\)th, at \(\gamma_{35}=111.874659177\), moves up, with \(\dot z_{35}(0)=+0.0595012982\), all \(35\) signs certified in ball arithmetic.

Earlier editions: Proposition 8.4.1, page 231 (edition 1).

Source: ch/ch21.tex, label prop:ch21:velocity.

One step of the backward heat flow of a real rooted polynomial under a Laguerre bound

Volume II, Lemma 4.1.3, page 57 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For a real polynomial \(p\) and a point \(x\) with \(p(x)\ne0\) let \(\mathcal L_p(x)=-(\log|p|)''(x)=(p'(x)^2-p(x)p''(x))/p(x)^2\) be Laguerre’s quantity. Let \(p\) be a real polynomial of degree \(n\ge2\) with only real simple zeros and critical points \(c_1<\dots<c_{n-1}\), and let \(s\) be real with \(s\,\mathcal L_p(c_i)<1\) for every \(i\). Then \(p+sp''\) has only real simple zeros, one in each of the intervals \((-\infty,c_1)\), \((c_1,c_2)\), , \((c_{n-1},\infty)\).

Source: ch/ch21.tex, label lem:wigbiane:onestep.

Critical point lower bound for the collision time of zeros under the backward heat flow

Volume II, Theorem 4.1.7, page 60 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_s\), \(s\ge0\), be real entire functions of genus at most one, smooth in \((z,s)\), with \(\partial_sf_s=\partial_z^2f_s\); for a real polynomial \(p\) and \(D=d/dz\) one such family is \(f_s=e^{sD^2}p=\sum_{k\ge0}s^kp^{(2k)}/k!\). For a real entire function \(f\) of genus at most one with zeros \(z_j\), Laguerre’s quantity is \(\mathcal{L}_f(x)=-\frac{d^2}{dx^2}\log|f(x)|=\big(f'(x)^2-f(x)f''(x)\big)/f(x)^2=\sum_j(x-z_j)^{-2}\), a real sum because nonreal zeros come in conjugate pairs. Fix a gap between two adjacent simple real zeros \(z_k(s)<z_{k+1}(s)\) of \(f_s\), which move continuously until they meet, if they do. When all zeros of \(f_s\) are real, \(f_s'/f_s\) falls from \(+\infty\) to \(-\infty\) across the gap and vanishes at exactly one point \(c(s)\), the critical point of the gap; put \(\ell(s)=\mathcal{L}_{f_s}(c(s))\). Let \(S>0\). If all zeros of \(f_\sigma\) are real and simple for \(0\le\sigma<S\), then \(\dot\ell\le\ell^2\) on \([0,S)\) and \(\ell(\sigma)\le\ell(0)/(1-\sigma\ell(0))\) for \(0\le\sigma<\min(S,1/\ell(0))\). In particular, if \(p\) is a real polynomial with only real simple zeros, and the first collision of any two zeros of \(e^{sD^2}p\) happens at \(s=s_1\), between the two zeros that bound the gap \((z_k,z_{k+1})\) of \(p\) at \(s=0\), then \(s_1\ge1/\mathcal{L}_p(c_k)\), where \(c_k\) is the critical point of \(p\) in that gap. The bound is attained by \(p(z)=z^2-a^2\).

Earlier editions: Theorem 8.4.7, page 243 (edition 1).

Source: ch/ch21.tex, label thm:wigbiane:riccati.

Sextic power sum inequality 3 S2 cubed minus 6 S2 S4 plus 4 S3 squared nonnegative

Volume II, Lemma 4.1.6, page 60 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For a real square summable sequence \((x_j)\) put \(S_k=\sum_jx_j^k\) for \(k\ge2\) (absolutely convergent sums). Then \(E=3S_2^3-6S_2S_4+4S_3^2\ge0\), and for a nonzero sequence equality holds exactly when the nonzero terms are one pair \(\{a,-a\}\).

Source: ch/ch21.tex, label lem:wigbiane:sextic.

Biane form lower bound for the collision time of zeros under the backward heat flow

Volume II, Theorem 4.1.9, page 61 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_s\), \(s\ge0\), be real entire functions of genus at most one, smooth in \((z,s)\), with \(\partial_sf_s=\partial_z^2f_s\); for a real polynomial \(p\) and \(D=d/dz\) one such family is \(f_s=e^{sD^2}p=\sum_{k\ge0}s^kp^{(2k)}/k!\). For a real entire function \(f\) of genus at most one with zeros \(z_j\), Laguerre’s quantity is \(\mathcal{L}_f(x)=-\frac{d^2}{dx^2}\log|f(x)|=\big(f'(x)^2-f(x)f''(x)\big)/f(x)^2=\sum_j(x-z_j)^{-2}\), a real sum because nonreal zeros come in conjugate pairs. Fix a gap between two adjacent simple real zeros \(z_k(s)<z_{k+1}(s)\) of \(f_s\), which move continuously until they meet, if they do. The quantity \(\mathcal{L}_{f_s}\) tends to \(+\infty\) at both ends of the gap; let \(m(s)\) be its minimum over the gap and \(D^+m\) the upper right Dini derivative of \(m\). Let \(S>0\). If all zeros of \(f_\sigma\) are real and simple for \(0\le\sigma<S\), then \(D^+m\le m^2\) on \([0,S)\) and \(m(\sigma)\le m(0)/(1-\sigma m(0))\) for \(0\le\sigma<\min(S,1/m(0))\). In particular, if \(p\) is a real polynomial with only real simple zeros, and the first collision of any two zeros of \(e^{sD^2}p\) happens at \(s=s_1\), between the two zeros that bound the gap \((z_k,z_{k+1})\) of \(p\) at \(s=0\), then \(s_1\ge1/\min_{(z_k,z_{k+1})}\mathcal{L}_p\). This bound is at least as strong as the critical point bound \(1/\mathcal{L}_p(c_k)\), where \(c_k\) is the critical point of \(p\) in the gap, because the minimum does not exceed the value at \(c_k\), and it is attained by \(p(z)=z^2-a^2\).

Earlier editions: Theorem 8.4.9, page 244 (edition 1).

Source: ch/ch21.tex, label thm:wigbiane:biane2.

Quartic power sum inequality 2 S4 at most S2 squared when S3 is zero

Volume II, Lemma 4.1.8, page 61 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For a real square summable sequence \((x_j)\) put \(S_k=\sum_jx_j^k\) for \(k\ge2\). If \(S_3=0\) then \(2S_4\le S_2^2\), and for a nonzero sequence equality holds exactly when the nonzero terms are one pair \(\{a,-a\}\).

Source: ch/ch21.tex, label lem:wigbiane:quartic.

Shadow inequalities for power sums and for a nonreal pair replaced by a double real point

Volume II, Lemma 4.1.10, page 62 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. (i) For every real square summable sequence with power sums \(S_k=\sum_jx_j^k\), \(S_4\le\frac12S_2^2+S_2^{1/2}|S_3|\), and the coefficient of the second term can be halved: \(S_4\le\frac12S_2^2+\frac12S_2^{1/2}|S_3|\), where \(\frac12\) is best possible, since a single nonzero term gives \(S_4-\frac12S_2^2=\frac12S_2^{1/2}|S_3|\). (ii) If \(w=1/(d-ib)\) with \(d\ne0\) and \(b\) real, then \(\big|2\operatorname{Re}w^k-2d^{-k}\big|\le k(k+1)\,b^2/|d|^{k+2}\) for every \(k\ge1\), and the bound is sharp as \(b/d\to0\), since \(\operatorname{Re}(1-i\epsilon)^{-k}=1-\frac12k(k+1)\epsilon^2+O(\epsilon^4)\).

Source: ch/ch21.tex, label lem:wigbiane:shadow.

Local lower bound for the closing of a gap with a correction from the nonreal zeros

Volume II, Theorem 4.1.12, page 63 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_s\), \(s\ge0\), be real entire functions of genus at most one, smooth in \((z,s)\), with \(\partial_sf_s=\partial_z^2f_s\), for example \(f_s=e^{sD^2}p\) with \(D=d/dz\) and \(p\) a real polynomial. Let \(\mathcal{L}(x)=-\frac{d^2}{dx^2}\log|f_s(x)|=\sum_j(x-z_j)^{-2}\), over the zeros \(z_j\) of \(f_s\), be Laguerre’s quantity (a real sum, because nonreal zeros come in conjugate pairs). Fix a gap between two adjacent simple real zeros of \(f_s\), which move continuously until they meet, if they do; \(\mathcal{L}\to+\infty\) at both ends of the gap. Let \(m(s)\) be the minimum of \(\mathcal{L}\) over the gap, \(D^+m\) its upper right Dini derivative, and at a time \(s\) let \(u\) be a point of the gap where the minimum is attained, with \(m=m(s)=\mathcal{L}(u)>0\). For the conjugate pairs \(a\pm ib\), \(b>0\), of nonreal zeros of \(f_s\), each pair counted once, put \(B_p=\sum b^2|u-a|^{-p}\), and assume that \(B_4\), \(B_5\) and \(B_6\) are finite. Then \(\mathcal{L}''(u)-3m^2\le\eta m^2\), where \(\eta=\big(36\,mB_4+108\,B_4^2+72\,(m+6B_4)^{1/2}B_5+120\,B_6\big)/m^2\), and \(D^+m(s)\le(1+\eta)m^2\). Hence, if the two zeros bounding the gap meet at a time \(s_c\), and \(m>0\) and \(\eta\le\bar\eta\) on \([0,s_c)\) for a constant \(\bar\eta\) (with \(\eta\) evaluated at a minimiser at each time), then \(s_c\ge1/\big((1+\bar\eta)\,m(0)\big)\). All of this holds with the coefficient \(72\) in \(\eta\) replaced by \(36\), by the sharp inequality \(S_4\le\tfrac12S_2^2+\tfrac12S_2^{1/2}|S_3|\) for the power sums \(S_k=\sum_jx_j^k\) of a real square summable sequence \((x_j)\).

Earlier editions: Theorem 8.4.12, page 245 (edition 1).

Source: ch/ch21.tex, label thm:wigbiane:local.

Lower bound for the closing of a gap from the quantity delta, with no knowledge of the nonreal zeros

Volume II, Corollary 4.1.13, page 63 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_s\), \(s\ge0\), be real entire functions of genus at most one, smooth in \((z,s)\), with \(\partial_sf_s=\partial_z^2f_s\), for example \(f_s=e^{sD^2}p\) with \(D=d/dz\) and \(p\) a real polynomial, and let \(\mathcal{L}(x)=-\frac{d^2}{dx^2}\log|f_s(x)|\) be Laguerre’s quantity. Fix a gap between two adjacent simple real zeros of \(f_s\), let \(m(s)\) be the minimum of \(\mathcal{L}\) over the gap, \(D^+m\) its upper right Dini derivative, and at a time \(s\) let \(u\) be a point of the gap where the minimum is attained, with \(m=m(s)=\mathcal{L}(u)>0\). For the conjugate pairs \(a\pm ib\), \(b>0\), of nonreal zeros of \(f_s\), each pair counted once, put \(B_p=\sum b^2|u-a|^{-p}\), assume that \(B_4\), \(B_5\) and \(B_6\) are finite, and put \(\eta=\big(36\,mB_4+108\,B_4^2+72\,(m+6B_4)^{1/2}B_5+120\,B_6\big)/m^2\). Put \(\delta=\mathcal{L}''(u)/m^2-3\), computed from \(f_s\) and its first four derivatives at \(u\). Then \(D^+m(s)\le(1+\delta)m^2\) and \(\delta\le\eta\), and \(\delta\le0\) when all zeros of \(f_s\) are real. Hence, if the two zeros bounding the gap meet at a time \(s_c\), and \(m>0\) and \(\delta\le\bar\delta\) on \([0,s_c)\) for a constant \(\bar\delta>-1\), then \(s_c\ge1/\big((1+\bar\delta)\,m(0)\big)\); if \(\delta\le0\) throughout, the collision cannot come before Biane’s time \(1/m(0)\), exactly as for functions with only real zeros.

Earlier editions: Corollary 8.4.13, page 246 (edition 1).

Source: ch/ch21.tex, label cor:wigbiane:delta.

Upper bound (deadline) for the closing time of a gap between real zeros

Volume II, Theorem 4.1.14, page 64 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_s\), \(s\ge0\), be real entire functions of genus at most one, smooth in \((z,s)\), with \(\partial_sf_s=\partial_z^2f_s\), for example \(f_s=e^{sD^2}p\) with \(D=d/dz\) and \(p\) a real polynomial, and let \(\mathcal{L}(x)=-\frac{d^2}{dx^2}\log|f_s(x)|=\sum_j(x-z_j)^{-2}\) be Laguerre’s quantity (a real sum, because nonreal zeros come in conjugate pairs). Fix a gap between two adjacent simple real zeros \(z_k<z_{k+1}\) of \(f_s\), let \(m(s)\) be the minimum of \(\mathcal{L}\) over the gap, and let \(u\) be a point where it is attained, so \(m=\mathcal{L}(u)\). Write \(m=P+Q\), where \(P=(u-z_k)^{-2}+(u-z_{k+1})^{-2}\) is the share of the two zeros bounding the gap and \(Q=\sum(u-z)^{-2}\) over all the other zeros \(z\) of \(f_s\), and let \(W=\sum|u-z|^{-4}\) over the nonreal zeros \(z\) of \(f_s\). While the two zeros bounding the gap are real, the right derivative \(m'(s^+)\) exists and is the least value of \(\mathcal{L}''(u)-2m^2\) over the minimisers \(u\), and at every minimiser \(\mathcal{L}''(u)-2m^2\ge m^2-6mQ+3Q^2-6W\). Suppose that, for constants \(\rho\) and \(\bar W\) with \(9\rho^2+6\bar W>0\), \(Q\le\rho\) and \(W\le\bar W\) at every minimiser for as long as the two zeros bounding the gap are real, and that \(m(0)>r_1\), where \(r_{1,2}=3\rho\pm(9\rho^2+6\bar W)^{1/2}\). Then the gap closes, one of the two zeros bounding it meeting another zero, at a time \(s_c\le s^*=\frac{1}{r_1-r_2}\log\frac{m(0)-r_2}{m(0)-r_1}\). If all zeros are real, then \(W=0\), so \(\bar W=0\) may be taken, and for \(\rho>0\) one has \(s^*=(6\rho)^{-1}\log\big(m(0)/(m(0)-6\rho)\big)=1/m(0)+3\rho/m(0)^2+12\rho^2/m(0)^3+\cdots\).

Earlier editions: Theorem 8.4.14, page 247 (edition 1).

Source: ch/ch21.tex, label thm:wigbiane:upper.

Deadline for the closing of a gap in terms of the quantity kappa

Volume II, Corollary 4.1.15, page 64 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f_s\), \(s\ge0\), be real entire functions of genus at most one, smooth in \((z,s)\), with \(\partial_sf_s=\partial_z^2f_s\), and let \(\mathcal{L}(x)=-\frac{d^2}{dx^2}\log|f_s(x)|=\sum_j(x-z_j)^{-2}\) be Laguerre’s quantity. Fix a gap between two adjacent simple real zeros \(z_k<z_{k+1}\) of \(f_s\), let \(m(s)\) be the minimum of \(\mathcal{L}\) over the gap, let \(u\) be a point where it is attained, and put \(x_j=1/(u-z_j)\) for the zeros \(z_j\) of \(f_s\), \(P=x_k^2+x_{k+1}^2\), \(Q=\sum_{j\ne k,k+1}x_j^2\), so that \(m=P+Q\), and \(W=\sum|x_j|^4\) over the nonreal zeros. Put \(\kappa=\mathcal{L}''(u)-3P^2\), computed from \(f_s\) at \(u\) and the two zeros bounding the gap. Then \(\kappa=3\big(x_k^2-x_{k+1}^2\big)^2+6\sum_{j\ne k,k+1}x_j^4\ge-6W\), with \(\kappa\ge0\) when all zeros are real. Put \(\bar\kappa=\max\big(0,\sup(-\kappa)\big)\), the supremum over the minimisers at the times before the collision. If \(Q\le\rho\) at every minimiser for as long as the two zeros bounding the gap are real, with \(9\rho^2+\bar\kappa>0\), and \(m(0)>r_1\), where \(r_{1,2}=3\rho\pm(9\rho^2+\bar\kappa)^{1/2}\), then the gap closes, one of the two zeros bounding it meeting another zero, at a time \(s_c\le\frac{1}{r_1-r_2}\log\frac{m(0)-r_2}{m(0)-r_1}\). This bound is never weaker than the same bound with \(6\bar W\) in place of \(\bar\kappa\), where \(\bar W\) is a constant with \(W\le\bar W\) at every minimiser for as long as the two zeros bounding the gap are real, since then \(\bar\kappa\le6\bar W\).

Earlier editions: Corollary 8.4.15, page 247 (edition 1).

Source: ch/ch21.tex, label cor:wigbiane:kappa.

Control of the nonreal zeros of the de Bruijn Newman function at negative times

Volume II, Problem 4.1, page 67 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For real \(t\) let \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) be the de Bruijn Newman function, with \(\Phi(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)\exp(-\pi n^2e^{4u})\), so that \(H_0(z)=\tfrac18\Xi(z/2)\) with \(\Xi(\theta)=\xi(\tfrac12+i\theta)\) for Riemann’s completed zeta function \(\xi\). As \(t\) decreases from \(0\) the zeros of \(H_t\) move by the backward heat flow, and the book bounds from below and from above, in terms of \(H_0\) alone, the time at which a gap between two adjacent real zeros closes, under hypotheses along the flow that the computation checks on a grid of times but cannot yet guarantee in advance. Problem: find a bound for how far the nonreal zeros of \(H_t\), \(t<0\), travel, and for how close to a gap between real zeros they come. No such bound is known, and such control would make the lower and upper bounds for the closing time of a gap unconditional statements about \(H_t\), computed from \(H_0\) alone.

Earlier editions: Problem 8.1, page 250 (edition 1).

Source: ch/ch21.tex, label prob:wigbiane:control.

Shielding deficit identity for a pair of adjacent real zeros of the de Bruijn Newman function (Newton 2026, Zenodo)

Volume II, Theorem 4.1.16, page 69 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a<b\) be adjacent simple real zeros of an entire function with zeros \(\rho\) (here the de Bruijn Newman function \(H_t\)), with gap \(g=b-a\), and put \(S_{a,b}=\sum_{\rho\ne a,b}\frac{1}{(a-\rho)(b-\rho)}\) and \(\mathrm{Def}=2-g^2S_{a,b}\). On each side of the pair number the exterior gaps outward, write them \(g(1+\varepsilon_m)\) with \(\varepsilon_m>-1\), and put \(D_0=0\), \(D_m=\sum_{j\le m}\varepsilon_j\), so that the \(m\)th exterior zero on that side is at distance \(g(m+D_m)\) from the nearer end of the pair. Then on one side, for every \(N\ge1\), \(1-\sum_{m=1}^N\frac{1}{(m+D_m)(m+1+D_m)}=\sum_{m=1}^N\frac{\varepsilon_m}{(m+D_{m-1})(m+D_m)}+\frac{1}{N+1+D_N}\). If all zeros are real and the exterior zeros tend to infinity on both sides, then \(\mathrm{Def}=\sum_{\text{sides}}\sum_{m\ge1}\frac{\varepsilon_m}{(m+D_{m-1})(m+D_m)}\). The result is due to Newton (2026, Zenodo).

Earlier editions: Theorem 8.4.16, page 252 (edition 1).

Source: ch/ch21.tex, label nw:thm:shield.

Consequences of the shielding deficit identity: lattice, truncated lattice, single enlarged gap, one sided bound and monotonicity of the smallest gap (Newton 2026, Zenodo)

Volume II, Corollary 4.1.17, page 70 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. In the setting of the shielding deficit identity (adjacent simple real zeros \(a<b\) with gap \(g\), \(\mathrm{Def}=2-g^2S_{a,b}\), exterior gaps \(g(1+\varepsilon_m)\) on each side, \(\varepsilon_m>-1\), and the flow \(\frac{d(g^2)}{dt}=4\,\mathrm{Def}\) for the de Bruijn Newman function): (a) if all \(\varepsilon_m=0\) then \(\mathrm{Def}=0\) and \(g\) is constant, as for the lattice of zeros of \(e^{\kappa^2t}\sin\kappa z\); (b) keeping \(R\) exterior zeros on each side of a regular lattice of gap \(g\) leaves \(\mathrm{Def}=2/(R+1)\); (c) a single enlarged exterior gap \(g(1+\delta)\) at position \(j\) contributes exactly \(\delta/(j(j+\delta))\); (d) with \(d_{\nu,j}\) the exterior gaps on the sides \(\nu=L,R\) and \([x]_+=\max(x,0)\), \([\mathrm{Def}]_+\le g^{-1}\sum_{j\ge1}j^{-2}\big[(d_{L,j}-g)_++(d_{R,j}-g)_+\big]\), and by (c) the constant \(j^{-2}\) is optimal to first order; (e) if the configuration has a smallest gap \(g>0\) (a finite configuration or a lattice) and \(g\) is that gap, then every \(\varepsilon_m\ge0\), so \(\mathrm{Def}\ge0\) and the smallest gap does not shrink forward in time, with equality only for the lattice. The book notes that (e) does not apply to \(H_t\), which has no smallest gap. The result is due to Newton (2026, Zenodo).

Earlier editions: Corollary 8.4.17, page 252 (edition 1).

Source: ch/ch21.tex, label nw:cor:shield.

Pair of zeros near height 185 still real at time minus 0.287979, certified

Volume II, Proposition 4.1.18, page 71 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and, for real \(t\) of either sign, \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), the de Bruijn Newman deformation of Riemann’s xi function (so \(\xi_0\) is the xi function), and \(H_t(z)=\frac18\xi_t(\frac12+\frac{iz}2)\). A zero \(\frac12+i\gamma\) of \(\xi_t\) on the critical line is the real zero \(z=2\gamma\) of \(H_t\), and while the real zeros are simple they move by \(dz_k/dt=2\sum_{j\ne k}(z_k-z_j)^{-1}\). Let \(\gamma_{71}<\gamma_{72}\) be the ordinates of the 71st and 72nd zeros of \(\xi\) in the upper half plane, the pair near height \(185\). At \(t=-0.287979\) the signs of \(\xi_t(\frac12+i\theta)\) at the \(131\) points \(\theta=184.6+0.01k\), \(k=0,\dots,130\), certified in ball arithmetic (Arb) at \(640\) bits from the trapezoid sum for the integral with an enclosure of its discretisation error, show two sign changes; hence the real function \(\theta\mapsto\xi_t(\frac12+i\theta)\) has at least two zeros in \([184.6,185.9]\), and the pair \((\gamma_{71},\gamma_{72})\) is still real at \(t=-0.287979\). This is the certified upper end of the book’s bracket \([-0.28803,-0.287979]\) for the time at which the pair collides when \(t\) decreases from \(0\); the lower end \(-0.28803\) rests on the asymptotic pair law \(D^2\approx8(t-t_*)\) for the distance \(D\) of the pair in the variable \(z\) near its collision time \(t_*\), and is not certified. Newton’s method on the same trapezoid sum puts the double zero of the pair at \(z_*=370.78087\), \(t_*=-0.287979837009\); a Krawczyk test certifies a real double zero of the trapezoid sum itself (a statement about the sum, not about \(H_t\)) in a box of half width \(2^{-400}\) about this point, and at \(t=t_*+1.2\times10^{-12}\) certified signs \(+,-,+\) of \(H_t\) at \(z_*-4\times10^{-6}\), \(z_*\) and \(z_*+4\times10^{-6}\) show that the pair is still real there, so the collision lies below \(-0.287979837008\). The floating point integration of the motion of the first \(600\) zeros that the book reports with this result (with the monotonicity of two gap functionals at the sampled times) is computed, not certified, and is not claimed here.

Earlier editions: Proposition 15.6.1, page 631 (edition 1).

Source: ch/ch21.tex, label cp:prop:gas.

Height of one quadruple of zeros of the de Bruijn Newman function decreases

Volume II, Proposition 4.1.19, page 72 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H_\lambda(\theta)\) be the family of entire functions with \(\partial_\lambda H=-\partial_\theta^2H\) (the de Bruijn Newman family \(H_t\) written in the ordinate \(\theta=z/2\) and the time \(\lambda=t/4\)), whose simple zeros \(\theta_j\) move by \(d\theta_j/d\lambda=2\sum_{k\ne j}(\theta_j-\theta_k)^{-1}\). Let \(x+ib\) with \(b>0\) be a nonreal zero whose other zeros are its quadruple partners \(\pm(x\pm ib)\) and real pairs \(\pm\gamma\) (\(\gamma>0\) running over the real pairs). Then \[\frac{db}{d\lambda}=-\frac1b-\frac b{b^2+x^2}-4b\sum_\gamma\frac{b^2+\gamma^2+x^2}{\big(b^2+(\gamma-x)^2\big)\big(b^2+(\gamma+x)^2\big)}<0 .\]

Source: ch/ch21.tex, label xi:prop:bdot.

Two imaginary quadruples of zeros, exact solution of the heat flow

Volume II, Proposition 4.1.21, page 73 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Under the flow \(\partial_\lambda H=-\partial_\theta^2H\), let \(H=\theta^4+p\theta^2+q\) with zeros \(\pm ib_1,\pm ib_2\), so \(p=b_1^2+b_2^2\) and \(q=b_1^2b_2^2\). Then \(\dot p=-12\) and \(\dot q=-2p\), hence \(p=p_0-12\lambda\), \(q=q_0-2p_0\lambda+12\lambda^2\), and the discriminant of \(H\) in \(w=\theta^2\) is \(p^2-4q=(p_0^2-4q_0)-16p_0\lambda+96\lambda^2\).

Source: ch/ch21.tex, label xi:prop:quartic.

Dilution at the maximal height in an equally spaced model

Volume II, Proposition 4.1.22, page 74 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Work in the model of an extreme nonreal zero at height \(B\), \(\theta=x+iB\), in the flow \(\partial_\lambda H=-\partial_\theta^2H\), with real background zeros at \(\theta=x+ks\), \(k\ne0\), \(s>0\), replaced by complex pairs with matching zero counts: a pair occupies two sites and pairs are spaced \(2s\). If every pair sits at height \(B\), the background pull on the extreme zero is exactly half of the all real pull, and \(\frac{dB}{d\lambda}=-\frac\pi s\coth\frac{\pi B}s\le-\frac\pi s\), so the extreme zero reaches the axis by \(\lambda=B_0s/\pi\), where \(B_0\) is its initial height, which is \(8B_0/\log(\gamma/2\pi)\) in standard time for the spacing \(s=2\pi/\log(\gamma/2\pi)\) at height \(\gamma\). The result holds in the model and is not a bound for the zeros of \(\zeta\).

Source: ch/ch21.tex, label xi:prop:dilution.

Small x asymptotic of the sum of exp(minus gamma x) over the ordinates of the zeros

Volume II, Proposition 4.1.24, page 75 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\gamma\) run over the ordinates of the zeros of \(\zeta\) in the upper half plane, with multiplicity, and let \(\gamma_{\mathrm E}\) be Euler’s constant. As \(x\downarrow0\), \(\sum_{\gamma>0}e^{-\gamma x}=\frac1{2\pi x}\log\frac1{2\pi e^{\gamma_{\mathrm E}}x}+O(1)\). No hypothesis is needed. The proof uses the Riemann and von Mangoldt formula and Littlewood’s bound \(\int_0^TS(t)\,dt=O(\log T)\) for the argument \(S\) of \(\zeta\) on the critical line.

Source: ch/ch21.tex, label xi:prop:levyasym.

The de Bruijn Newman function as a Gaussian average and the reciprocal transform of the tilted clock as a Gaussian smoothing of xi

Volume II, Proposition 4.1.25, page 76 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s completed zeta function, \(\Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\bigr)\exp(-\pi n^2e^{4u})\), \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) for real \(t\) (so that \(H_0(z)=\frac18\xi(\frac12+\frac{iz}2)\)) and \(R_t(s)=H_t(0)/H_t(2i\sqrt s)\) for \(s\ge0\). The function \(R_0(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\) is the Laplace transform of the clock \(T\), which exists with no hypothesis on the zeros, and for \(t\) at least the de Bruijn Newman constant \(\Lambda\), \(R_t\) is the Laplace transform of a generalized gamma convolution \(T_t\), the tilted clock. Let \(Z\) be a standard normal variable. Then for every \(t>0\), \(H_t(z)=\mathbb{E}\,H_0\bigl(z+i\sqrt{2t}\,Z\bigr)\) for all \(z\in\mathbb{C}\), and \(\dfrac1{R_t(\sigma^2)}=\dfrac{\mathbb{E}\,\xi\bigl(\frac12+\sigma+\sqrt{t/2}\,Z\bigr)}{\mathbb{E}\,\xi\bigl(\frac12+\sqrt{t/2}\,Z\bigr)}\) for \(\sigma\ge0\). So for \(t>0\) the tilt is Gaussian convolution, along the real axis of \(\sigma\), of the even function \(\sigma\mapsto\xi(\frac12+\sigma)\), with variance \(t/2\), normalised at \(\sigma=0\).

Earlier editions: Proposition 8.4.18, page 253 (edition 1).

Source: ch/ch21.tex, label xg:prop:gauss.

The mean of the tilted clock increases with the tilt

Volume II, Result 4.1.11, page 77 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(v)=2\sum_{n\ge1}\bigl(2\pi^2n^4e^{9v/2}-3\pi n^2e^{5v/2}\bigr)\exp(-\pi n^2e^{2v})\) be the Pólya kernel, \(\Phi_{\mathrm{dBN}}(u)=\frac12\Phi(2u)\), and \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\) the de Bruijn Newman function, so that \(8H_t(2\theta)=\int_{\mathbb{R}}\Phi(v)e^{tv^2/4}\cos(\theta v)\,dv\). Let \(M_k(t)=\int_{\mathbb{R}}v^{2k}e^{tv^2/4}\Phi(v)\,dv\), and for those \(t\) at which \(R_t(u)=H_t(0)/H_t(2i\sqrt u)\) is the Laplace transform of a law \(T_t\) (the tilted clock, which exists at \(t=0\) and for every \(t\ge\Lambda\), so for every \(t\ge0.2\), with \(\Lambda\) the de Bruijn Newman constant), with mean \(\mathbb{E}T_t=M_1(t)/(2M_0(t))\). Since \(M_k'(t)=\frac14M_{k+1}(t)\), \(\frac{d}{dt}\mathbb{E}T_t=\frac{M_0(t)M_2(t)-M_1(t)^2}{8M_0(t)^2}>0\) by the Cauchy Schwarz inequality, at every \(t\) at which \(T_t\) exists.

Earlier editions: Result 8.4.8, page 254 (edition 1).

Source: ch/ch21.tex, label res:ch21:meanup.

The tilted clock is not monotone under the common exponential coupling

Volume II, Result 4.1.11, page 77 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. For \(t\ge0.2\) every zero of the de Bruijn Newman function \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) with \(\Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\bigr)\exp(-\pi n^2e^{4u})\) is real (Platt and Trudgian 2021), so the tilted clock is \(T_t=\sum_jE_j/a_j(t)\) with \(a_j(t)\) the squares of the half zeros \(z_j(t)/2\) (the Thorin atoms) and \(E_j\) independent standard exponential variables, the same for every \(t\) (the common coupling). By the equation of motion of the zeros, computed with Gauss Hermite quadrature in double precision (not certified; the two velocities are unchanged to \(3\times10^{-17}\) between \(60\) and \(90\) nodes), \(\dot z_{64}(0.2)=+0.26897\) and \(\dot z_{92}(0.2)=+0.13385\), the only two rising zeros among the first \(100\), so \(a_{64}\) and \(a_{92}\) increase. Hence for \(h>0\) small enough, \(T_{0.2+h}<T_{0.2}\) with positive probability, since \(T_{0.2+h}-T_{0.2}=cE_{64}+Y\) with \(c<0\) and \(Y\) independent of \(E_{64}\) and the exponential variable \(E_{64}\) is unbounded; so the coupling is not almost surely monotone on any right neighbourhood of \(t=0.2\). At \(t=0\) the same argument with \(\dot z_{35}(0)>0\) of the velocity formula (certified in ball arithmetic) gives the statement on every right neighbourhood of \(0\), conditional on RH, which the coupling at \(t=0\) needs. It does not decide whether the tilted clock is stochastically increasing in \(t\).

Earlier editions: Result 8.4.8, page 255 (edition 1).

Source: ch/ch21.tex, label res:ch21:coupling.

Problem: is the tilted clock stochastically increasing in the tilt

Volume II, Problem 4.2, page 77 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) be the de Bruijn Newman function, with \(\Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\bigr)\exp(-\pi n^2e^{4u})\), \(\Lambda\) the de Bruijn Newman constant, and for \(t\ge\Lambda\) (so unconditionally for \(t\ge0.2\)) let \(T_t\) be the tilted clock, the nonnegative variable with Laplace transform \(R_t(u)=H_t(0)/H_t(2i\sqrt u)\). Problem: is \(T_t\) stochastically increasing in \(t\) on \(t\ge\Lambda\), that is, is \(P(T_t\le x)\), whose Laplace transform in \(x\) is \(R_t(u)/u\), nonincreasing in \(t\) for every \(x>0\)? The failure of the common exponential coupling does not decide it, since another coupling may be monotone, and an increasing mean is necessary but not sufficient.

Earlier editions: Problem 8.2, page 255 (edition 1).

Source: ch/ch21.tex, label prob:ch21:order.

Large tilt of the clock: the limit is twice the Brownian exit time

Volume II, Proposition 4.1.27, page 78 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\bigr)\exp(-\pi n^2e^{4u})\), \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) (so that \(H_0(z)=\frac18\xi(\frac12+\frac{iz}2)\) for Riemann’s completed zeta function \(\xi\)) and \(R_t(s)=H_t(0)/H_t(2i\sqrt s)\). Let \(\Lambda\) be the de Bruijn Newman constant, so that \(H_t\) has only real zeros exactly when \(t\ge\Lambda\). For \(t\ge\Lambda\), in particular for \(t\ge0.2\) (the bound \(\Lambda\le0.2\) of Platt and Trudgian 2021), let \(0<z_1(t)\le z_2(t)\le\cdots\) be the positive zeros of \(H_t\); then \(R_t\) is the Laplace transform of a generalized gamma convolution \(T_t\), the tilted clock, with Thorin atoms \(z_j(t)^2/4\). For large \(t\) let \(u_t\) be the mode of \(u\mapsto e^{tu^2}\Phi(u)\) on \((0,\infty)\), so that \(e^{4u_t}\sim tu_t/(2\pi)\) as \(t\to\infty\). Let \(C\) be the exit time of standard Brownian motion from \((-1,1)\), with \(\mathbb{E}e^{-\theta C}=1/\cosh\sqrt{2\theta}\). Then as \(t\to\infty\), \(R_t(s/u_t^2)\to1/\cosh(2\sqrt s)\) for every \(s\ge0\), \(T_t/u_t^2\) converges in law to \(2C\) (whose Laplace transform is \(1/\cosh(2\sqrt s)\)), and \(u_tz_j(t)\to\pi(j-\frac12)\) for every \(j\ge1\).

Earlier editions: Proposition 8.4.19, page 255 (edition 1).

Source: ch/ch21.tex, label xg:prop:large.

The tilted clock is increasing in the Laplace transform order

Volume II, Proposition 4.1.26, page 78 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mathbb{E}_t\) and \(\operatorname{Var}_t\) be expectation and variance under the probability density proportional to \(e^{tu^2}\Phi_{\rm dBN}(u)\) on \((0,\infty)\), let \(R_t(s)=1/\mathbb{E}_t\cosh(2\sqrt s\,u)\) be the Laplace transform of the tilted clock \(T_t\), and let \(\Lambda\) be the de Bruijn Newman constant. For every \(s>0\) the function \(t\mapsto R_t(s)\) is strictly decreasing on \(\mathbb R\), and at every \(t\) at which \(T_t\) exists, \(\frac d{dt}\mathbb{E} T_t=2\operatorname{Var}_t(u^2)>0\). Hence for \(t\ge\Lambda\) (so, unconditionally, for \(t\ge0.2\)) the clock \(T_t\) is strictly increasing in \(t\) in the Laplace transform order: \(\mathbb{E} e^{-sT_t}\) decreases in \(t\) for every \(s>0\).

Source: ch/ch21.tex, label prop:ch21:ltorder.

Gaussian smoothing of a reciprocal GGC transform is again one

Volume II, Theorem 4.1.28, page 79 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A generalized gamma convolution (GGC) with Thorin measure \(U\ge0\) and no drift is a law on \([0,\infty)\) with Laplace transform \(\exp\bigl(-\int_{(0,\infty)}\log(1+s/z)\,U(dz)\bigr)\). Let \(a_j>0\) with \(\sum_ja_j^{-1}<\infty\), and let \(M(\sigma)=\prod_j(1+\sigma^2/a_j)\), an even entire function of genus zero in \(\sigma^2\), so that \(1/M(\sqrt s)\) is the Laplace transform of the GGC with the discrete Thorin measure \(\sum_j\delta_{a_j}\). Let \(Z\) be a standard normal variable and, for \(c>0\), \(M_c(\sigma)=\mathbb{E}\,M(\sigma+cZ)\big/\mathbb{E}\,M(cZ)\). Then \(M_c(\sigma)=\prod_j(1+\sigma^2/a_j(c))\) with real \(a_j(c)>0\), so that \(1/M_c(\sqrt s)\) is the Laplace transform of a GGC \(S_\lambda\) with the Thorin measure \(\sum_j\delta_{a_j(c)}\), where \(\lambda=2c^2\). While the atoms are simple they obey, as functions of \(\lambda\), the motion equation \(\dfrac{da_j}{d\lambda}=\dfrac12+2\sum_{k\ne j}\dfrac{a_j}{a_j-a_k}\), and \(\dfrac{d}{d\lambda}\mathbb{E}S_\lambda=(\mathbb{E}S_\lambda)^2-\dfrac32\operatorname{Var}S_\lambda\).

Earlier editions: Theorem 8.4.20, page 256 (edition 1).

Source: ch/ch21.tex, label xg:thm:smooth.

Concentration of the Laguerre expressions at the origin along the whole de Bruijn Newman flow, with positivity of \(L_{n,t}\) on every window for large \(n\) (Newton 2026, Zenodo)

Volume II, Theorem 4.1.29, page 80 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}\) be the de Bruijn Newman kernel, the even function with \(H_t(z)=\int_{\mathbb{R}}\nu_t(u)e^{izu}\,du\), \(\nu_t(u)=\frac12\Phi_{\mathrm{dBN}}(u)e^{tu^2}\), and let \(L_{n,t}\) be the Laguerre expressions of \(H_t\) of the book, so that \(L_{n,t}(x)/L_{n,t}(0)=\mathbb{E}_{n,t}\cos(xw)\), where under \(\mathbb{E}_{n,t}\) the pair \((u,v)\) has the density proportional to \(b^{2n}\nu_t(u)\nu_t(v)\), where \(b=u-v\) and \(w=u+v\). Then \(\lim_{n\to\infty}\sup_{0\le t\le1/2}\mathbb{E}_{n,t}[w^2]=0\). Consequently, for every \(X<\infty\) there is \(N_X\) with \(L_{n,t}(x)>0\) for all \(n\ge N_X\), \(|x|\le X\) and \(0\le t\le\frac12\). The index \(N_X\) is not effective. The result is due to Newton (2026, Zenodo).

Earlier editions: Theorem 8.4.21, page 258 (edition 1).

Source: ch/ch21.tex, label nw:thm:conc.

Compact uniform asymptotic of the Laguerre ratio \(k_nL_{n+1,t}(x)/L_{n,t}(x)\) along the whole de Bruijn Newman flow (Newton 2026, Zenodo)

Volume II, Theorem 4.1.31, page 81 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(L_{n,t}\) be the Laguerre expressions of the de Bruijn Newman function \(H_t\), \(k_n\) their normalising constants and \(c_n(t)=k_nL_{n+1,t}(0)/L_{n,t}(0)\) (as in the book). For every \(X<\infty\), uniformly in \(0\le t\le\frac12\) and \(|x|\le X\), \(\frac{k_nL_{n+1,t}(x)}{L_{n,t}(x)}=c_n(t)(1+o(1))=\frac18(\log n)^2(1+o(1))\) as \(n\to\infty\). The first equality follows from the concentration theorem for \(\mathbb{E}_{n,t}[w^2]\) and the second from the Lambert centring \(c_n(t)=\frac18\mathcal{W}(n/\pi)^2+o(\log n)\), which the book proves only by a sketch. The result is due to Newton (2026, Zenodo).

Earlier editions: Theorem 8.4.23, page 259 (edition 1).

Source: ch/ch21.tex, label nw:thm:compact.

Splitting of a multiple real zero of the de Bruijn Newman function by Hermite roots

Volume II, Lemma 4.1.32, page 82 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\rm dBN}(u)\cos(zu)\,du\) be the de Bruijn Newman family, with \(\Phi_{\rm dBN}\) the Pólya kernel, entire in \((z,t)\). Let \(c\) be a real zero of multiplicity \(m\ge1\) of \(H_{t^*}\). For \(t<t^*\) close to \(t^*\) the \(m\) zeros of \(H_t\) near \(c\) are \(c+i\sqrt{2(t^*-t)}\,x_k+o(\sqrt{t^*-t})\), \(k=1,\dots,m\), with \(x_k\) the roots of the Hermite polynomial \(\mathrm{He}_m\), so at most one of them is real; for \(t>t^*\) close to \(t^*\) all \(m\) of them are real and simple. Consequently real zeros of \(H_t\) never collide as \(t\) increases, a real zero stays real, a simple real zero stays simple except at times when zeros arriving from the complex plane meet it, and nonreal zeros of \(H_\tau\) cannot accumulate at a real point as \(\tau\) decreases to a limit \(\tau_1\).

Source: ch/ch21.tex, label ft:lem:split.

If the de Bruijn Newman constant is positive, the zeros at that time include a multiple real zero

Volume II, Corollary 4.1.33, page 83 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)\exp(-\pi n^2e^{4u})\) and \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\), entire in \(z\) and \(t\), so that \(H_0(z)=\frac18\Xi(z/2)\) with \(\Xi(t)=\xi(\frac12+it)\) and \(\xi\) Riemann’s xi function. Let \(\Lambda\) be the de Bruijn Newman constant, so that \(H_t\) has only real zeros exactly for \(t\ge\Lambda\) (Newman; \(\Lambda\le\frac12\) by de Bruijn). If \(\Lambda>0\), then \(H_\Lambda\) has a multiple real zero. Hence the Riemann Hypothesis holds if and only if \(H_t\) has only simple zeros for every \(t\in(0,0.2]\). The proof uses Theorem 1.5 of the Polymath project (for an absolute constant \(C\) and every \(0<t\le\frac12\) every zero \(x+iy\) of \(H_t\) with \(|x|\ge e^{C/t}\) is real and simple) and the bound \(\Lambda\le0.2\) of Platt and Trudgian when the Riemann Hypothesis fails.

Earlier editions: Corollary 8.4.25, page 260 (edition 1).

Source: ch/ch21.tex, label ft:cor:collide.

Real simple zeros of the de Bruijn Newman function below a height from the count barrier hypothesis

Volume II, Proposition 4.1.34, page 83 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the hypothesis B(X) of a count barrier on rectangles at grid times, which the book states and has not verified at any height.

Statement. Let \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\rm dBN}(u)\cos(zu)\,du\) be the de Bruijn Newman family, with \(\Phi_{\rm dBN}\) the Pólya kernel. For a height \(X\) let \(0=t_0<t_1<\dots<t_M=0.2\) and let \(Q_i=[u_i,v_i]\times[-1,1]\) be rectangles in the \(z\) plane with \(u_i<X<v_i\). The hypothesis \(B(X)\) is: (B0) every zero of \(H_0\) with \(|\operatorname{Re}z|\le u_0\) is real and simple; (B1) for each \(i\), \(H_{t_i}\) has no zero on \(\partial Q_i\), and the number of zeros of \(H_{t_i}\) in \(Q_i\) (counted with multiplicity by the argument principle) is at most the number of sign changes of \(H_{t_i}\) observed on \([u_i,v_i]\); (B2) for each \(i\) and every \(t\in[t_i,t_{i+1}]\), \(|H_t-H_{t_i}|<|H_{t_i}|\) on \(\partial Q_i\). Assume \(B(X)\). Then for every \(\lambda\in[0,0.2]\) every zero of \(H_\lambda\) with \(|\operatorname{Re}z|\le X\) is real and simple. The book states that \(B(X)\) has not been verified at any height \(X\) and claims no computation.

Source: ch/ch21.tex, label ft:prop:below.

Local Rouché criterion for backward persistence of an adjacent pair of real simple zeros of the de Bruijn Newman function (Newton 2026, Zenodo)

Volume II, Proposition 4.1.35, page 84 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H_t(z)=\int_{\mathbb{R}}\nu_t(u)e^{izu}\,du\) be the de Bruijn Newman function with \(\nu_t(u)=\frac12\Phi_{\mathrm{dBN}}(u)e^{tu^2}\) and \(\Phi_{\mathrm{dBN}}\) the even de Bruijn Newman kernel, let \(t_*\ge0\), and let \(a<b\) be adjacent real simple zeros of \(H_{t_*}\) with \(g_*=b-a\). Let \(0<r<g_*/2\) and let \(\Omega\) be a conjugation invariant Jordan domain containing the closed discs \(D_a\), \(D_b\) of radius \(r\) about \(a\), \(b\) and the segment between them, with no other zero of \(H_{t_*}\) in \(\overline\Omega\). Put \(\Gamma=\partial D_a\cup\partial D_b\cup\partial\Omega\), \(Y=\max_\Gamma|\operatorname{Im}z|\), \(m=\min_\Gamma|H_{t_*}|\) and \(\psi_Y(\sigma)=\int_0^\infty\Phi_{\mathrm{dBN}}(u)e^{t_*u^2+Yu}(1-e^{-\sigma u^2})\,du\). If \(\psi_Y(\sigma)<m\), then for \(0\le\tau\le\sigma\) the function \(H_{t_*-\tau}\) has exactly one zero in each of \(D_a\), \(D_b\), both real and simple, and no other zero in \(\Omega\); so the gap \(g\) of the pair satisfies \(g\ge g_*-2r\) and, with \(\mathrm{Def}\) the shielding deficit and \(\frac{d(g^2)}{dt}=4\,\mathrm{Def}\), \(4\int\mathrm{Def}\,dt\le g_*^2-(g_*-2r)^2\) over the interval. The result is due to Newton (2026, Zenodo).

Earlier editions: Proposition 8.4.27, page 262 (edition 1).

Source: ch/ch21.tex, label nw:prop:local.

Obstruction at height: the absolute value Rouché criterion certifies only backward steps of size \(O(e^{-\delta a})\) at a real zero \(a\) of the de Bruijn Newman function (Newton 2026, Zenodo, mechanism corrected)

Volume II, Proposition 4.1.36, page 85 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With the notation of the local Rouché criterion (\(H_t\) the de Bruijn Newman function, \(\Phi_{\mathrm{dBN}}\) its even kernel, \(\psi_Y(\sigma)=\int_0^\infty\Phi_{\mathrm{dBN}}(u)e^{t_*u^2+Yu}(1-e^{-\sigma u^2})\,du\), \(m=\min_\Gamma|H_{t_*}|\) on a contour \(\Gamma\)), fix \(t_*\ge0\), \(\sigma>0\) and \(0<\delta<\pi/8\), and let \(C(\delta,t)=\frac12\int_{\mathbb{R}}|\Phi_{\mathrm{dBN}}(u+i\delta)|e^{t(u^2-\delta^2)+|u|}\,du<\infty\). For every real zero \(a\ge1\) of \(H_{t_*}\) and every contour \(\Gamma\) containing a point \(a+i\eta\) with \(0<\eta\le1\), \(\min_\Gamma|H_{t_*}|\le C(\delta,t_*)e^{-\delta a}<\psi_0(\sigma)\le\psi_Y(\sigma)\) as soon as \(a>\delta^{-1}\log\big(C(\delta,t_*)/\psi_0(\sigma)\big)\). So no criterion of the form \(\psi_Y(\sigma)<m\) certifies a fixed step \(\sigma>0\) at every height, and the step it can certify at height \(a\) is \(O(e^{-\delta a})\). The book notes that this concerns bounds through the absolute value of the integrand only. The result is due to Newton (2026, Zenodo), with the mechanism corrected in the book.

Earlier editions: Proposition 8.4.28, page 262 (edition 1).

Source: ch/ch21.tex, label nw:prop:height.

Volume II, Chapter 5. The Thorin phase diagram

Threshold theorem along the flow for the shifted clocks of the heat flow function

Volume II, Theorem 5.2.3, page 91 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\), \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\) (so \(H_0(z)=\frac18\xi(\frac12+\frac{iz}2)\)) and \(\xi_t(s)=8H_t(-2i(s-\frac12))=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), so that \(\xi_0=\xi\). For \(t\ge0\) and \(\alpha\ge\frac12\) let \(\varphi_{t,\alpha}(u)=\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\) for \(u\ge0\), the shifted time \(t\) clock (for \(\alpha=\frac12\) it is \(R_t(u)=H_t(0)/H_t(2i\sqrt u)\), the transform of the centre clock of the flow). A generalized gamma convolution (GGC) is a random variable \(Y\ge0\) with \(\mathbb{E}e^{-uY}=\exp\{-au-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\}\) for \(u>0\), a drift \(a\ge0\) and a Thorin measure \(U\). Then \(\varphi_{t,\alpha}\) is the Laplace transform of a GGC if and only if \(\alpha\ge\sup_\rho\operatorname{Re}\rho\) over the zeros \(\rho\) of \(\xi_t\). In that case there is no drift, and the Thorin measure, written in the variable \(\theta=\sqrt z\) on \(\theta>0\), is \(\sum_{\rho=\beta+i\gamma,\ \gamma>0,\ \beta<\alpha}\bigl[P_{\alpha-\beta}(\theta-\gamma)+P_{\alpha-\beta}(\theta+\gamma)\bigr]\,d\theta+\sum_{\rho=\alpha+i\gamma,\ \gamma>0}m_\rho\,\delta_\gamma\), where the sums run over the zeros \(\rho=\beta+i\gamma\) of \(\xi_t\) with \(\gamma>0\) (in the first sum each zero is repeated according to its multiplicity \(m_\rho\)), \(P_a(x)=a/(\pi(a^2+x^2))\) is the Cauchy density of scale \(a>0\) and \(\delta_\gamma\) is the point mass at \(\gamma\). For \(t=0\) this is the threshold theorem for the shifted clock of \(\xi\).

Earlier editions: Theorem 15.2.3, page 615 (edition 1).

Source: ch/cp2.tex, label cp:thm:transfer.

Minimum at the origin of the Thorin density of the shifted clocks along the heat flow

Volume II, Proposition 5.2.5, page 92 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\) the heat flow deformation of \(\xi\) (so \(\xi_0=\xi\)). Let \(\Lambda\le0.2\) (Platt and Trudgian 2021) be the de Bruijn Newman constant, and let \(t\ge\Lambda\), so that the zeros of \(\xi_t\) are \(\frac12\pm i\theta_k\) with \(0<\theta_1\le\theta_2\le\cdots\) (all \(\theta_k=\theta_k(t)\) real, counted with multiplicity). Put \(T_t\) for the positive random variable with \(\mathbb{E}e^{-uT_t}=\prod_k(1+u/\theta_k^2)^{-1}=\xi_t(\frac12)/\xi_t(\frac12+\sqrt u)\), so that \(\mathbb{E}T_t=\sum_k\theta_k^{-2}\) and \(\operatorname{Var}T_t=\sum_k\theta_k^{-4}\), and \(V_t=(\log\xi_t)''(\frac12)=2\sum_k\theta_k^{-2}=2\mathbb{E}T_t\). For \(\alpha>\frac12\) let \(a=\alpha-\frac12\), \(v_t(\alpha,\theta)=\operatorname{Re}(\xi_t'/\xi_t)(\alpha+i\theta)\), \(b_t(\alpha)=v_t(\alpha,0)\), \(v(\theta)=v_t(\alpha,\theta)\) and \(b=b_t(\alpha)\). The shifted clock is \(\varphi_{t,\alpha}(u)=\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\), and \(\tilde\varphi_{t,\alpha}(u)=\varphi_{t,\alpha}(u)e^{b\sqrt u}\) is the shifted clock with the constant part of its Thorin density (which is \(v(\theta)/\pi\) in the variable \(\theta=\sqrt z\)) removed; \(\tilde\varphi_{t,\alpha}\) is the Laplace transform of a generalized gamma convolution (GGC) exactly when \(v(\theta)\ge b\) for every \(\theta\). Then (i) \(b\le aV_t\). (ii) If \(a^2\le\frac35\theta_1^2\), then \(v(\theta)\ge b\) for \(|\theta|\le\theta_1\); if \(a\le\frac12\) and \(V_t<4\), then \(v(\theta)>b\) at every \(\theta\) within \(D_t=(1/V_t-\frac14)^{1/2}\) of some \(\theta_k\). (iii) Let \(a\le\frac12\) and \(V_t<4\). If, for some \(T_*\ge\theta_1\), every \(\theta\in[\theta_1,T_*]\) is within \(D_t\) of some \(\theta_k\), and for every \(\theta>T_*\) some \(w>0\) has \(\#\{k:|\theta_k-\theta|\le w\}>V_t(w^2+\frac14)\), then \(v\ge b\) (and then \(\tilde\varphi_{t,\alpha}\) is a GGC transform for every \(\alpha\in[\frac12,1]\)). (iv) Let \(\alpha'>\frac12\) and \(a'=\alpha'-\frac12\) with \(a'^2\le\frac35\theta_1^2\). If \(v_t(\alpha',\theta)\ge a'V_t\) for every \(\theta\ge\theta_1\), then \(v_t(\alpha,\cdot)\ge b_t(\alpha)\) for every \(\alpha\in(\frac12,\alpha']\) (and then \(\tilde\varphi_{t,\alpha}\) is a GGC transform for every \(\alpha\in[\frac12,\alpha']\)). (v) As \(\theta\to0\), \((v(\theta)-b)/\theta^2\to\sum_k2a(3\theta_k^2-a^2)/(a^2+\theta_k^2)^3=6a\operatorname{Var}T_t+O(a^3)\). For \(\alpha=\frac12\), \(b_t(\frac12)=0\) and \(\tilde\varphi_{t,1/2}=\varphi_{t,1/2}\).

Earlier editions: Proposition 15.2.5, page 616 (edition 1).

Source: ch/cp2.tex, label cp:prop:minflow.

Heat flow function as a shifted Gaussian average of Riemann’s xi function

Volume II, Lemma 5.2.6, page 93 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=g(s)\zeta(s)\) be Riemann’s xi function with \(g(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\), put \(\ell=g'/g\), let \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\) with \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) be the heat flow function (\(\xi_0=\xi\)), and let \(Z\) be standard normal. For \(t\ge0\) and \(\operatorname{Im}s>0\), \(\xi_t(s)=g(s)\,e^{t\ell(s)^2/4}\,\zeta_t(s)\) with \(\zeta_t(s)=\mathbb{E}\,\Psi\bigl(s,\frac t2\ell(s)+\sqrt{t/2}\,Z\bigr)\) and \(\Psi(s,h)=\xi(s+h)/(g(s)e^{h\ell(s)})\). Moreover \(\Psi(s,h)=\zeta(s+h)e^{r(s,h)}\) with \(r(s,h)=h^2\int_0^1(1-u)\,\ell'(s+uh)\,du\), and if \(Y=\min\bigl(\operatorname{Im}s,\operatorname{Im}s+\frac t2\operatorname{Im}\ell(s)\bigr)>0\), then \(|r(s,h)|\le|h|^2\bigl(\frac1{4Y}+\frac7{4Y^2}\bigr)\) for every \(h\in\frac t2\ell(s)+\mathbb{R}\).

Earlier editions: Lemma 15.2.6, page 617 (edition 1).

Source: ch/cp2.tex, label cp:lem:shift.

Zero count of the heat flow function and a Backlund type bound on its argument term

Volume II, Proposition 5.2.7, page 94 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=g(s)\zeta(s)\) be Riemann’s xi function with \(g(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\), \(\ell=g'/g\), and \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\) with \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) the heat flow function (\(\xi_0=\xi\)), and \(\zeta_t=\xi_t/(g\,e^{t\ell^2/4})\). Let \(t\ge0\) and let \(N_t(T)\) be the number of zeros \(\rho\) of \(\xi_t\) with \(0<\operatorname{Im}\rho\le T\), counted with multiplicity whether or not they lie on the critical line. For \(T>0\) not an ordinate let \(\pi S_t(T)\) be the change of a continuous argument of \(\zeta_t\) along the segments from \(2\) to \(2+iT\) and from \(2+iT\) to \(\frac12+iT\), starting from \(\arg\zeta_t(2)=0\) (at ordinates take right limits); \(S_0=S\) is the usual argument term of the Riemann zeta function. (i) \(N_t(T)=1+\vartheta(T)/\pi+\frac t{4\pi}\operatorname{Im}\bigl(\ell(\frac12+iT)^2\bigr)+S_t(T)\), where \(\vartheta\) is the Riemann Siegel theta function. (ii) The main term is \(\frac T{2\pi}\log\frac T{2\pi e}+\frac78+\frac t{16}\log\frac T{2\pi}+O\bigl((1+t\log T)/T\bigr)\). (iii) Let \(T>\frac52\). If \(\operatorname{Re}\zeta_t(2+iy)>0\) for \(0\le y\le T\), and \(M(\omega)\ge\max\bigl(|\zeta_t(2+iT+\frac52e^{i\omega})|,|\zeta_t(2+iT+\frac52e^{-i\omega})|\bigr)\) for \(0\le\omega\le2\pi\), then \(|S_t(T)|\le\frac12+\frac1{\log(5/3)}\Bigl(\frac1{2\pi}\int_0^{2\pi}\log M(\omega)\,d\omega-\log\operatorname{Re}\zeta_t(2+iT)\Bigr)\).

Earlier editions: Proposition 15.2.7, page 618 (edition 1).

Source: ch/cp2.tex, label cp:prop:count.

GGC property at time 0.2 of the shifted clock with the constant part of its Thorin density removed, for basepoints up to 10, conditional on a computed bound

Volume II, Corollary 5.2.8, page 95 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The inequality \(v_{0.2}(10,\theta)\ge\frac{19}2V_{0.2}\) for every \(\theta\ge\theta_1(0.2)\), where \(v_{0.2}(10,\theta)=\operatorname{Re}(\xi_{0.2}'/\xi_{0.2})(10+i\theta)\). The book computes it only for \(\theta_1(0.2)\le\theta\le10^4\) on a grid of step \(0.05\) with a Gauss Hermite quadrature whose error is not bounded, so it is not certified there and not proved for larger \(\theta\) (Problem on the bound for \(\zeta_{0.2}'/\zeta_{0.2}\) on the line \(\operatorname{Re}s=10\)). The proof also uses \(\frac35\theta_1(0.2)^2\ge(\frac{19}2)^2\), which the book takes from the tabulated value \(\operatorname{Var}T_{0.2}=3.77203\times10^{-5}\) (computed, not certified) through \(\theta_1(0.2)\ge(\operatorname{Var}T_{0.2})^{-1/4}=12.76\).

Statement. Let \(\xi_t\) be the heat flow function (\(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\) with \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\), \(\xi_0=\xi\) Riemann’s xi function). Its zeros at time \(0.2\) are \(\frac12\pm i\theta_k(0.2)\), \(0<\theta_1(0.2)\le\theta_2(0.2)\le\cdots\) (since the de Bruijn Newman constant satisfies \(\Lambda\le0.2\), Platt and Trudgian 2021), and \(V_{0.2}=(\log\xi_{0.2})''(\frac12)=2\sum_k\theta_k(0.2)^{-2}\). For \(\alpha\ge\frac12\) let \(v_{0.2}(\alpha,\theta)=\operatorname{Re}(\xi_{0.2}'/\xi_{0.2})(\alpha+i\theta)\), \(b_{0.2}(\alpha)=v_{0.2}(\alpha,0)\) and \(\tilde\varphi_{0.2,\alpha}(u)=\frac{\xi_{0.2}(\alpha)}{\xi_{0.2}(\alpha+\sqrt u)}e^{b_{0.2}(\alpha)\sqrt u}\). Suppose that \(v_{0.2}(10,\theta)\ge\frac{19}2V_{0.2}\) for every \(\theta\ge\theta_1(0.2)\). Then \(\tilde\varphi_{0.2,\alpha}\) is the Laplace transform of a generalized gamma convolution for every \(\alpha\in[\frac12,10]\). The proof needs only \(\theta_1(0.2)\ge12.76\), which follows from \(\operatorname{Var}T_{0.2}=\sum_k\theta_k(0.2)^{-4}\ge\theta_1(0.2)^{-4}\), \(T_{0.2}\) being the positive variable with \(\mathbb{E}e^{-uT_{0.2}}=\xi_{0.2}(\frac12)/\xi_{0.2}(\frac12+\sqrt u)\), and the tabulated value \(\operatorname{Var}T_{0.2}=3.77203\times10^{-5}\) (computed, not certified). The hypothesis is computed, not certified, for \(\theta_1(0.2)\le\theta\le10^4\): on a grid of step \(0.05\), with a Gauss Hermite quadrature whose error is not bounded, the least margin is \(v_{0.2}(10,\theta)-\frac{19}2V_{0.2}=0.1163\), at \(\theta=14.0768\).

Earlier editions: Corollary 15.2.8, page 619 (edition 1).

Source: ch/cp2.tex, label cp:cor:minflow.

Phase diagram theorem, de Bruijn’s strip theorem in Thorin form

Volume II, Theorem 5.3.2, page 96 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be Riemann’s xi function, \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\), \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\) (so \(H_0(z)=\frac18\xi(\frac12+\frac{iz}2)\)) and \(\xi_t(s)=8H_t(-2i(s-\frac12))\), so that \(\xi_0=\xi\) and a zero \(x+iy\) of \(H_t\) is the zero \(\frac12-\frac y2+\frac{ix}2\) of \(\xi_t\). For \(t\ge0\) and \(\alpha\ge\frac12\) let \(\varphi_{t,\alpha}(u)=\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\) for \(u\ge0\) (the shifted time \(t\) clock), and let the GGC region \(\mathcal{G}\) be the set of points \((t,\alpha)\) with \(t\ge0\) and \(\alpha\ge\frac12\) at which \(\varphi_{t,\alpha}\) is the Laplace transform of a generalized gamma convolution; by the threshold theorem for the clocks of \(\xi_t\) this holds exactly when \(\xi_t\) has no zero with real part above \(\alpha\), that is, when every zero \(x+iy\) of \(H_t\) has \(|y|\le2(\alpha-\frac12)\). Let \(\Lambda\) be the de Bruijn Newman constant, so that \(H_t\) has only real zeros exactly when \(t\ge\Lambda\). (i) If \((t_0,\alpha_0)\in\mathcal{G}\), then \((t,\alpha)\in\mathcal{G}\) for every \(t\ge t_0\) and every \(\alpha\ge\frac12\) with \((\alpha-\frac12)^2+\frac t2\ge(\alpha_0-\frac12)^2+\frac{t_0}2\). (ii) \(\Lambda=\min\{t+2(\alpha-\frac12)^2:(t,\alpha)\in\mathcal{G}\}\). (iii) The Riemann Hypothesis (every zero of \(\xi\) has real part \(\frac12\)) holds if and only if \((0,\frac12)\in\mathcal{G}\), and if and only if for every \(\delta>0\) there is \(t\in[0,\delta]\) with \((t,\frac12+\sqrt{(\delta-t)/2})\in\mathcal{G}\). Part (i) is de Bruijn’s strip theorem (if the zeros of \(H_{t_0}\) lie in \(|\operatorname{Im}z|\le\Delta\), then the zeros of \(H_t\) lie in \((\operatorname{Im}z)^2\le\max(\Delta^2-2(t-t_0),0)\) for \(t\ge t_0\)) read through the threshold theorem, and parts (ii) and (iii) also use the theorem of Rodgers and Tao that \(\Lambda\ge0\).

Earlier editions: Theorem 15.3.2, page 620 (edition 1).

Source: ch/cp3.tex, label cp:thm:phase.

Winding number certificate for the point (0.2, 0.6) of the phase diagram

Volume II, Theorem 5.5.6, page 101 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), the de Bruijn Newman deformation of Riemann’s xi function \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) (so \(\xi_0=\xi\)), and \(H_t(z)=\frac18\xi_t(\frac12+\frac{iz}2)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\). The kernel \(\Phi_{\mathrm{dBN}}\) continues analytically to \(|\operatorname{Im}u|<\pi/8\). For \(0<\eta<\pi/8\) and real \(x\) put \(K_\eta(0.2,x)=4\int_{-\infty}^{\infty}\bigl|e^{0.2(u+i\eta)^2}\bigr|\,|\Phi_{\mathrm{dBN}}(u+i\eta)|\,e^{2xu}\,du\) and \(\bar K_\eta(0.2,R)=\max_{|x|\le R}K_\eta(0.2,x)\); they give the bound \(|\xi_{0.2}(\frac12+x+i\theta)|\le K_\eta(0.2,x)e^{-2\eta\theta}\) for \(\theta\ge0\). Let \(X>0\) be a barrier for the point \((0.2,0.6)\) in the variable of \(H_{0.2}\), as Polymath15 provide one (no script of the book checks it), that is, a number such that \(H_{0.2}\) has no zero \(z\) with \(\operatorname{Re}z\ge X\) and \(\operatorname{Im}z>0.2\), so that \(\xi_{0.2}\) has no zero with \(\operatorname{Re}s>0.6\) and \(\operatorname{Im}s\ge X/2\), and let \(Q=\{s:0.6\le\operatorname{Re}s\le0.9,\ 0\le\operatorname{Im}s\le X/2\}\). Let \(s_0,\dots,s_M\) be points of the left, top and right edges of \(Q\), in order along the boundary of \(Q\) from one bottom corner, \(s_0\), to the other, \(s_M\), and including the two top corners, and let \(B_j\) be ball enclosures of \(\xi_{0.2}(s_j)\) with radii \(\varepsilon_j\). Suppose that for every \(j\) there are \(r>0\) and a tilt \(\eta\in(0,\pi/8)\) (both may depend on \(j\)) such that \(|s_{j+1}-s_j|\,\frac{\bar K_\eta(0.2,\,0.4+r)}{r}\,e^{-2\eta(\theta_j-r)}+\varepsilon_j+\varepsilon_{j+1}<\min|B_j|\), where \(\theta_j=\min(\operatorname{Im}s_j,\operatorname{Im}s_{j+1})\), \(\min|B_j|\) is the smallest modulus of a point of the ball \(B_j\) and the factor \(e^{-2\eta(\theta_j-r)}\) may be replaced by \(1\) when \(\theta_j\le r\), and that \(\sum_j\operatorname{Arg}(B_{j+1}/B_j)\) (principal values), evaluated in ball arithmetic, lies within \((-\pi,\pi)\). (The bottom edge needs no points: \(\xi_t(\sigma)>0\) for every real \(\sigma\) and every \(t\ge0\), so the argument of \(\xi_{0.2}\) does not change along it.) Then \(\xi_{0.2}\) has no zero in \(Q\), hence no zero with real part above \(0.6\): the point \((0.2,0.6)\) lies in the GGC region (the shifted clock \(\xi_{0.2}(0.6)/\xi_{0.2}(0.6+\sqrt u)\) is the Laplace transform of a generalized gamma convolution), and the de Bruijn Newman constant satisfies \(\Lambda\le0.22\).

Earlier editions: Theorem 15.5.6, page 625 (edition 1).

Source: ch/cp5.tex, label cp:thm:cert.

Certificate in Thorin form, positive logarithmic derivative on a rectangle excludes zeros

Volume II, Proposition 5.5.7, page 102 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), the de Bruijn Newman deformation of Riemann’s xi function, and for real \(\sigma,\theta\) with \(\xi_t(\sigma+i\theta)\ne0\) put \(v_t(\sigma,\theta)=\operatorname{Re}\bigl(\xi_t'(\sigma+i\theta)/\xi_t(\sigma+i\theta)\bigr)\) (when the shifted clock \(\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\) is a generalized gamma convolution, \(v_t(\alpha,\theta)/\pi\) is its Thorin density in the variable \(\theta=\sqrt z\), away from atoms). Let \(t\ge0\), \(\frac12<\alpha<\sigma_1\) and \(T>0\), and suppose \(v_t(\sigma,\theta)>0\) for every \((\sigma,\theta)\in[\alpha,\sigma_1]\times[0,T]\) at which \(\xi_t(\sigma+i\theta)\ne0\). Then \(\xi_t\) has no zero \(\beta+i\gamma\) with \(\alpha<\beta\le\sigma_1\) and \(0<\gamma<T\).

Earlier editions: Proposition 15.5.7, page 626 (edition 1).

Source: ch/cp5.tex, label cp:prop:thorincert.

No zero of the time 0.2 function in the rectangle up to height 160, certified

Volume II, Theorem 5.5.8, page 104 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), the de Bruijn Newman deformation of Riemann’s xi function \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) (so \(\xi_0=\xi\)). Then \(\xi_{0.2}\) has no zero \(s\) with \(0.6\le\operatorname{Re}s\le0.9\) and \(0\le\operatorname{Im}s\le160\). This is certified by a computation in ball arithmetic (Arb, called from Python): \(\xi_{0.2}\) is evaluated by the trapezoid rule for its integral with an enclosure of the error of the rule, and the boundary of the rectangle \([0.6,0.9]\times[0,160]\) is walked counterclockwise, up the right edge, along the top and down the left edge, in \(23{,}219\) steps, with the step condition of the winding number certificate for the point \((0.2,0.6)\) (the step length times an explicit bound for \(|\xi_{0.2}'|\) along the step, plus the radii of the enclosures at its two ends, is below the smallest modulus of the enclosure at its start) verified at every step. The bottom edge is not walked, since \(\xi_{0.2}\) is real and positive on the real axis. The winding number of \(\xi_{0.2}\) around the rectangle is enclosed in \([0\pm3.47\times10^{-7}]\), so it is \(0\).

Earlier editions: Theorem 15.5.8, page 628 (edition 1).

Source: ch/cp5.tex, label cp:thm:rig.

Open problem on the candidate Thorin densities at times 0 and 0.2 near the Polymath15 barrier

Volume II, Problem 5.5, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), the de Bruijn Newman deformation of Riemann’s xi function (so \(\xi_0\) is the xi function), with \(H_t(z)=\frac18\xi_t(\frac12+\frac{iz}2)\), and put \(v_t(\alpha,\theta)=\operatorname{Re}\bigl(\xi_t'(\alpha+i\theta)/\xi_t(\alpha+i\theta)\bigr)\); when the shifted clock \(\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\) is the Laplace transform of a generalized gamma convolution, \(v_t(\alpha,\theta)/\pi\) is its Thorin density in the variable \(\theta=\sqrt z\), away from atoms. Problem: compute \(v_t(\alpha,\theta)\) at \(t=0\) and at \(t=0.2\) near the barrier of Polymath15, at heights \(\theta\) near \(3\times10^{10}\), with a Riemann Siegel type evaluator of \(H_{0.2}\) such as the effective approximation of Polymath15, and measure how the difference between the functions \(v_{0.2}/\pi\) and \(v_0/\pi\) grows with the height. The book computes this difference only for \(\theta\le100\), from the Pólya integral.

Earlier editions: Problem 15.1, page 635 (edition 1).

Source: ch/cp6.tex, label cp:prob:high.

Open problem on the boundary of the GGC region for 0 at most t at most 0.2

Volume II, Problem 5.6, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\) be the de Bruijn Newman deformation of Riemann’s xi function, with \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\). The GGC region \(\mathcal{G}\) is the set of points \((t,\alpha)\), \(t\ge0\), \(\alpha\ge\frac12\), at which the shifted clock \(\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\) is the Laplace transform of a generalized gamma convolution; this holds exactly when \(\alpha\ge\alpha^*(t)\), where \(\alpha^*(t)=\max(\frac12,\sup\operatorname{Re}\rho)\) over the zeros \(\rho\) of \(\xi_t\). Problem: describe the boundary \(\alpha^*(t)\) of \(\mathcal{G}\) for \(0\le t\le0.2\) as far as computation and the known zero free regions allow; is \(\alpha^*\) continuous, and is \((\alpha^*(t)-\frac12)^2+\frac t2\) strictly decreasing in \(t\) where \(\alpha^*(t)>\frac12\)? (By the phase diagram theorem it is nonincreasing there.)

Earlier editions: Problem 15.2, page 635 (edition 1).

Source: ch/cp6.tex, label cp:prob:boundary.

Open problem on a coordinate on the phase diagram for the symmetrised truncations

Volume II, Problem 5.7, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\) be the de Bruijn Newman deformation of Riemann’s xi function, with \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\), and let the GGC region \(\mathcal{G}\) be the set of points \((t,\alpha)\), \(t\ge0\), \(\alpha\ge\frac12\), at which \(\xi_t\) has no zero with real part above \(\alpha\), equivalently at which the shifted clock \(\xi_t(\alpha)/\xi_t(\alpha+\sqrt u)\) is the Laplace transform of a generalized gamma convolution. The symmetrised truncations are \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-1/2}F_n(\frac12-w)\) with \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\), where \(e_k,e_k'\) are independent standard exponential variables; Conjecture S says that, for every \(n\), all zeros of \(K_n\) in the strip \(0<\operatorname{Re}w<\frac12\) lie on the line \(\operatorname{Re}w=\frac14\), and at \(t=0\), \(\alpha=\frac12\) the Hurwitz argument shows that this for infinitely many \(n\) would imply the Riemann Hypothesis. The known part of \(\mathcal{G}\) consists of the points \((0,1)\) (the critical strip), \((0.2,0.6)\) (Polymath15) and a point on the level set \(t+2(\alpha-\frac12)^2=0.2\) (Polymath15 with the verification of Platt and Trudgian), conditionally on an announced zero free half plane also \((0,\frac78)\), and every point \((t,\alpha)\) with \(\alpha\ge\frac12\), \(t\ge t_0\) and \((\alpha-\frac12)^2+\frac t2\ge(\alpha_0-\frac12)^2+\frac{t_0}2\) for one of these points \((t_0,\alpha_0)\) (de Bruijn’s strip theorem in Thorin form). Problem: give the symmetrised truncations a coordinate on the phase diagram: for some \((t,\alpha)\) outside the known part, is there a family of explicit truncations of \(\xi_t\), built from finitely many terms of the Pólya integral or of a Dirichlet series and making no use of the zeros, as the \(K_n\) are, that converges to \(\xi_t\) uniformly on compact sets, together with heights \(h_n\to\infty\) such that for infinitely many \(n\) the \(n\)th truncation has no zero with \(\operatorname{Re}s>\alpha\) and \(|\operatorname{Im}s|<h_n\)? Hurwitz’s theorem would then make \((t,\alpha)\) a new point of \(\mathcal{G}\); at \(t=0\) and \(\alpha=\frac12\) the question for the \(K_n\) is Conjecture S, and for the finite stages of the Pólya density as a maximum of exponential variables it is the open problem posed with those stages.

Earlier editions: Problem 15.3, page 635 (edition 1).

Source: ch/cp6.tex, label cp:prob:trunc.

Open problem on the Cholesky factor of the Weil Gram matrix

Volume II, Problem 5.8, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(h\) smooth enough, with \(\hat h(r)=\int h(x)e^{irx}\,dx\), let \(W(h)=\hat h(\frac i2)+\hat h(-\frac i2)-\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\bigl(h(\log n)+h(-\log n)\bigr)+\frac1{2\pi}\int_{\mathbb{R}}\hat h(r)\operatorname{Re}\psi\bigl(\frac14+\frac{ir}2\bigr)\,dr-h(0)\log\pi\), where \(\Lambda\) is von Mangoldt’s function and \(\psi=\Gamma'/\Gamma\); by Weil’s explicit formula \(W(h)=\sum_\rho\hat h(\gamma_\rho)\) over the zeros \(\rho=\frac12+i\gamma_\rho\) of Riemann’s xi function. For \(g\) supported in \([-L/2,L/2]\) and \(h=g*\tilde g\) with \(\tilde g(x)=\overline{g(-x)}\), under the Riemann Hypothesis (every \(\gamma_\rho\) real) \(W(h)=\sum_\rho|\hat g(\gamma_\rho)|^2\ge0\) (Weil’s criterion). Take the hat functions \(g_j(x)=\max(0,1-|x-c_j|/w)\) at nodes \(c_j\) spaced \(w\) apart inside \([-L/2,L/2]\), \(N=L/w-1\) functions in all, and the Weil Gram matrix \(G_{jk}=W(g_j*\tilde g_k)\), which is Hermitian. Problem: compute the Cholesky factor of \(G\) for decreasing \(w\) and increasing \(L\), and decide whether its entries converge to an object built from the arithmetic, as the first test of a factorisation \(W(g*\tilde g)=\|Ag\|^2\) with \(A\) built from the arithmetic.

Earlier editions: Problem 15.4, page 635 (edition 1).

Source: ch/cp6.tex, label cp:prob:chol.

Open problem on a global functional of the zeros that is monotone along the flow

Volume II, Problem 5.9, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\), the de Bruijn Newman deformation of Riemann’s xi function (so \(H_0(z)=\frac18\xi(\frac12+\frac{iz}2)\)); while simple, its zeros \(z_k(t)\) move by \(dz_k/dt=2\sum_{j\ne k}(z_k-z_j)^{-1}\), like a log gas. Problem: find a global functional of the zeros of \(H_t\), insensitive to individual close pairs, that is monotone along the flow and whose positivity at \(t=0\) excludes collisions at positive time. Biane’s gap functional, the minimum over one gap of Laguerre’s quantity, is local; the functional asked for should instead be one quantity for all the zeros.

Earlier editions: Problem 15.5, page 635 (edition 1).

Source: ch/cp6.tex, label cp:prob:gas.

Open problem on explicit bounds for the heat flow zeta factor and the argument term up to time 0.2

Volume II, Problem 5.10, page 120 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi_t\) be the heat flow function (\(\xi_t(s)=8\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cosh((2s-1)u)\,du\), \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\), \(\xi_0=\xi\)), \(V_{0.2}=(\log\xi_{0.2})''(\frac12)\), \(\theta_1(0.2)\) the least positive ordinate of a zero of \(\xi_{0.2}\), \(v_{0.2}(10,\theta)=\operatorname{Re}(\xi_{0.2}'/\xi_{0.2})(10+i\theta)\), \(\zeta_t=\xi_t/(g\,e^{t\ell^2/4})\) with \(g(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\) and \(\ell=g'/g\), and \(S_t(T)\) the argument term of the zero count of \(\xi_t\) (the change of \(\arg\zeta_t\) along the segments from \(2\) to \(2+iT\) and \(2+iT\) to \(\frac12+iT\), divided by \(\pi\)). Problem (a): prove that \(v_{0.2}(10,\theta)\ge\frac{19}2V_{0.2}\) for every \(\theta\ge\theta_1(0.2)\), which would make the shifted clock of \(\xi_{0.2}\), with the constant part of its Thorin density removed, a GGC transform for every basepoint in \([\frac12,10]\); it suffices to bound \(|\zeta_{0.2}-1|\) on the discs of radius \(5\) about \(10+i\theta\) by a small constant uniform in \(\theta\) (computed: at the four sampled heights the largest of \(256\) sampled values of \(|\zeta_{0.2}-1|\) on each boundary circle is at most \(0.0326\), which is not a bound on the discs). Problem (b): for \(0<t\le0.2\) prove an explicit bound \(|S_t(T)|\le c_1\log T+c_2\) for \(T\ge T_1\), for instance by proving, with an explicit majorant \(M\), the two hypotheses of the Backlund type bound for \(S_t\) (\(\operatorname{Re}\zeta_t(2+iy)>0\) for \(0\le y\le T\), and \(M(\omega)\ge|\zeta_t(2+iT+\frac52e^{\pm i\omega})|\) for \(0\le\omega\le2\pi\)) (computed: the minimum of \(\operatorname{Re}\zeta_{0.2}(2+iy)\) over \(0\le y\le200\) is \(0.69006\), near \(y=14.015\)). At \(t=0\) much sharper explicit bounds are known (Hasanalizade, Shen and Wong 2022).

Earlier editions: Problem 15.6, page 635 (edition 1).

Source: ch/cp6.tex, label cp:prob:rvm.

Volume II, Chapter 6. The Cauchy deconvolution operator

Uniqueness of Cauchy deconvolution for admissible measures

Volume II, Proposition 6.1.1, page 127 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=(a/\pi)/(a^2+x^2)\) be the Cauchy density of width \(a>0\). Call a signed measure \(\mu\) on \(\mathbb{R}\) admissible when \(\int(1+x^2)^{-1}d|\mu|(x)<\infty\). If \(\mu_1,\mu_2\) are admissible and \(P_a*\mu_1=P_a*\mu_2\) then \(\mu_1=\mu_2\). The proof applies the semigroup property \(P_a*P_b=P_{a+b}\) to the difference, shows its Poisson extension vanishes for heights above \(a\) and hence on the whole upper half plane, and uses that an admissible measure is the vague boundary value of its Poisson integral.

Earlier editions: Proposition 33.1.1, page 1167 (edition 1).

Source: ch/cd1.tex, label cd:prop:unique.

The range of convolution with a Cauchy density on L2, unboundedness of the inverse, and a positive bounded datum outside the range

Volume II, Proposition 6.1.2, page 127 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=(a/\pi)/(a^2+x^2)\), \(a>0\), with Fourier transform \(\widehat f(\omega)=\int f(x)e^{-i\omega x}dx\), so \(\widehat{P_a}=e^{-a|\omega|}\), and let \(H^s\) be the Sobolev space of order \(s\). (a) For \(f\in L^2(\mathbb{R})\), \(\int e^{2a|\omega|}|\widehat{P_a*f}|^2d\omega=\int|\widehat f|^2d\omega\), and every \(u\) with \(\int e^{2a|\omega|}|\widehat u|^2d\omega<\infty\) equals \(P_a*f\) for some \(f\in L^2\). (b) For all real \(s,t\), \(\sup_f\|f\|_{H^t}/\|P_a*f\|_{H^s}=\infty\). (c) The function \(u(x)=2+\cos(x^2)\) is positive and bounded, its Poisson extension continues to an entire harmonic function, and yet \(u\ne P_a*\mu\) for every \(a>0\) and every measure \(\mu\) with \(\int(1+t^2)^{-1}d|\mu|(t)<\infty\).

Earlier editions: Proposition 33.1.2, page 1168 (edition 1).

Source: ch/cd1.tex, label cd:prop:range.

A positive datum with a signed source: the two atom example and its critical width

Volume II, Example 6.2.1, page 128 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=(a/\pi)/(a^2+x^2)\), \(d>0\), \(0<q<1\), \(a>0\) and \(\mu=\delta_0-q\delta_d\). Then \(P_a*\mu=P_a-qP_a(\cdot-d)\ge0\) on \(\mathbb{R}\) if and only if \(q\le(r-d)/(r+d)\) with \(r=\sqrt{d^2+4a^2}\). By uniqueness of the deconvolution, the signed measure \(\mu\) is then the only source of this datum at width \(a\). The partial deconvolution \(P_{a-c}*\mu\), \(0\le c<a\), is nonnegative if and only if \(q\le(r_c-d)/(r_c+d)\) with \(r_c=\sqrt{d^2+4(a-c)^2}\), that is, if and only if \(c\le c^*=a-\frac12\sqrt{m^2-d^2}\) with \(m=\frac{1+q}{1-q}d\). For \(q=9-4\sqrt5\), \(a=\frac12\) and \(d=1\) this gives \(c^*=\frac14\) exactly.

Earlier editions: Example 33.2.1, page 1169 (edition 1).

Source: ch/cd1.tex, label cd:ex:signed.

The critical width of a positive datum for Cauchy deconvolution is attained when finite

Volume II, Proposition 6.2.2, page 128 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_c(x)=(c/\pi)/(c^2+x^2)\) with \(P_0*\nu=\nu\), and let \(u\ge0\) satisfy \(\int u(x)(1+x^2)^{-1}dx<\infty\). Put \(c^*(u)=\sup\{c\ge0:\ u=P_c*\nu\text{ for some measure }\nu\ge0\}\in[0,\infty]\), so that \(c=0\) always qualifies. Then a positive source at width \(c\) exists for every finite \(c\le c^*(u)\) and for no \(c>c^*(u)\); in particular the supremum is attained when it is finite. The proof uses that the admissible widths form an interval and the Herglotz and Riesz characterisation of nonnegative harmonic functions on a half plane.

Earlier editions: Proposition 33.2.2, page 1169 (edition 1).

Source: ch/cd1.tex, label cd:prop:critical.

Share of the level one density explained by a positive source at width one half

Volume II, Proposition 6.2.3, page 129 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mathfrak u=u_1\,dv\) with \(u_1(v)=\pi^{-1}\Re(\xi'/\xi)(1+iv)\), let \(P_y\) be the Poisson kernel of width \(y\), let \(N(T)\) count the zeros \(\rho=\beta+i\gamma\) of \(\xi\) with \(0<\gamma\le T\) and \(N(\sigma,T)\) those among them with \(\beta>\sigma\). Then \(\mathfrak u=P_{1/2}*\nu+\mathfrak d\) with \(\nu\ge0\) and \(\mathfrak d\ge0\), where, summing over all zeros with multiplicity, \(\nu=\sum_{\beta<1/2}P_{1/2-\beta}(\cdot-\gamma)+\sum_{\beta=1/2}\delta_\gamma+\sum_{\beta>1/2}2(1-\beta)\,\delta_\gamma\) and \(\mathfrak d=\sum_{\beta>1/2}[P_{1-\beta}-2(1-\beta)P_{1/2}](\cdot-\gamma)\). Moreover \(\mathfrak d([-T,T])=4\int_{1/2}^1N(\sigma,T)\,d\sigma+O(\log^2T)=O(T)\) and \(\mathfrak u([-T,T])=2N(T)+O(\log^2T)\sim\frac T\pi\log T\). So the part of the mass of \(\mathfrak u\) on \([-T,T]\) explained by a positive source at width \(\frac12\) is \(1-O(1/\log T)\), unconditionally.

Source: ch/cd1.tex, label cd:prop:share.

A positive source exists at width c if and only if the harmonic extension continues as a nonnegative harmonic function to the half plane y > -c

Volume II, Theorem 6.2.4, page 130 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(u\ge0\) be continuous with \(\int u(x)(1+x^2)^{-1}dx<\infty\) and let \(h(x,y)=(P_y*u)(x)\), \(y>0\), be its Poisson (harmonic) extension to the upper half plane, with \(P_y\) the Cauchy density of width \(y\). For \(c>0\) the following are equivalent: (i) \(u=P_c*\nu\) for a measure \(\nu\ge0\); (ii) \(h\) extends to a nonnegative harmonic function on the half plane \(y>-c\). The proof of (ii) implies (i) uses the Herglotz and Riesz representation of nonnegative harmonic functions on a half plane and shows that its linear coefficient vanishes.

Earlier editions: Theorem 33.2.3, page 1170 (edition 1).

Source: ch/cd1.tex, label cd:thm:herglotz.

Gram decomposition of the Pick matrix of the logarithmic derivative of xi

Volume II, Lemma 6.2.5, page 131 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\lambda\in[\frac12,1)\), \(f=\xi'/\xi\), let \(z_j=1+i\tau_j\) (\(j=1,\dots,n\)) be distinct nodes and \(P^{(\lambda)}_{jk}=\dfrac{f(z_j)+\overline{f(z_k)}}{z_j+\overline{z_k}-2\lambda}\). For a zero \(\rho=\beta_\rho+i\gamma_\rho\) let \(k_\rho=((z_j-\rho)^{-1})_j\), let \(J\) be the all ones matrix, \(S_{jk}=(z_j+\overline{z_k}-2\lambda)^{-1}\), and let \(\circ\) be the entrywise product. Then, with the zeros summed in conjugate pairs, \(P^{(\lambda)}=\sum_\rho(k_\rho k_\rho^{*})\circ(J+2(\lambda-\beta_\rho)S)\), each entry converging absolutely, and \(S\) is positive semidefinite.

Source: ch/cd1.tex, label cd:lem:gram.

Pick matrix criterion for a zero free half plane of xi

Volume II, Theorem 6.2.6, page 131 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\lambda\in[\frac12,1)\) and let \(P^{(\lambda)}\) be the Pick matrix of \(f=\xi'/\xi\) at nodes \(z_j=1+i\tau_j\), with entries \(P^{(\lambda)}_{jk}=(f(z_j)+\overline{f(z_k)})/(z_j+\overline{z_k}-2\lambda)\). The following are equivalent. (i) \(\xi\) has no zero \(\rho\) with \(\Re\rho>\lambda\). (ii) \(u_1\,dv=P_{1-\lambda}*\nu\) for a measure \(\nu\ge0\), where \(u_1(v)=\pi^{-1}\Re f(1+iv)\) and \(P_y\) is the Poisson kernel of width \(y\). (iii) \(P^{(\lambda)}\succeq0\) for every finite set of distinct nodes on the line \(\Re z=1\). (iv) \(P^{(\lambda)}\succeq0\) for every finite set of distinct nodes in one segment \(\{1+i\tau:a\le\tau\le b\}\) with \(a<b\).

Source: ch/cd1.tex, label cd:thm:pick.

A negative Pick eigenvalue proves a zero of xi to the right of level lambda

Volume II, Corollary 6.2.7, page 132 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\lambda\in[\frac12,1)\) and let \(P^{(\lambda)}\) be the Pick matrix of \(\xi'/\xi\) at finitely many distinct nodes on the line \(\Re z=1\) (Theorem on the Pick criterion). If \(P^{(\lambda)}\) has a negative eigenvalue for some finite set of nodes, then \(\xi\) has a zero \(\rho\) with \(\Re\rho>\lambda\). Evaluated in ball arithmetic the test would be a certified proof of that statement.

Source: ch/cd1.tex, label cd:cor:cert.

Hölder stability of Cauchy deconvolution between levels by the three lines theorem, with sharp exponent

Volume II, Theorem 6.3.1, page 133 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_y\) be the Cauchy density of width \(y>0\), with \(P_0*\mu=\mu\), let \(b_0<b<a\) and let \(\mu\) be a signed measure with \(P_{b_0}*\mu\in L^2(\mathbb{R})\), where for \(b_0<0\) this means \(\mu=P_{|b_0|}*\lambda\) with \(\lambda\in L^2\) and \(P_{b_0}*\mu\) stands for \(\lambda\); for \(y>b_0\), \(P_y*\mu\) means \(P_{y-b_0}*(P_{b_0}*\mu)\). Then \(\|P_b*\mu\|_2\le\|P_a*\mu\|_2^{\kappa}\,\|P_{b_0}*\mu\|_2^{1-\kappa}\) with \(\kappa=(b-b_0)/(a-b_0)\). In particular, if two sources have data at width \(a\) within \(\varepsilon\) in \(L^2\) and both are bounded by \(M\) at width \(b_0\), they agree at width \(b\) to within \(\varepsilon^\kappa(2M)^{1-\kappa}\). The exponent \(\kappa\) is attained in rate. The proof is Hölder’s inequality on the Fourier side, equivalently convexity of \(y\mapsto\log\|P_y*\mu\|_2\).

Earlier editions: Theorem 33.3.1, page 1171 (edition 1).

Source: ch/cd1.tex, label cd:thm:holder.

Sharp Hölder exponent (b+w)/(a+w) for Cauchy deconvolution of finite signed sources

Volume II, Proposition 6.3.2, page 134 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_y\) be the Cauchy density of width \(y>0\) with \(P_0*\nu=\nu\). Let \(\nu\) be a finite signed measure of total variation at most \(B\), let \(w\ge0\), \(h=P_w*\nu\) (with \(P_b*h\) meaning \(P_{b+w}*\nu\) when \(b<0\)) and let \(-w<b<a\). Then \(\|P_b*h\|_2\le\sqrt2\,D^{1-\kappa}\|P_a*h\|_2^{\kappa}\) with \(\kappa=(b+w)/(a+w)\) and \(D=B/\sqrt{2\pi(b+w)}\), and no larger exponent holds uniformly over this class. The upper bound is a split of the frequencies at a cutoff chosen as a function of the data error, and the sharpness is shown by the sources \(B\phi(x)\cos(\Omega x)dx\) with \(\Omega\to\infty\) for a fixed nonnegative kernel \(\phi\) with transform \((1-|\omega|)_+\).

Earlier editions: Proposition 33.3.2, page 1172 (edition 1).

Source: ch/cd1.tex, label cd:prop:finite.

The stability exponent is attained inside the positive cone

Volume II, Proposition 6.3.3, page 134 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_s\) be the Poisson kernel of width \(s\) and \(0<b<a\). There is a constant \(k>0\) depending only on \(a,b\) such that for every small \(\delta>0\) there are positive measures \(\nu_0,\nu_1\) of mass at most \(2\) with \(\|P_a*(\nu_1-\nu_0)\|_2\le\delta\) and \(\|P_b*(\nu_1-\nu_0)\|_2\ge k\,\delta^{b/a}\).

Source: ch/cd1.tex, label cd:prop:conelower.

Two level Hölder bound for positive sources of Cauchy deconvolution with data in L2 on the line

Volume II, Corollary 6.3.4, page 135 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_y\) be the Cauchy density of width \(y>0\). If \(\mu_1,\mu_2\ge0\) are measures, \(u_j=P_a*\mu_j\in L^2(\mathbb{R})\) and \(\|u_1-u_2\|_2\le\varepsilon\), then for \(0<d<b<a\), \(\|P_b*(\mu_1-\mu_2)\|_2\le\varepsilon^\kappa\bigl((a/d)(\|u_1\|_2+\|u_2\|_2)\bigr)^{1-\kappa}\) with \(\kappa=(b-d)/(a-d)\). The proof uses the pointwise bound \(P_d\le(a/d)P_a\) and the three lines bound. The statement is for data in \(L^2\) on the whole line; the case of data on a window is a problem of the chapter.

Earlier editions: Corollary 33.3.3, page 1172 (edition 1).

Source: ch/cd1.tex, label cd:cor:positive.

Logarithmic stability of Cauchy deconvolution for sources in a Sobolev ball, sharp in order and in the constant

Volume II, Theorem 6.3.5, page 135 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a\) be the Cauchy density of width \(a>0\), \(s>0\), and \(H^s\) the Sobolev space of order \(s\); \(C_s\) denotes a constant depending only on \(s\). Suppose \(\|\mu\|_{H^s}\le M\) and \(\|P_a*\mu\|_2\le\varepsilon\), with \(\varepsilon/M\) small enough depending on \(a\) and \(s\). Then \(\|\mu\|_2\le C_sM\bigl(2a/\log(M/\varepsilon)\bigr)^s\), and no bound of smaller order holds uniformly over the class: the modulus of continuity of the inverse is of this order. The sharp modulus \(\sup\{\|\mu\|_2:\|\mu\|_{H^s}\le M,\|P_a*\mu\|_2\le\varepsilon\}\) is \((1+o(1))M(a/\log(M/\varepsilon))^s\) as \(\varepsilon/M\to0\).

Earlier editions: Theorem 33.3.4, page 1173 (edition 1).

Source: ch/cd1.tex, label cd:thm:log.

Sharp logarithmic rate of Cauchy deconvolution for positive measures of bounded mass

Volume II, Theorem 6.3.6, page 136 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=a/(\pi(a^2+x^2))\) be the Cauchy kernel and \(d_{\mathrm{BL}}(\mu,\mu')=\sup\{\int\varphi\,d(\mu-\mu'):\|\varphi\|_\infty\le1,\ \mathrm{Lip}(\varphi)\le1\}\) the bounded Lipschitz distance. There are absolute constants \(c_0,c_1,c_2>0\) such that the following holds. Let \(\mu,\mu'\) be positive measures on \(\mathbb{R}\) of mass at most \(M\), let \(a>0\), and let \(\eta=\|P_a*(\mu-\mu')\|_{L^1}\) be small enough that \(L=\log(1/\eta)\ge e^2\). Then \(d_{\mathrm{BL}}(\mu,\mu')\le\frac{c_1Ma}{L-2\log L}+\frac{c_1(1+L)^{1/2}}{L^2}\). Conversely, for every \(\Omega\ge2\) there are positive measures \(\mu,\mu'\) of mass \(M\) with \(\|P_a*(\mu-\mu')\|_{L^1}\le c_2M(1+a)^{1/2}e^{-a(\Omega-1)}\) and \(d_{\mathrm{BL}}(\mu,\mu')\ge c_0M/\Omega\). So positivity does not improve the rate \(a/\log(1/\eta)\).

Source: ch/cd1.tex, label cd:thm:rate.

Saturation of positive kernels, bias of order the inverse square of the log

Volume II, Proposition 6.3.7, page 137 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(R\ge0\) be even with \(\int R=1\) and Fourier transform \(\widehat R\) supported in \([-1,1]\), and put \(R_\Omega(x)=\Omega R(\Omega x)\). Then \(1-\widehat R(u)\ge c_Ru^2\) for \(|u|\le1\) with \(c_R>0\). Consequently, for every \(\mu\in L^2\) with \(\mu\ne0\) there are \(c>0\) and \(\Omega_0\) with \(\|R_\Omega*\mu-\mu\|_{L^2}\ge c\,\Omega^{-2}\) for \(\Omega\ge\Omega_0\). If the bandwidth is limited to \(\Omega\le a^{-1}\log(1/\delta)\), the level at which noise of size \(\delta\) amplified by \(e^{a\Omega}\) reaches one, the bias of any such positive linear method is at least of order \((a/\log(1/\delta))^2\).

Source: ch/cd1.tex, label cd:prop:sat.

The positive part of an estimate is a pointwise contraction

Volume II, Proposition 6.3.8, page 137 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\ge0\) be a function and \(f\) any real estimate of it, with \(f_+=\max(f,0)\). Then \(|f_+-\mu|\le|f-\mu|\) pointwise, so \(f_+\ge0\) satisfies every \(L^p\) error bound that \(f\) does.

Source: ch/cd1.tex, label cd:prop:pospart.

Polynomial upper rate for density deconvolution with Cauchy noise and a target analytic in a strip

Volume II, Proposition 6.4.1, page 138 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Y_1,\dots,Y_n\) be i.i.d. observations \(Y_j=X_j+E_j\), with \(X_j\) of density \(f\) independent of \(E_j\), Cauchy noise of width \(a>0\), with \(\int e^{2w|\omega|}|\widehat f(\omega)|^2d\omega\le2\pi M^2\) for some \(w>0\), where \(\widehat f(\omega)=\int f(x)e^{-i\omega x}dx\). The spectral cutoff estimator \(\widehat f_n(x)=(2\pi)^{-1}\int_{|\omega|\le\Omega}e^{a|\omega|}\widehat\phi_n(\omega)e^{i\omega x}d\omega\), with \(\widehat\phi_n(\omega)=n^{-1}\sum_{j\le n}e^{-i\omega Y_j}\) and \(\Omega=\log n/(2(a+w))\), has root mean integrated squared error \(O(n^{-w/(2(a+w))})\). This is an upper bound only; no lower bound is claimed.

Earlier editions: Proposition 33.4.1, page 1174 (edition 1).

Source: ch/cd1.tex, label cd:prop:strip.

A three atom cluster imitates a pair of atoms after Cauchy blurring

Volume II, Proposition 6.5.1, page 140 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a>0\), \(P_a(x)=a/(\pi(a^2+x^2))\), \(\pi_\delta=\delta_{-\delta/2}+\delta_{\delta/2}\) and \(\kappa_\delta=\frac14\delta_{-\delta}+\frac32\delta_0+\frac14\delta_\delta\). Both are positive measures of mass \(2\) with the same moments of orders \(0\) to \(3\), and \(\frac{\|P_a*(\pi_\delta-\kappa_\delta)\|_\infty}{2P_a(0)}\le\frac5{16}\big(\frac\delta a\big)^4\). Hence if the blurred data of a pair are known to relative accuracy \(\eta\), that is to \(\eta\) times the peak height \(2P_a(0)\), they cannot exclude the three atom cluster when \(\delta\le a(16\eta/5)^{1/4}\); for \(a=\frac12\) and \(\eta=2\times10^{-4}\) this is about \(0.08\) (the bound proved, the number computed).

Source: ch/cd1.tex, label cd:prop:cluster.

Problem: estimating the critical width of noisy positive data

Volume II, Problem 6.1, page 141 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(u\ge0\) with \(\int u(x)(1+x^2)^{-1}dx<\infty\) and \(c^*(u)=\sup\{c\ge0:u=P_c*\nu\text{ for some measure }\nu\ge0\}\), with \(P_c\) the Cauchy density of width \(c\). Given noisy data \(u+\text{noise}\), estimate \(c^*(u)\). For atomic sources with positive weights \(c^*\) is the smallest width among the poles, estimable by a matrix pencil on a class with bounded order, weights bounded below, separation and no aliasing; without weights bounded below there is no uniform rate, since \(c^*\) is discontinuous (\((1-\delta)P_A+\delta P_r\) has \(c^*=r<A\) for every \(\delta>0\)). For general sources \(c^*\) is a boundary of a convex cone in an infinite dimensional space. Status: open.

Earlier editions: Problem 33.1, page 1177 (edition 1).

Source: ch/cd1.tex, label cd:prob:width.

Problem: positivity and rates for the Thorin family

Volume II, Problem 6.2, page 141 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(P_a\) be the Cauchy density of width \(a>0\). Positive sources of the form \(P_w*\nu\) with \(\nu\ge0\) of mass at most \(B\) have the polynomial exponent \(w/(a+w)\) for deconvolution at \(b=0\), but this comes from the smoothing and the mass bound, not from positivity, and the zero counting measure of the Riemann zeta function has infinite mass. Positivity does not improve the logarithmic minimax rate over smoothness classes of densities in the worst case, and prescribing a number of sign changes of the Fourier transform does not help. For positive measures of mass at most \(M\) the logarithmic rate is sharp: in the bounded Lipschitz distance positivity does not improve the rate \(a/\log(1/\eta)\), where \(\eta\) is the \(L^1\) norm of the difference of the blurred measures. Open: is the Thorin family, the family of the solutions \(\mu\ge0\) of \(P_{1-\theta}*\mu=u_1\), \(\frac12\le\theta<1\), with \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\) and \(\xi\) the completed zeta function, in a class on which positivity alone gives a polynomial rate?

Earlier editions: Problem 33.2, page 1177 (edition 1).

Source: ch/cd1.tex, label cd:prob:rates.

Problem: Hölder bound between levels from data on a window for positive sources

Volume II, Problem 6.3, page 142 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(P_y\) be the Cauchy density of width \(y>0\). The three lines bound needs an a priori bound at a level below the target, and for nonnegative sources with data in \(L^2\) on the whole line this bound is removed. For data known only on a window, the mass in a window of a positive source is controlled by the data alone. Open: does positivity give a Hölder bound between levels from data on a window, without any a priori bound below the target, with an exponent determined by the data?

Earlier editions: Problem 33.3, page 1178 (edition 1).

Source: ch/cd1.tex, label cd:prob:three.

Close pairs, is the cluster bound sharp

Volume II, Problem 6.4, page 142 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(a>0\), \(P_a(x)=a/(\pi(a^2+x^2))\) the Cauchy kernel, \(\pi_\delta=\delta_{-\delta/2}+\delta_{\delta/2}\) and \(d_{\mathrm{BL}}\) the bounded Lipschitz distance. Open problem: prove that the bound of the cluster proposition, which says that \(\pi_\delta\) and a three atom cluster with the same first four moments have blurred versions differing by at most \(\frac5{16}(\delta/a)^4\) relative to the peak, is sharp, that is, that any positive measure of mass \(2\) near \(\pi_\delta\) whose blurred version \(P_a*\cdot\) fits that of \(\pi_\delta\) to relative accuracy \(\eta\) lies within \(C(a/\delta)^4\eta\) of \(\pi_\delta\) in the bounded Lipschitz distance.

Source: ch/cd1.tex, label cd:prob:cluster.

Volume II, Chapter 7. Three families of generalised gamma convolutions and the deconvolution obstruction

Tilt family of the clock: the Thorin property at one tilt is equivalent to RH

Volume II, Theorem 7.1.2, page 145 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(G(u)=\xi(\tfrac12+\sqrt u)/\xi(\tfrac12)\), an entire function of \(u\). Each pair \(\{\rho,1-\rho\}\) of zeros \(\rho=\beta+i\gamma\) of \(\xi\) gives the number \(a_\rho=-(\rho-\tfrac12)^2=\gamma^2-(\beta-\tfrac12)^2-2i\gamma(\beta-\tfrac12)\), and Hadamard’s product gives the absolutely convergent product \(G(u)=\prod(1+u/a_\rho)\). For \(u_0\ge0\) put \(\widetilde\varphi_{u_0}(u)=G(u_0)/G(u_0+u)\) for \(u\ge0\), the Laplace transform of the clock \(T\) of \(\xi\) (the variable with \(\mathbb{E}e^{-uT}=\xi(\tfrac12)/\xi(\tfrac12+\sqrt u)\)) tilted by \(e^{-u_0T}\); for \(u_0=0\) it is the Laplace transform of \(T\) itself. A generalized gamma convolution (GGC) Laplace transform is a function \(\exp\bigl(-du-\int_{(0,\infty)}\log(1+u/z)\,U(dz)\bigr)\) with drift \(d\ge0\) and a positive Thorin measure \(U\). Then: (a) for \(u_1\ge0\), \(\widetilde\varphi_{u_0+u_1}(u)=\widetilde\varphi_{u_0}(u_1+u)/\widetilde\varphi_{u_0}(u_1)\), so the tilt by \(u_1\) is the Esscher transform by \(e^{-u_1\widetilde T_{u_0}}\), where \(\widetilde T_{u_0}\) has transform \(\widetilde\varphi_{u_0}\); it maps GGC transforms to GGC transforms and moves the Thorin measure up by \(u_1\); backward, if \(\widetilde\varphi_{u_0+u_1}\) is a GGC transform with Thorin measure carried by \((u_1,\infty)\), then \(\widetilde\varphi_{u_0}(u)=\widetilde\varphi_{u_0+u_1}(u-u_1)/\widetilde\varphi_{u_0+u_1}(-u_1)\) is a GGC transform with the Thorin measure moved down by \(u_1\), the denominator being \(G(u_0+u_1)/G(u_0)\), finite and positive. (b) \(\widetilde\varphi_{u_0}\) is the Laplace transform of a GGC if and only if every \(u_0+a_\rho\) is real and positive; the imaginary part of \(u_0+a_\rho\) is \(-2\gamma(\beta-\tfrac12)\), which does not depend on \(u_0\). (c) The following are equivalent: \(\widetilde\varphi_{u_0}\) is a GGC transform for one \(u_0\ge0\); it is a GGC transform for every \(u_0\ge0\); the Riemann Hypothesis holds.

Earlier editions: Theorem 19.1.2, page 727 (edition 1).

Source: ch/tf1.tex, label tf:thm:tilt.

Poisson semigroup of the basepoint family and its converse

Volume II, Theorem 7.2.1, page 146 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function with zeros \(\rho=\beta+i\gamma\) counted with multiplicity. For real \(\varepsilon\) and \(\theta\) such that \(\tfrac12+\varepsilon+i\theta\) is not a zero put \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\tfrac12+\varepsilon+i\theta)\), the Thorin density at level \(\varepsilon\) (in the chart \(\theta=\sqrt z\) it is the Thorin density of the shifted clock \(\xi(\tfrac12+\varepsilon)/\xi(\tfrac12+\varepsilon+\sqrt u)\) for \(\varepsilon>0\), whenever that is a generalized gamma convolution and no zero lies on the line \(\operatorname{Re}s=\tfrac12+\varepsilon\)). Let \(P_a(x)=a/(\pi(a^2+x^2))\) for real \(a\ne0\), so that \(P_a\) is the Cauchy density of scale \(a\) for \(a>0\), \(P_{-a}=-P_a\) and \(P_a*P_b=P_{a+b}\) for \(a,b>0\). Hadamard’s product gives the absolutely convergent expansion \(\rho_\varepsilon(\theta)=\sum_\rho P_{\varepsilon-(\beta-\frac12)}(\theta-\gamma)\) over all zeros with multiplicity, for \(\varepsilon\ne\beta-\tfrac12\). (a) If \(\varepsilon_0>\beta-\tfrac12\) for every zero, then \(\rho_{\varepsilon_0+a}=P_a*\rho_{\varepsilon_0}\) for every \(a>0\) (convolution in \(\theta\)) and \(\rho_\varepsilon>0\) for every \(\varepsilon\ge\varepsilon_0\); this hypothesis holds, with no assumption on the zeros, for every \(\varepsilon_0\ge\tfrac12\). (b) If \(\varepsilon_0\ge0\) lies at positive distance from the set \(\{\beta-\tfrac12\}\) of the offsets of the zeros and some zero has \(\beta-\tfrac12>\varepsilon_0\), then \(\rho_{\varepsilon_0+a}\ne P_a*\rho_{\varepsilon_0}\) for some \(a>0\).

Earlier editions: Theorem 19.2.1, page 728 (edition 1).

Source: ch/tf2.tex, label tf:thm:poisson.

Defect formula for the Poisson semigroup of the level densities of xi

Volume II, Proposition 7.2.2, page 146 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_\varepsilon(\theta)=\pi^{-1}\Re\frac{\xi'}\xi(\frac12+\varepsilon+i\theta)\), let \(P_a(x)=a/(\pi(a^2+x^2))\) be the Poisson kernel, \(a>0\), \(\varepsilon_0\ge\frac12\), \(\varepsilon_1=\varepsilon_0-a\ge0\), and suppose no zero \(\rho=\beta+i\gamma\) has \(\beta-\frac12=\varepsilon_1\). Then \(P_a*\rho_{\varepsilon_1}\) is a function and \(\rho_{\varepsilon_0}-P_a*\rho_{\varepsilon_1}=\sum_{\varepsilon_1<\beta-\frac12<\varepsilon_0}\big[P_{\varepsilon_0-(\beta-\frac12)}(\cdot-\gamma)+P_{2a-(\varepsilon_0-(\beta-\frac12))}(\cdot-\gamma)\big]\ge0\), where the two widths of each term add up to \(2a\).

Source: ch/tf2.tex, label tf:prop:defect.

Deconvolution form of RH

Volume II, Theorem 7.2.4, page 147 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function with zeros \(\rho=\beta+i\gamma\) of multiplicity \(m_\rho\), let \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\tfrac12+\varepsilon+i\theta)\) be the Thorin density at level \(\varepsilon\), and let \(P_a(x)=a/(\pi(a^2+x^2))\) be the Cauchy density of scale \(a>0\); \(*\) denotes convolution in \(\theta\). Let \(\varepsilon_0>\beta-\tfrac12\) for every zero (for instance \(\varepsilon_0\ge\tfrac12\), which needs no assumption) and let \(0\le\varepsilon_1<\varepsilon_0\). (a) There is a positive measure \(\nu\) on \(\mathbb{R}\) with \(\int(1+x^2)^{-1}\nu(dx)<\infty\) and \(\rho_{\varepsilon_0}=P_{\varepsilon_0-\varepsilon_1}*\nu\) if and only if \(\beta-\tfrac12\le\varepsilon_1\) for every zero. When it exists, \(\nu\) is unique: it is \(\rho_{\varepsilon_1}\,d\theta\) if \(\beta-\tfrac12<\varepsilon_1\) for every zero, and otherwise the vague limit of \(\rho_\varepsilon\,d\theta\) as \(\varepsilon\downarrow\varepsilon_1\), which carries the point masses \(m_\rho\delta_\gamma\) at the zeros with \(\beta-\tfrac12=\varepsilon_1\). (b) In particular RH holds if and only if \(\rho_{\varepsilon_0}=P_{\varepsilon_0}*\mu\) for a positive measure \(\mu\) on \(\mathbb{R}\) (the condition \(\int(1+x^2)^{-1}\mu(dx)<\infty\) is then automatic, because \(P_{\varepsilon_0}*\mu=\rho_{\varepsilon_0}\) is finite), and then \(\mu=\sum_\gamma\delta_\gamma\), the sum over the ordinates of all zeros, of both signs, with multiplicity.

Earlier editions: Theorem 19.2.2, page 729 (edition 1).

Source: ch/tf2.tex, label tf:thm:deconv.

Local bound on the Poisson semigroup defect from the verified zero height

Volume II, Proposition 7.2.3, page 147 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(H=3\cdot10^{12}\), so that every zero with \(|\gamma|\le H\) has \(\beta=\frac12\) (Platt and Trudgian), let \(0<a<\frac12\) and suppose no zero has \(\beta=1-a\), with \(\rho_\varepsilon\) and \(P_a\) as in the defect formula. For \(|\theta|\le H/2\), \(0\le\rho_{1/2}(\theta)-(P_a*\rho_{1/2-a})(\theta)\le\frac{8a}\pi\sum_{|\gamma|>H}\frac1{\gamma^2}\le7.54\times10^{-12}\,a\). The numerical constant is computed from the bound for \(N(T)\) of Hasanalizade, Shen and Wong, and the error terms of that bound change only its sixth digit.

Source: ch/tf2.tex, label tf:prop:local.

Window rigidity of the positive deconvolution to the critical line (Polson note)

Volume II, Theorem 7.2.6, page 149 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}\frac{\xi'}\xi(\frac12+\varepsilon+i\theta)\) with \(\xi\) the completed zeta function, \(P_a(x)=a/(\pi(a^2+x^2))\), and let \(\varepsilon_0>\beta-\frac12\) for every zero \(\beta+i\gamma\) and \(0\le\varepsilon_1<\varepsilon_0\), \(c=\varepsilon_0-\varepsilon_1\). For \(C>0\) let \(\mathcal M_C\) be the set of positive measures \(\mu\) on \(\mathbb{R}\) with \(\mu((-T,T))\le C(1+T)\log(2+T)\) for all \(T>0\), with \(C\) large enough that the measures \(\rho_\varepsilon d\theta\), \(\varepsilon_1\le\varepsilon\le\varepsilon_0\), and the deconvolution measure \(\nu\) with \(\rho_{\varepsilon_0}=P_c*\nu\) lie in it. For a nonempty open interval \(W\) put \(D_W(c)=\inf_{\mu\in\mathcal M_C}\sup_{\theta\in W}|(P_c*\mu)(\theta)-\rho_{\varepsilon_0}(\theta)|\). The infimum is attained, and \(D_W(c)=0\) if and only if \(\beta-\frac12\le\varepsilon_1\) for every zero.

Source: ch/tf2.tex, label tf:thm:rigid.

RH holds if and only if the deconvolution misfit vanishes on any window (Polson note)

Volume II, Corollary 7.2.7, page 149 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With the notation of window rigidity (the misfit \(D_W(\varepsilon_0)\) of the best positive deconvolution of \(\rho_{\varepsilon_0}=\pi^{-1}\operatorname{Re}\frac{\xi'}\xi(\frac12+\varepsilon_0+i\theta)\) by the Cauchy kernel \(P_{\varepsilon_0}\) over the class \(\mathcal M_C\) of positive measures with \(\mu((-T,T))\le C(1+T)\log(2+T)\), \(C\) large, on a window \(W\)), let \(\varepsilon_0\ge\frac12\) and let \(W\) be any open interval. Then the Riemann Hypothesis holds if and only if \(D_W(\varepsilon_0)=0\). In particular, if RH fails, the positive deconvolution to the critical line misfits the data on every window, however far the window lies from the offending zero. The result is qualitative: the misfit on a window at height \(Y\) can be astronomically small.

Source: ch/tf2.tex, label tf:cor:window.

Stability of positive deconvolutions on a window (Polson note)

Volume II, Theorem 7.2.9, page 149 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function, \(\rho_\varepsilon=\pi^{-1}\operatorname{Re}\frac{\xi'}\xi(\frac12+\varepsilon+i\theta)\), \(P_c\) the Cauchy density of width \(c=\varepsilon_0-\varepsilon_1\), \(\varepsilon_0>\beta-\frac12\) for every zero and \(0\le\varepsilon_1<\varepsilon_0\), and let \(\nu\) be the positive measure with \(\rho_{\varepsilon_0}=P_c*\nu\) (it exists exactly when every zero has \(\beta-\frac12\le\varepsilon_1\)). Let \(\beta-\frac12\le\varepsilon_1\) for every zero, let \(\mu_n\) be positive measures with \(\mu_n((-T,T))\le C(1+T)\log(2+T)\) for all \(T\), and suppose \(\sup_{\theta\in W}|P_c*\mu_n-\rho_{\varepsilon_0}|\to0\) on some open interval \(W\). Then \(\mu_n\to\nu\) vaguely. This is a consistency statement under exact data and the exact far field.

Source: ch/tf2.tex, label tf:thm:stab.

Thorin density near the critical line and Laguerre’s quantity

Volume II, Proposition 7.3.1, page 152 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(\theta)=\xi(\tfrac12+i\theta)\), and for real \(\varepsilon\) and \(\theta\) such that \(\tfrac12+\varepsilon+i\theta\) is not a zero put \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\tfrac12+\varepsilon+i\theta)\). Let \(\theta\) be real and not the ordinate of a zero on the critical line. Then \(\rho_\varepsilon(\theta)=\frac{\varepsilon}{\pi}\mathcal L_\Xi(\theta)+O(\varepsilon^3)\) as \(\varepsilon\to0\), where \(\mathcal L_\Xi(\theta)=-\frac{d^2}{d\theta^2}\log|\Xi(\theta)|=\sum_\rho(\theta-\gamma_\rho)^{-2}\) is Laguerre’s quantity, \(\gamma_\rho=-i(\rho-\tfrac12)=\gamma-i(\beta-\tfrac12)\) runs over the zeros of \(\Xi\), and the sum is real because the zeros of \(\Xi\) come in conjugate pairs. Under RH, \(\mathcal L_\Xi(\theta)=\sum_\gamma(\theta-\gamma)^{-2}>0\). A quadruple of zeros \(\tfrac12\pm\delta\pm i\gamma^*\) with \(\delta>0\) and \(\gamma^*>0\) contributes \(2\operatorname{Re}(\theta-\gamma^*+i\delta)^{-2}+2\operatorname{Re}(\theta+\gamma^*+i\delta)^{-2}\) to \(\mathcal L_\Xi(\theta)\), which equals \(-2/\delta^2+O(\gamma^{*-2})\) at \(\theta=\gamma^*\).

Earlier editions: Proposition 19.3.1, page 732 (edition 1).

Source: ch/tf3.tex, label tf:prop:nearline.

Visibility curve of a hypothetical off line quadruple and its closed forms in three lattice models

Volume II, Proposition 7.3.2, page 154 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. No off line zero of \(\xi\) is known; consider a hypothetical zero set (zeros \(\rho=\beta+i\gamma\) counted with multiplicity, for instance the zeros of \(\xi\) with a quadruple added by hand) that contains a quadruple \(\tfrac12\pm\delta\pm i\gamma^*\) with \(\delta>0\) and \(\gamma^*>0\). Let \(P_a(x)=a/(\pi(a^2+x^2))\) for real \(a\ne0\) and let \(\rho_\varepsilon(\theta)=\sum_\rho P_{\varepsilon-(\beta-\frac12)}(\theta-\gamma)\) be the Thorin density at level \(\varepsilon\) of the zero set (for the zeros of \(\xi\) it equals \(\pi^{-1}\operatorname{Re}(\xi'/\xi)(\tfrac12+\varepsilon+i\theta)\)). Let \(\rho^0_\varepsilon\) be the same sum over all zeros other than the two zeros \(\tfrac12\pm\delta+i\gamma^*\), so that the images \(\tfrac12\pm\delta-i\gamma^*\) of the quadruple at \(-\gamma^*\) are included in it, and suppose \(\rho^0_\varepsilon(\gamma^*)>0\). Then for \(0<\varepsilon<\delta\), \(\rho_\varepsilon(\gamma^*)<0\) if and only if \(\delta<\delta_{\mathrm{det}}(\varepsilon)\), where \(\delta_{\mathrm{det}}(\varepsilon)^2=\varepsilon^2+\frac{2\varepsilon}{\pi\rho^0_\varepsilon(\gamma^*)}\), which depends on \(\delta\) through the images; so the level \(\varepsilon\) sees the quadruple at its ordinate exactly for the distances \(\varepsilon<\delta<\delta_{\mathrm{det}}(\varepsilon)\). In the lattice models the other zeros are the following points of the critical line and nothing else (the images at \(-\gamma^*\) and the distant zeros are neglected), with spacing \(\Delta>0\): in model (ii) the points \(\gamma^*\pm(k+\tfrac12)\Delta\), \(k\ge0\) (the quadruple sits at the midpoint of a gap); in model (i) the points \(\gamma^*\pm k\Delta\), \(k\ge1\) (the quadruple replaces the zero at \(\gamma^*\)); in model (iii) the points \(\gamma^*\pm(k+\tfrac12)\Delta\), \(k\ge1\) (the quadruple replaces the two zeros that bound a gap and sits at its midpoint). Then [_{}()²⁼2+ \[\begin{cases}\dfrac{2\varepsilon\Delta}{\pi\tanh(\pi\varepsilon/\Delta)} & \text{in (ii)},\\[2.5ex] \dfrac{2\varepsilon}{\frac\pi\Delta\coth\frac{\pi\varepsilon}\Delta-\frac1\varepsilon} & \text{in (i)},\\[2.5ex] \dfrac{2\varepsilon}{\frac\pi\Delta\tanh\frac{\pi\varepsilon}\Delta-\frac{2\varepsilon}{\varepsilon^2+\Delta^2/4}} & \text{in (iii)}.\end{cases}\]

] As \(\varepsilon\to0\) these tend to \(\delta_c^2=2\Delta^2/\pi^2\), \(6\Delta^2/\pi^2\) and \(2\Delta^2/(\pi^2-8)\) respectively; for \(\varepsilon\ge\Delta\) all three satisfy \(\delta_{\mathrm{det}}(\varepsilon)^2-\varepsilon^2=2\varepsilon\Delta/\pi+O(\Delta^2)\), that is \(\delta_{\mathrm{det}}(\varepsilon)-\varepsilon=(\Delta/\pi)\bigl(1+O(\Delta/\varepsilon)\bigr)\); and in all three \(\delta_{\mathrm{det}}\) is increasing in \(\varepsilon\).

Earlier editions: Proposition 19.3.2, page 734 (edition 1).

Source: ch/tf3.tex, label tf:prop:curve.

Number of levels needed in the lattice models, one quarter of log h

Volume II, Corollary 7.3.3, page 156 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Consider the three lattice models of a single hypothetical off line quadruple \(\tfrac12\pm\delta\pm i\gamma^*\) among zeros of the critical line with spacing \(\Delta=2\pi/\log(h/2\pi)\) at height \(h\): the other zeros are the points \(\gamma^*\pm(k+\tfrac12)\Delta\), \(k\ge0\), in model (ii), the points \(\gamma^*\pm k\Delta\), \(k\ge1\), in model (i), and the points \(\gamma^*\pm(k+\tfrac12)\Delta\), \(k\ge1\), in model (iii), and the images of the quadruple at \(-\gamma^*\) and the distant zeros are neglected. Let \(S(\varepsilon)=\sum(\varepsilon^2+(\gamma-\gamma^*)^2)^{-1}\) over the other zeros \(\tfrac12+i\gamma\), \(\delta_{\mathrm{det}}(\varepsilon)=\sqrt{\varepsilon^2+2/S(\varepsilon)}\) the visibility distance (the level \(\varepsilon\) detects, at the ordinate \(\gamma^*\), the quadruple at distance \(\delta\), that is \(\rho_\varepsilon(\gamma^*)<0\) for the Thorin density \(\rho_\varepsilon(\theta)=\sum_\rho P_{\varepsilon-(\beta-\frac12)}(\theta-\gamma)\) of the zero set, with \(P_a(x)=a/(\pi(a^2+x^2))\), exactly when \(\varepsilon<\delta<\delta_{\mathrm{det}}(\varepsilon)\)), and \(\delta_c=\delta_{\mathrm{det}}(0)=\sqrt{2/S(0)}\), which equals \((\sqrt2/\pi)\Delta\), \((\sqrt6/\pi)\Delta\) and \(\sqrt{2/(\pi^2-8)}\,\Delta\) in models (ii), (i), (iii) (the first order test is the sign of Laguerre’s quantity \(\mathcal L_\Xi(\gamma^*)=-(\log|\Xi|)''(\gamma^*)\) with \(\Xi(\theta)=\xi(\tfrac12+i\theta)\), which in the model is the sign of \(S(0)-2/\delta^2\), so that it detects the quadruple exactly for \(\delta<\delta_c\)). Put \(\varepsilon_1=\delta_c\) and \(\varepsilon_{k+1}=\delta_{\mathrm{det}}(\varepsilon_k)\), and let \(K\) be the least \(k\ge0\) with \(\varepsilon_{k+1}>\tfrac12\), so that \(K=0\) if \(\delta_c>\tfrac12\). (a) The sign of \(\mathcal L_\Xi(\gamma^*)\), together with positivity of \(\rho_\varepsilon(\gamma^*)\) at the levels \(\varepsilon_1,\dots,\varepsilon_K\), each moved down slightly so that consecutive bands overlap, detects a single off line quadruple at every distance in \((0,\tfrac12]\). (b) No test made at the ordinate, that is the sign of \(\mathcal L_\Xi(\gamma^*)\) together with the sign of \(\rho_\varepsilon(\gamma^*)\) at finitely many levels, that detects a single off line quadruple at every distance in \((0,\tfrac12]\) uses fewer than \(K\) levels. (c) \(K=\tfrac14\log(h/2\pi)+O(\log\log h)\) as \(h\to\infty\).

Earlier editions: Corollary 19.3.3, page 736 (edition 1).

Source: ch/tf3.tex, label tf:cor:levels.

Hyperbolic splitting of the Cauchy kernel (Polson note)

Volume II, Theorem 7.3.4, page 158 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=a/(\pi(a^2+x^2))\) and, for \(0<a<1\) and \(\varepsilon_0>0\), \(H_a(t)=\sin(\pi a)/(2\varepsilon_0(\cosh(\pi t/\varepsilon_0)+\cos\pi a))\), a positive density of mass \(a\) with Fourier transform \(\sinh(a\varepsilon_0\omega)/\sinh(\varepsilon_0\omega)\). For \(0<h<\varepsilon_0\) and \(r=h/\varepsilon_0\), \(P_h=P_{\varepsilon_0}*H_r+H_{1-r}\). Both terms are positive, and for \(e^{\pi|u|/\varepsilon_0}>2\), \(H_{1-r}(u)=\sin\pi r/(2\varepsilon_0(\cosh(\pi u/\varepsilon_0)-\cos\pi r))\le(\sin\pi r/\varepsilon_0)\,e^{-\pi|u|/\varepsilon_0}/(1-2e^{-\pi|u|/\varepsilon_0})\).

Source: ch/tf3.tex, label tf:thm:split.

A far off line zero quadruple is exponentially hidden from the deconvolution at fixed depth (Polson note)

Volume II, Corollary 7.3.5, page 158 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function, \(\varepsilon_0\ge\frac12\), and assume, as a model and not as a claim about \(\xi\), that the only zeros off the critical line are \(\rho_0=\beta_0+i\gamma_0\) and its images under \(\rho\mapsto\bar\rho\) and \(\rho\mapsto1-\rho\), with \(\kappa=\beta_0-\frac12\in(0,\varepsilon_0)\). Let \(W\) be a window at distance at least \(D\) from \(\pm\gamma_0\), with \(e^{\pi D/\varepsilon_0}>2\), and \(D_W(\varepsilon_0)\) the misfit of the best positive deconvolution of \(\pi^{-1}\operatorname{Re}\frac{\xi'}\xi(\frac12+\varepsilon_0+i\theta)\) by the Cauchy kernel of width \(\varepsilon_0\) on \(W\) over the measures of at most the growth of the zero counting measure. Then \(D_W(\varepsilon_0)\le\frac{2\sin(\pi\kappa/\varepsilon_0)}{\varepsilon_0}\,\frac{e^{-\pi D/\varepsilon_0}}{1-2e^{-\pi D/\varepsilon_0}}\). At \(\varepsilon_0=\frac12\) this is \(4\sin(2\pi\kappa)e^{-2\pi D}(1+O(e^{-2\pi D}))\), so a window five units away from the quadruple has misfit below \(10^{-13}\).

Source: ch/tf3.tex, label tf:cor:hidden.

Ceiling on window certificates in the one quadruple model (Polson note)

Volume II, Proposition 7.3.6, page 159 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Work in the model of a single off line quadruple of zeros: the only zeros off the critical line are \(\rho_0=\beta_0+i\gamma_0\) and its images under \(\rho\mapsto\bar\rho\) and \(\rho\mapsto1-\rho\), with \(\kappa=\beta_0-\frac12\in(0,\varepsilon_0)\), and \(\rho_{\varepsilon}(\theta)=\pi^{-1}\operatorname{Re}\frac{\xi'}\xi(\frac12+\varepsilon+i\theta)\), \(P_s\) the Cauchy density of width \(s\). A certificate is a set of points \(y_1,\dots,y_J\) in a window at distance at least \(D\) from \(\pm\gamma_0\) and weights \(\varphi\in\mathbb{R}^J\) with \(\sum_j|\varphi_j|\le1\) and \(\sum_j\varphi_jP_{\varepsilon_0}(y_j-x)\ge0\) for all real \(x\). Every certificate satisfies \(-\sum_j\varphi_j\rho_{\varepsilon_0}(y_j)\le\frac{12\kappa^2}{\pi((\varepsilon_0-\kappa)^2+D^2)^{3/2}}\). For the data \(P_{\varepsilon_0-\kappa}+P_{\varepsilon_0+\kappa}\) with no other zeros, a floating point linear program with \(41\) points on a window \([D,D+4]\) attains between \(0.9\) and \(1.3\) per cent of the one sided ceiling \(6\kappa^2/(\pi((\varepsilon_0-\kappa)^2+D^2)^{3/2})\) at \(D=\varepsilon_0\) (computed).

Source: ch/tf3.tex, label tf:prop:ceiling.

No averaging stable property carries real zeros

Volume II, Corollary 7.4.2, page 161 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(\theta)=\xi(\tfrac12+i\theta)\), \(\Phi(u)=2\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\bigr)\exp(-\pi n^2e^{2u})\) the Pólya kernel, and for real \(t\) let \(H_t(z)=\tfrac12\int_0^\infty e^{tu^2}\Phi(2u)\cos(zu)\,du\) be the de Bruijn Newman function, defined for every real \(t\) because \(\Phi(2u)\) decays like \(e^{9u}\exp(-\pi e^{4u})\), with \(H_0(z)=\tfrac18\Xi(z/2)\). For \(s>0\) and \(Z\) a standard normal variable, \(H_{t-s}(z)=\mathbb{E}\bigl[H_t(z+\sqrt{2s}\,Z)\bigr]\) for complex \(z\). Let \(\mathcal P\) be a property of entire functions that is preserved by the Gaussian average \(f\mapsto\mathbb{E}f(\cdot+\sqrt{2s}\,Z)\) for some \(s>0\). If \(\mathcal P\) implies that all zeros are real, then \(H_t\) has the property \(\mathcal P\) for no real \(t\).

Earlier editions: Corollary 19.4.2, page 739 (edition 1).

Source: ch/tf4.tex, label tf:cor:average.

Question: a certificate with a few levels for the actual zeros

Volume II, Problem 7.1, page 162 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(\theta)=\xi(\tfrac12+i\theta)\), \(\rho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\tfrac12+\varepsilon+i\theta)\) the Thorin density at level \(\varepsilon\) and \(\mathcal L_\Xi=-(\log|\Xi|)''\) Laguerre’s quantity. In a lattice model with one hypothetical off line quadruple among regularly spaced zeros of the critical line, tested only at its ordinate, about \(\tfrac14\log(h/2\pi)\) levels are needed at height \(h\). The problem is to turn this into a certificate for the actual zeros: find levels \(\varepsilon_1<\dots<\varepsilon_K\) such that positivity of every \(\rho_{\varepsilon_k}\) on \([0,h]\), together with positivity of \(\mathcal L_\Xi\), excludes all off line zeros of \(\xi\) up to height \(h\), allowing several off line zeros, irregular spacing and minima of \(\rho_\varepsilon\) off the ordinates of the zeros. Is \(K\) still of order \(\log h\)? The book notes that such a certificate would be finite information and a different verification of RH up to height \(h\), from \(\xi\) on lines to the right of the critical line rather than from sign changes on it.

Earlier editions: Problem 19.1, page 740 (edition 1).

Source: ch/tf4.tex, label tf:prob:levels.

Question: stability with an a priori bound that comes from the Thorin structure

Volume II, Problem 7.2, page 162 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and suppose that the strip \(\tfrac12+\varepsilon_1\le\operatorname{Re}s\le\tfrac12+\varepsilon_0\) is free of zeros. Hadamard’s three lines theorem bounds the error of an approximation \(g\) to \(\xi'/\xi\) at level \(\varepsilon\) (on the line \(\operatorname{Re}s=\tfrac12+\varepsilon\)) by \(\eta^{\lambda}M^{1-\lambda}\) with \(\lambda=(\varepsilon-\varepsilon_1)/(\varepsilon_0-\varepsilon_1)\), where \(\eta\) is the error at level \(\varepsilon_0\) and \(M\) is a bound at level \(\varepsilon_1\). The bound \(M\) presupposes the zero free strip. Is there a formulation in which the a priori bound comes from the Thorin structure itself, for instance from positivity of the Thorin density \(\rho_\varepsilon=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\tfrac12+\varepsilon+i\,\cdot)\) at the levels already certified?

Earlier editions: Problem 19.2, page 740 (edition 1).

Source: ch/tf4.tex, label tf:prob:three.

Question: which positivity statements fail for negative heat time by a mechanism visible at finite height

Volume II, Problem 7.3, page 162 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and \(H_t\) the de Bruijn Newman function, with \(H_0(z)=\tfrac18\Xi(z/2)\) for \(\Xi(\theta)=\xi(\tfrac12+i\theta)\) and \(R_t(s)=H_t(0)/H_t(2i\sqrt s)\), so that \(R_0(s)=\xi(\tfrac12)/\xi(\tfrac12+\sqrt s)\) is the Laplace transform of the clock \(T\) of \(\xi\). Since \(H_t\) has nonreal zeros for every \(t<0\) (Rodgers and Tao), a property of \(H_0\) that implies real zeros fails for \(H_t\) at every \(t<0\). Which of the positivity statements of the book, for instance the generalized gamma convolution (GGC) property of the clock, whose Laplace transform is \(R_0\), or the Pólya frequency property of the density of \(Z\sqrt{2T}\) (with \(Z\) a standard normal variable independent of \(T\)), can be shown to fail for \(t<0\) by a mechanism visible at finite height, as Lehmer pairs make the de Bruijn Newman constant \(\Lambda\) close to \(0\) visible (Csordas, Smith and Varga 1994)?

Earlier editions: Problem 19.3, page 740 (edition 1).

Source: ch/tf4.tex, label tf:prob:heat.

Question: the window of levels below delta, its distribution over heights and the sine kernel

Volume II, Problem 7.4, page 162 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and add by hand a hypothetical quadruple \(\tfrac12\pm\delta\pm i\gamma^*\) to its zeros, at an ordinate \(\gamma^*\). Let \(\rho^0_\varepsilon(\gamma^*)\) be the Thorin density at level \(\varepsilon\) and at the ordinate \(\gamma^*\) of the other zeros, \(\delta_{\mathrm{det}}(\varepsilon)^2=\varepsilon^2+2\varepsilon/(\pi\rho^0_\varepsilon(\gamma^*))\) the visibility curve (the level \(\varepsilon\) sees the quadruple at its ordinate exactly when \(\varepsilon<\delta<\delta_{\mathrm{det}}(\varepsilon)\)) and \(\delta_c=\lim_{\varepsilon\to0}\delta_{\mathrm{det}}(\varepsilon)\) the first order visibility distance. For a quadruple at distance \(\delta>\delta_c\) the levels at which the test detects it form a set contained in \((0,\delta)\), an interval \((\varepsilon_c,\delta)\) in the examples of the book; at the ordinate \(\varepsilon_c\) is the inverse of the visibility curve. Describe the distribution of \(\delta_{\mathrm{det}}\), and of the onset \(\varepsilon_c(\delta,\gamma^*)\), over heights in terms of the local zero statistics, and compare it with the prediction of the sine kernel.

Earlier editions: Problem 19.4, page 740 (edition 1).

Source: ch/tf4.tex, label tf:prob:window.

Volume II, Chapter 8. Deconvolution as a damped Weil criterion

The deconvolved Thorin functional is Weil’s distribution

Volume II, Theorem 8.1.1, page 166 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, with nontrivial zeros \(\rho=\beta+i\gamma\) counted with multiplicity, and put \(\gamma_\rho=-i(\rho-\frac12)=\gamma-i(\beta-\frac12)\), so that the \(\gamma_\rho\) are the zeros of \(\Xi(\theta)=\xi(\frac12+i\theta)\). For real \(\alpha\) and \(\theta\) let \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\). Call \(\varepsilon_0>0\) to the right of the zeros if \(\beta<\frac12+\varepsilon_0\) for every zero; every \(\varepsilon_0\ge\frac12\) is, because \(\zeta\) has no zero with \(\beta\ge1\), so no hypothesis on the zeros is needed. For such \(\varepsilon_0\) let \(\varrho_{\varepsilon_0}(\theta)=\pi^{-1}v_{1/2+\varepsilon_0}(\theta)\), the Thorin density of the shifted clock at the basepoint \(\frac12+\varepsilon_0\) in the chart \(\theta=\sqrt z\) (the Poisson smoothing of the zeros). Write \(P_a(x)=a/(\pi(a^2+x^2))\) for the Cauchy density of scale \(a>0\) and \(\hat f(t)=\int f(x)e^{itx}\,dx\), so that \(\hat P_a(x)=e^{-a|x|}\). Let \(\mathcal{B}\) be the space of even functions \(\varphi(\theta)=\frac1{2\pi}\int\hat\varphi(x)e^{-i\theta x}\,dx\) with \(\hat\varphi\) of compact support, Lipschitz, and with \(\hat\varphi'\) of bounded variation; each such \(\varphi\) is entire of exponential type, with \(\varphi(\theta+is)=O(\theta^{-2})\) uniformly for \(|s|\le1\). For \(a\ge0\) let \(P_a^{-1}\varphi\in\mathcal{B}\) be the function with Fourier transform \(\hat\varphi(x)e^{a|x|}\). Let \(\varepsilon_0\) be to the right of the zeros and, for \(\varphi\in\mathcal{B}\), put \(D(\varphi)=\int\varrho_{\varepsilon_0}(\theta)\,(P_{\varepsilon_0}^{-1}\varphi)(\theta)\,d\theta\). Then \(D(\varphi)\) does not depend on \(\varepsilon_0\), and with \(g(x)=\hat\varphi(x)/2\pi\), so that \(\varphi(\theta)=\int g(x)e^{i\theta x}dx=\hat g(\theta)\), [ D()=()=g(i2)+g(-i2)-(_E+),g(0)+0^,dx-2{n},g(n), ] where the sum over \(\rho\) runs over all zeros with multiplicity, in conjugate pairs, and converges absolutely, \(\gamma_E\) is Euler’s constant and \(\Lambda\) the von Mangoldt function. The second expression is Weil’s explicit formula on \(\mathcal{B}\), so \(D\) is computed from the gamma factor, the pole and the primes \(n\) with \(\log n\) in the support of \(g\); it is the action on \(g\) of Weil’s distribution \(\sum_\rho e^{(\rho-1/2)x}\), that is, \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\). Moreover, through the \(x\) form of its archimedean term, the Thorin functional \(U\) of the clock (the linear functional with \(\langle U,\psi\rangle=\sum_k\psi(a_k)\) over the exponents \(a_k=-(\rho_k-\frac12)^2\), one for each pair of zeros \(\{\rho,1-\rho\}\), defined by the explicit formula, in which no zero appears) extends to \(\mathcal{B}\), and then \(D(\varphi)=2\langle U,\varphi(\sqrt{\cdot})\rangle\).

Earlier editions: Theorem 20.1.1, page 744 (edition 1).

Source: ch/dc1.tex, label dc:thm:weil.

Damped Weil criterion

Volume II, Theorem 8.1.2, page 167 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with nontrivial zeros \(\rho=\beta+i\gamma\) counted with multiplicity, \(\gamma_\rho=-i(\rho-\frac12)\), and for \(\varepsilon>0\) let \(\varrho_\varepsilon(\theta)=\pi^{-1}\operatorname{Re}\frac{\xi'}{\xi}(\frac12+\varepsilon+i\theta)\), the Poisson smoothing of the zeros. Weil’s distribution \(D\) acts on an even Lipschitz function \(g\) of compact support with \(g'\) of bounded variation by \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\), where \(\hat g(t)=\int g(x)e^{itx}\,dx\) (Weil’s explicit formula computes this from the gamma factor, the pole of \(\zeta\) and the primes), and on a function that is not even through its even part. For a bounded function \(h\) of bounded variation and compact support let \(\tilde h(x)=\overline{h(-x)}\), so that \(h*\tilde h\) is Lipschitz with \(\widehat{h*\tilde h}=|\hat h|^2\), and for \(\varepsilon\ge0\) put \(Q_\varepsilon(h)=\langle D,e^{-\varepsilon|x|}(h*\tilde h)\rangle\), Weil’s form damped by \(e^{-\varepsilon|x|}\). Call \(\varepsilon>0\) to the right of the zeros if \(\beta<\frac12+\varepsilon\) for every zero. Let \(0\le\varepsilon\le\frac12\). Then \(Q_\varepsilon(h)\ge0\) for every \(h\in C_c^\infty(\mathbb{R})\) (real \(h\) suffice) if and only if no nontrivial zero of \(\zeta\) has \(\beta>\frac12+\varepsilon\). When \(\varepsilon>0\) is to the right of the zeros, \(Q_\varepsilon(h)=\int_{\mathbb{R}}|\hat h(\theta)|^2\varrho_\varepsilon(\theta)\,d\theta\) for every bounded \(h\) of bounded variation and compact support. In particular: at \(\varepsilon=0\) this is Weil’s criterion for the Riemann Hypothesis; at \(\varepsilon=\frac12\) it holds with no hypothesis on the zeros; and the zero free half plane \(\operatorname{Re}s>\frac78\) announced in October 2026, which has not been refereed, is equivalent to \(Q_{3/8}\ge0\).

Earlier editions: Theorem 20.1.2, page 745 (edition 1).

Source: ch/dc1.tex, label dc:thm:damped.

Ladder theorem: least eigenvalue of the damped Weil form on a window

Volume II, Theorem 8.2.1, page 169 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Parts (a) and (b), for a level \(0<\varepsilon<\frac12\): that the closed half plane \(\operatorname{Re}s\ge\frac12+\varepsilon\) contains no zero of \(\zeta\) (that \(\varepsilon\) is to the right of the zeros). This zero free half plane is not proved for any \(\varepsilon<\frac12\); the half plane \(\operatorname{Re}s>\frac78\) announced in October 2026 and not refereed would give it for \(\varepsilon>\frac38\) only.; Parts (a) and (b), for a level \(0<\varepsilon<\frac12\): that \(v_{1/2+\varepsilon}(\theta)>v_{1/2+\varepsilon}(0)\) for every real \(\theta\ne0\) and \(v_{1/2+\varepsilon}''(0)>0\), that is, that the Thorin density has a strict, nondegenerate global minimum at the origin. The book certifies this only up to a finite height, and the strict minimum at the origin on every line \(\frac12<\alpha<1\) at once is equivalent to RH.; Parts (a) and (b), for a level \(0<\varepsilon<\frac12\): that \(\liminf_{|\theta|\to\infty}v_{1/2+\varepsilon}(\theta)>v_{1/2+\varepsilon}(0)\), that is, that the minimum is not approached at infinity. This is not known for \(0<\varepsilon<\frac12\) without RH, under which it holds.; Part (c), first half (the nonnegativity and the decay of \(\lambda_{\min}(\frac12,a)\)): the Riemann Hypothesis. All hypotheses of (a) and (b) hold at \(\varepsilon=\frac12\) with no assumption on the zeros, and the second half of (c), the negativity for large \(a\) if RH fails, needs no hypothesis. For \(\varepsilon>\frac12\) the chapter does not discuss whether the hypotheses on \(v_{1/2+\varepsilon}\) hold.

Statement. Let \(\xi\) be the completed Riemann zeta function with nontrivial zeros \(\rho=\beta+i\gamma\) counted with multiplicity and \(\gamma_\rho=-i(\rho-\frac12)\). For real \(\alpha\) and \(\theta\) put \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\) and \(b_\alpha=v_\alpha(0)=\frac{\xi'}{\xi}(\alpha)\). Weil’s distribution \(D\) acts on an even Lipschitz function \(g\) of compact support with \(g'\) of bounded variation by \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\), where \(\hat g(t)=\int g(x)e^{itx}\,dx\) (Weil’s explicit formula computes this from the gamma factor, the pole of \(\zeta\) and the primes). For a bounded real function \(h\) of bounded variation and compact support let \(\tilde h(x)=h(-x)\) and, for \(\varepsilon\ge0\), \(Q_\varepsilon(h)=\langle D,e^{-\varepsilon|x|}(h*\tilde h)\rangle\); for \(a>0\) let \(\lambda_{\min}(\frac12+\varepsilon,a)\) be the infimum of \(Q_\varepsilon(h)/\|h\|_2^2\) over all such \(h\ne0\) supported in \([-a,a]\) (only the prime powers \(n\le e^{2a}\) enter \(Q_\varepsilon(h)\) for these \(h\)). Let \(\varepsilon>0\) be to the right of the zeros, that is, \(\beta<\frac12+\varepsilon\) for every zero, and suppose that \(v_{1/2+\varepsilon}(\theta)>v_{1/2+\varepsilon}(0)\) for every \(\theta\ne0\) (a strict global minimum at the origin), that \(v_{1/2+\varepsilon}''(0)>0\) (nondegenerate), and that \(\liminf_{|\theta|\to\infty}v_{1/2+\varepsilon}(\theta)>v_{1/2+\varepsilon}(0)\) (not approached at infinity). Then: (a) \(\lambda_{\min}(\frac12+\varepsilon,a)\) is nonincreasing in \(a\), and \(\lambda_{\min}(\frac12+\varepsilon,a)\ge2b_{1/2+\varepsilon}>0\) for every \(a>0\); (b) \(\lambda_{\min}(\frac12+\varepsilon,a)\to2b_{1/2+\varepsilon}\) as \(a\to\infty\), and there are constants \(c,a_0>0\) such that for all \(a\ge a_0\), \(\frac c{a^2}\le\lambda_{\min}(\frac12+\varepsilon,a)-2b_{1/2+\varepsilon}\le\frac{\pi^2v_{1/2+\varepsilon}''(0)/4+o(1)}{a^2}\), where \(o(1)\to0\) as \(a\to\infty\); (c) at \(\varepsilon=0\), with none of the hypotheses above: if the Riemann Hypothesis holds, then \(\lambda_{\min}(\frac12,a)\ge0\) for every \(a\) and, for every \(N\), \(\lambda_{\min}(\frac12,a)=O(a^{-N})\) as \(a\to\infty\), so that \(\lambda_{\min}(\frac12,a)\to0\); if the Riemann Hypothesis fails, then \(\lambda_{\min}(\frac12,a)<0\) for all large \(a\).

Earlier editions: Theorem 20.2.1, page 747 (edition 1).

Source: ch/dc2.tex, label dc:thm:ladder.

A Schur bound for the step in the level of the damped Weil form

Volume II, Proposition 8.2.2, page 174 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Q_c\) denote the damped Weil quadratic form at level \(c\) (damping \(c-\frac12\), as in Theorem on the damped Weil criterion), let \(\frac12<c\le1\) and \(0<\eta<c-\frac12\). On real functions supported in an interval of length \(L\) the operator norm of \(Q_{c-\frac12-\eta}-Q_{c-\frac12}\), in the indexing of the damping, is at most \(\eta B_c(L)\), where for an absolute constant \(C_0\), \(B_c(L)\le2\sum_{n\le e^L}\frac{\Lambda(n)\log n}{n^{c-\eta}}+4L\int_0^L\cosh\tfrac x2\,e^{-(c-\eta-\frac12)x}\,dx+C_0\). Consequently, if the smallest eigenvalue \(\lambda_{\min}(c,L/2)\ge\mu>0\), then \(\lambda_{\min}(c-\eta,L/2)\ge0\) for \(0<\eta<c-\frac12\) with \(\eta B_c(L)\le\mu\).

Source: ch/dc2.tex, label dc:prop:schur.

Certified rungs of the ladder

Volume II, Problem 8.1, page 177 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho\) and \(\gamma_\rho=-i(\rho-\frac12)\). Weil’s distribution \(D\) acts on an even Lipschitz function \(g\) of compact support with \(g'\) of bounded variation by \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\), \(\hat g(t)=\int g(x)e^{itx}\,dx\) (Weil’s explicit formula computes this from the gamma factor, the pole and the primes). For bounded real \(h\) of bounded variation and compact support let \(\tilde h(x)=h(-x)\) and \(Q_\varepsilon(h)=\langle D,e^{-\varepsilon|x|}(h*\tilde h)\rangle\), and for \(a>0\) let \(\lambda_{\min}(\frac12+\varepsilon,a)\) be the infimum of \(Q_\varepsilon(h)/\|h\|_2^2\) over all such \(h\ne0\) supported in \([-a,a]\). Certify, in interval arithmetic, that \(\lambda_{\min}(\frac12+\varepsilon,a)>0\) for \(\varepsilon\in\{0.05,0.1,0.2,0.375\}\) and \(a=4\), which by monotonicity in \(a\) covers every \(a\le4\), and determine how far in \(a\) a certificate can be pushed at fixed \(\varepsilon\). Each certificate would be unconditional, and at \(\varepsilon=\frac38\) it would agree with what the announced zero free half plane \(\operatorname{Re}s>\frac78\) gives, namely \(\lambda_{\min}\ge0\), and could be checked independently of it. Certifying one rung needs a lower bound for the bottom of the spectrum, not the variational upper bounds and extrapolated values the book computes; the book has not done this.

Earlier editions: Problem 20.1, page 754 (edition 1).

Source: ch/dc3.tex, label dc:prob:cert.

The exact rate of the ladder

Volume II, Problem 8.2, page 177 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\beta+i\gamma\) and \(\gamma_\rho=-i(\rho-\frac12)\), and put \(v_\alpha(\theta)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\theta)\) and \(b_\alpha=v_\alpha(0)=\frac{\xi'}{\xi}(\alpha)\). Weil’s distribution \(D\) acts on an even Lipschitz function \(g\) of compact support with \(g'\) of bounded variation by \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\), \(\hat g(t)=\int g(x)e^{itx}\,dx\). For bounded real \(h\) of bounded variation and compact support let \(\tilde h(x)=h(-x)\) and \(Q_\varepsilon(h)=\langle D,e^{-\varepsilon|x|}(h*\tilde h)\rangle\), and for \(a>0\) let \(\lambda_{\min}(\frac12+\varepsilon,a)\) be the infimum of \(Q_\varepsilon(h)/\|h\|_2^2\) over all such \(h\ne0\) supported in \([-a,a]\). Let \(\varepsilon>0\) be to the right of the zeros (\(\beta<\frac12+\varepsilon\) for every zero), and suppose that \(v_{1/2+\varepsilon}(\theta)>v_{1/2+\varepsilon}(0)\) for every \(\theta\ne0\), that \(v_{1/2+\varepsilon}''(0)>0\), and that \(\liminf_{|\theta|\to\infty}v_{1/2+\varepsilon}(\theta)>v_{1/2+\varepsilon}(0)\). Prove that \(a^2\big(\lambda_{\min}(\frac12+\varepsilon,a)-2b_{1/2+\varepsilon}\big)\to\pi^2v_{1/2+\varepsilon}''(0)/4\) as \(a\to\infty\), and find the next term. Under these hypotheses the book proves only that this scaled excess is at most \(\pi^2v_{1/2+\varepsilon}''(0)/4+o(1)\) and at least a positive constant \(c\). The book’s numerical fits of converged, extrapolated values are consistent with the limit and with a positive correction of order \(1/a\), so that \(\pi^2v_{1/2+\varepsilon}''(0)/4\) would not be an upper bound at finite \(a\); neither the limit nor the order of the correction is proved.

Earlier editions: Problem 20.2, page 754 (edition 1).

Source: ch/dc3.tex, label dc:prob:rate.

The transition of the ladder at the line

Volume II, Problem 8.3, page 177 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho\) and \(\gamma_\rho=-i(\rho-\frac12)\), and put \(b_\alpha=\frac{\xi'}{\xi}(\alpha)\) for real \(\alpha\) and \(V_\xi=(\log\xi)''(\frac12)\). Weil’s distribution \(D\) acts on an even Lipschitz function \(g\) of compact support with \(g'\) of bounded variation by \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\), \(\hat g(t)=\int g(x)e^{itx}\,dx\). For bounded real \(h\) of bounded variation and compact support let \(\tilde h(x)=h(-x)\) and \(Q_\varepsilon(h)=\langle D,e^{-\varepsilon|x|}(h*\tilde h)\rangle\), and for \(a>0\) let \(\lambda_{\min}(\frac12+\varepsilon,a)\) be the infimum of \(Q_\varepsilon(h)/\|h\|_2^2\) over all such \(h\ne0\) supported in \([-a,a]\). Describe \(\lambda_{\min}(\frac12+\varepsilon,a)\) jointly as \(a\to\infty\) and \(\varepsilon\to0\). For fixed \(\varepsilon>0\), under the hypotheses of the ladder theorem (the level \(\varepsilon\) to the right of the zeros, and the Thorin density \(\pi^{-1}\operatorname{Re}\frac{\xi'}{\xi}(\frac12+\varepsilon+i\theta)\) having a strict, nondegenerate global minimum at \(\theta=0\) that is not approached at infinity), the limit is \(2b_{1/2+\varepsilon}\), which is at most \(2V_\xi\,\varepsilon\) and asymptotic to it as \(\varepsilon\downarrow0\), and the rate is algebraic; at \(\varepsilon=0\), under RH, the decay is faster than any power of \(a\). The questions are: on which curves \(\varepsilon=\varepsilon(a)\to0\) does \(\lambda_{\min}(\frac12+\varepsilon,a)\) stay comparable to \(2b_{1/2+\varepsilon}\), and does the crossover occur at \(\varepsilon\) of the order of the mean spacing of the zeros at height \(e^{2a}\)?

Earlier editions: Problem 20.3, page 754 (edition 1).

Source: ch/dc3.tex, label dc:prob:transition.

Uniformity of the damped Weil form from structure

Volume II, Problem 8.4, page 177 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho\) and \(\gamma_\rho=-i(\rho-\frac12)\). Weil’s distribution \(D\) acts on an even Lipschitz function \(g\) of compact support with \(g'\) of bounded variation by \(\langle D,g\rangle=\sum_\rho\hat g(\gamma_\rho)\), \(\hat g(t)=\int g(x)e^{itx}\,dx\) (Weil’s explicit formula computes this from the gamma factor, the pole and the primes). For bounded real \(h\) of bounded variation and compact support let \(\tilde h(x)=h(-x)\) and \(Q_\varepsilon(h)=\langle D,e^{-\varepsilon|x|}(h*\tilde h)\rangle\). Find a structural reason, valid for all \(a\), why \(Q_\varepsilon(h)\ge0\) for every such \(h\) supported in \([-a,a]\) (that is, \(Q_\varepsilon\ge0\) on \(L^2[-a,a]\)) for some fixed \(\varepsilon<\frac12\), using properties of the archimedean and pole terms that dominate the primes up to \(e^{2a}\) uniformly in \(a\). By the damped Weil criterion, any such argument proves the zero free half plane \(\operatorname{Re}s>\frac12+\varepsilon\).

Earlier editions: Problem 20.4, page 754 (edition 1).

Source: ch/dc3.tex, label dc:prob:uniform.

Problem of rigorous visibility of off line zeros in a finite window

Volume II, Problem 8.5, page 177 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(b>0\) let \(K_b(x,y)=e^{b|x-y|}\) on an interval of length \(L\); it has one large positive and one large negative eigenvalue, both about \(\pm e^{bL}/(2b)\) when \(bL\) is large, and the others of order \(1\). A pair of zeros off the line with real parts \(\frac12\pm(\varepsilon+b)\) adds to the damped Weil form \(Q_\varepsilon\) at level \(c=\frac12+\varepsilon\) a term of about \(+\frac12\langle h,K_bh\rangle\) at the frequency of the zero, so heuristically a window of length \(L\) sees a zero of height \(\gamma\) once \(e^{bL}\gtrsim2b\log\gamma\), that is \(L\gtrsim\log\log\gamma/b\). Problem: prove that positivity of \(Q_\varepsilon\) on intervals of length \(L\) excludes every zero with \(\beta>\frac12+\varepsilon+C\log\log\gamma/L\) at every height \(\gamma\), controlling clusters of off line zeros by zero density estimates.

Source: ch/dc3.tex, label dc:prob:visible.

Volume II, Chapter 9. The level one measure and the backward Cauchy flow

The Thorin measure of reciprocal xi is absolutely continuous with the explicit density (1/pi) Re xi’/xi at every level from one

Volume II, Theorem 9.1.1, page 180 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(\rho=\sigma_\rho+i\gamma_\rho\) run over its zeros with multiplicity, \(\Lambda\) the von Mangoldt function, and for \(\theta>\sigma_\rho\) for every \(\rho\) (in particular for every \(\theta>1\), and also for \(\theta=1\) because \(\zeta(1+it)\ne0\)) let \(\mathfrak U_\theta=\sum_\rho\mathrm{Cauchy}(\gamma_\rho,\theta-\sigma_\rho)\) be the positive measure on \(\mathbb{R}\) with \(\int\mathfrak U_\theta(dv)/(1+v^2)<\infty\), where \(\mathrm{Cauchy}(\gamma,c)\) has density \((c/\pi)/(c^2+(v-\gamma)^2)\). Then \(\mathfrak U_\theta\) is absolutely continuous with real analytic density \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)=\frac1\pi\partial_\sigma\log|\xi(\sigma+iv)|\) at \(\sigma=\theta\), and \(\mathfrak U_\theta((0,V])=\frac1\pi\arg\xi(\theta+iV)\) for the branch with \(\arg\xi(\theta)=0\). At \(\theta=1\), for every real \(v\), \(u_1(v)=\frac1\pi\bigl[\frac1{1+v^2}-\frac{\log\pi}2+\frac12\operatorname{Re}\frac{\Gamma'}{\Gamma}(\frac{1+iv}2)-\lim_{X\to\infty}\bigl(\sum_{n\le X}\frac{\Lambda(n)}n\cos(v\log n)-\frac{\sin(v\log X)}{v}\bigr)\bigr]\), where \(\sin(v\log X)/v\) means \(\log X\) at \(v=0\); the limit exists locally uniformly, equals \(\operatorname{Re}C(1+iv)\) with \(C(s)=-\zeta'(s)/\zeta(s)-1/(s-1)\), and \(u_1(0)=(2+\gamma_E-\log4\pi)/(2\pi)\) with \(\gamma_E\) Euler’s constant, while the cosine series itself has no limit. For \(\theta>1\), \(u_\theta(v)=\frac1\pi\bigl[\frac\theta{\theta^2+v^2}+\frac{\theta-1}{(\theta-1)^2+v^2}-\frac{\log\pi}2+\frac12\operatorname{Re}\frac{\Gamma'}{\Gamma}(\frac{\theta+iv}2)-\sum_{n\ge2}\Lambda(n)n^{-\theta}\cos(v\log n)\bigr]\) with the prime sum absolutely convergent.

Earlier editions: Theorem 30.1.1, page 1051 (edition 1).

Source: ch/lo1.tex, label lo:thm:density.

The Fourier transform of the level one Thorin measure carries the prime powers as atoms, an explicit formula for one test function

Volume II, Proposition 9.2.1, page 182 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With \(\mathfrak U_1=\sum_\rho\mathrm{Cauchy}(\gamma_\rho,1-\sigma_\rho)\) the level one Thorin measure of \(\xi\) (zeros \(\rho=\sigma_\rho+i\gamma_\rho\)), \(\Lambda\) the von Mangoldt function, \(P_c(v)=(c/\pi)/(c^2+v^2)\), \(\widehat f(\omega)=\int e^{-i\omega v}f(v)\,dv\), and for \(\theta\ge1\) \(g_\theta(v)=\frac1\pi[\theta/(\theta^2+v^2)-\frac12\log\pi+\frac12\operatorname{Re}\frac{\Gamma'}\Gamma(\frac{\theta+iv}2)]\): as tempered distributions \(\widehat{\mathfrak U_1}(\omega)=\widehat{g_1}(\omega)+1-\sum_{n\ge2}\frac{\Lambda(n)}n(\delta_{\log n}+\delta_{-\log n})(\omega)\), the constant \(1\) being the transform of the atom at \(0\) that the pole of \(\zeta\) contributes in the limit \(\theta\downarrow1\), and on the zero side \(\widehat{\mathfrak U_1}(\omega)=\sum_\rho e^{-(1-\sigma_\rho)|\omega|}e^{-i\omega\gamma_\rho}\) as the limit of the sums over \(|\gamma_\rho|<T\). Equivalently, for every Schwartz function \(\varphi\), \(\sum_\rho(P_{1-\sigma_\rho}*\varphi)(\gamma_\rho)=\int g_1\varphi\,dv+\varphi(0)-\frac1{2\pi}\sum_{n\ge2}\frac{\Lambda(n)}n(\widehat\varphi(\log n)+\widehat\varphi(-\log n))\). For \(\theta\ge1\), \(\mathfrak U_\theta=P_{\theta-1}*\mathfrak U_1\) and \(\widehat{\mathfrak U_\theta}(\omega)=e^{-(\theta-1)|\omega|}\widehat{\mathfrak U_1}(\omega)\), which replaces \(\Lambda(n)/n\) by \(\Lambda(n)n^{-\theta}\) on the prime side.

Earlier editions: Proposition 30.2.1, page 1053 (edition 1).

Source: ch/lo1.tex, label lo:prop:spectrum.

Dichotomy for the band limited backward Cauchy flow from level one: convergence at level theta’ if and only if zeta has no zeros to the right of theta’

Volume II, Theorem 9.3.1, page 185 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12\le\theta'<1\), \(a=1-\theta'\), \(\mathfrak U_\theta=\sum_\rho\mathrm{Cauchy}(\gamma_\rho,\theta-\sigma_\rho)\) the Thorin measure at level \(\theta\) of \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), over its zeros \(\rho=\sigma_\rho+i\gamma_\rho\) with multiplicity (\(\mathrm{Cauchy}(\gamma,c)\) the Cauchy law of centre \(\gamma\) and width \(c\), \(\mathrm{Cauchy}(\gamma,0)=\delta_\gamma\)), \(\Phi_\Omega(v)=(1-\cos\Omega v)/(\pi\Omega v^2)\) the Fejér kernel, and \(K_{\theta',\Omega}\) the inverse Fourier transform of \((1-|\omega|/\Omega)_+e^{a|\omega|}\). The band limited backward flow is \(B_{\theta',\Omega}\mathfrak U_1=\mathfrak U_1*K_{\theta',\Omega}\). (i) If \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta'\), then \(B_{\theta',\Omega}\mathfrak U_1=\mathfrak U_{\theta'}*\Phi_\Omega\), which converges vaguely to \(\mathfrak U_{\theta'}\) as \(\Omega\to\infty\), with \(\int B_{\theta',\Omega}\mathfrak U_1(v)\,dv/(v^2+s)\to\int\mathfrak U_{\theta'}(dv)/(v^2+s)\) for every \(s>0\); if moreover \(\theta'>\sigma_\rho\) for every zero the convergence is locally uniform as densities. (ii) If \(\zeta\) has a zero \(\rho\) with \(\sigma_\rho>\theta'\), there is no signed measure \(\mu\) with \(\int|\mu|(dv)/(1+v^2)<\infty\) such that \(\int B_{\theta',\Omega}\mathfrak U_1(v)\,dv/(v^2+s)\to\int\mu(dv)/(v^2+s)\) as \(\Omega\to\infty\) for all large \(s\). Hence the Riemann hypothesis is equivalent to convergence of the flow at every \(\theta'>\frac12\).

Earlier editions: Theorem 30.3.1, page 1056 (edition 1).

Source: ch/lo1.tex, label lo:thm:dichotomy.

Computation: the Fourier backward flow from samples of the level one density on Re s = 1 places a peak within 0.035 of each of 109 zeros

Volume II, Result 9.3.1, page 186 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. The computation samples the level one Thorin density \(u_1(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(1+iv)\), \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), on \([0,400]\) at spacing \(0.05\) from \(\zeta\) and \(\zeta'\) on the line \(\operatorname{Re}s=1\) only (the Arb library (python flint) balls at \(128\) bits, with the analytic value of \(u_1(0)\)), applies a Hann window and the backward multiplier \(e^{(1-\theta')|\omega|}\) with a cutoff at \(\Omega\), and inverts. A peak is a local maximum of the output in \(100<v<300\) of height at least a quarter of the maximum, and the \(109\) zeros of \(\zeta\) in that range are those counted by \(N(300)-N(100)=138-29\). With a sharp cutoff the output at \(\theta'=0.75\), \(\Omega=40\) has \(109\) peaks, each within \(0.035\) of a zero, and is positive; at \(\theta'=0.5\), \(\Omega=40\) it has \(109\) peaks within \(0.027\) of the zeros and minimum \(-3.2\); at \(\theta'=0.5\), \(\Omega=60\) it has \(213\) peaks, \(109\) at the zeros, error \(0.042\) and minimum \(-16.5\), which is ringing of the cutoff. With the Fejér window at \(\theta'=0.5\), \(\Omega=60\) it has \(109\) peaks, error \(0.025\) and minimum \(-0.020\). The statement covers only these runs, and the sharp cutoff is not covered by the convergence theorem for the backward flow.

Earlier editions: Result 30.3.1, page 1057 (edition 1).

Source: ch/lo1.tex, label lo:res:flow.

A positive solution of the Cauchy deconvolution of the level one density exists exactly when zeta has the zero free half plane, and the total Kullback Leibler residual detects it

Volume II, Proposition 9.4.1, page 189 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\theta'\in[\frac12,1)\), \(a=1-\theta'\), \(P_a(v)=(a/\pi)/(a^2+v^2)\), \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) with zeros \(\rho=\sigma_\rho+i\gamma_\rho\), \(\mathfrak U_\theta=\sum_\rho\mathrm{Cauchy}(\gamma_\rho,\theta-\sigma_\rho)\) its Thorin measure at level \(\theta\) (defined when no zero has \(\sigma_\rho>\theta\); \(\mathrm{Cauchy}(\gamma,0)=\delta_\gamma\)), and \(u_1(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(1+iv)\) the density of \(\mathfrak U_1\). (a) There is a positive measure \(x\) on \(\mathbb{R}\) with \(\int x(dv)/(1+v^2)<\infty\) and \(P_a*x=u_1\) if and only if \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta'\); it is then unique and equals \(\mathfrak U_{\theta'}\). (b) Let \(h_k\to0\) and \(L_k\to\infty\) with \(h_k^2L_k\log L_k\to0\), let \(v_i\in h_k\mathbb{Z}\cap[-L_k,L_k]\), \(y_i=u_1(v_i)\), and \(D_k=\inf_xh_k\sum_i\bigl(y_i\log\frac{y_i}{m_i(x)}-y_i+m_i(x)\bigr)\), with the infimum over positive measures \(x\) on the grid \(h_k\mathbb{Z}\) with \(\sum_jx_j/(1+j^2h_k^2)<\infty\) and \(m_i(x)=(P_a*x)(v_i)\). Then \(D_k\to0\) if \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta'\), and \(\liminf_kD_k>0\) if it has one.

Earlier editions: Proposition 30.4.1, page 1060 (edition 1).

Source: ch/lo2.tex, label lo:prop:emres.

Computation: the EM inversion of the level one density separates an injected off line quartet of zeros from a control

Volume II, Result 9.4.4, page 191 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. The computation runs the EM iteration for the Cauchy deconvolution of the level one Thorin density \(u_1(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(1+iv)\) of \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) on the data window \([100,300]\) (unknown on \([80,320]\), spacing \(0.05\)) and on \([150,250]\), with the total Kullback Leibler residual as the plain sum over the data points. The two zeros nearest the centre of the window (\(\gamma=198.0153\) and \(201.2648\)) are replaced by a quartet \(\rho,1-\bar\rho,\bar\rho,1-\rho\) with real part \(\sigma\) at their midpoint, total mass kept to \(10^{-4}\), and compared with a control of two line zeros at the midpoint. At \(\theta'=0.75\), \(\sigma=0.85\) the quartet residual after \(10^4\) steps is \(7.03\times10^{-3}\) on both windows to \(0.3\) per cent, a floor, while the control and the real data fall below \(10^{-7}\). At \(\theta'=0.5\), \(\sigma=0.75\) the quartet residual is \(1.85\times10^{-2}\) against \(1.08\times10^{-3}\) for the control and has not reached a floor. At \(\theta'=0.75=\sigma\) the residual keeps decaying, more slowly than the control. The statement covers only these runs: a finite run certifies neither the existence nor the nonexistence of a positive solution.

Earlier editions: Result 30.4.4, page 1063 (edition 1).

Source: ch/lo2.tex, label lo:res:inject.

Computation: generative Bayesian computation under a GUE prior counts the zeros of zeta in unit bins from the level one Thorin density

Volume II, Result 9.5.2, page 192 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(u_1(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(1+iv)\) its level one Thorin density. The computation trains, for two seeds, a quantile network (two hidden layers of \(256\) rectified units, pinball loss, quantile levels \(0.1\), \(0.5\), \(0.9\)) on \(24000\) simulated windows in which the ordinates of the zeros are eigenvalues of a GUE matrix of size \(400\) unfolded to a density drawn uniformly from \([0.45,0.8]\), all zeros on the critical line; the input is the sum of Cauchy densities of width \(\frac12\) at the simulated ordinates on a grid of spacing \(0.1\), plus the Cauchy smoothing of the mean density for the zeros beyond, and the targets are the counts of zeros in eight unit bins at the centre of a window of length \(16\). On the \(25\) windows of the real \(u_1\) centred at \(v=150,158,\dots,342\) (\(200\) bins) the rounded posterior median equals the true count in \(196\) and \(197\) bins and is within one of it in all \(200\); on held out simulations the interval \([q_{0.1},q_{0.9}]\) covers \(0.806\) and \(0.796\) of the counts (nominal \(0.8\)), and \(0.805\) and \(0.775\) on the real windows. The formulation is heuristic and the statement covers only these runs: the prior puts every zero on the critical line, and the network counts zeros in bins and does not locate them.

Earlier editions: Result 30.5.2, page 1064 (edition 1).

Source: ch/lo2.tex, label lo:res:gbc.

Open problem: the data augmentation sampler and a generative Bayesian inversion with the EM reconstruction as a feature

Volume II, Problem 9.1, page 197 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(u_1(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(1+iv)\) be the level one Thorin density of \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\). Implement a data augmentation Gibbs sampler for the Cauchy deconvolution of \(u_1\) and compare its posterior spread at \(\theta'=0.75\) and \(\theta'=0.5\) with the EM point estimate; then build a generative Bayesian computation inversion with the EM reconstruction as an input feature and the positions of the zeros as continuous outputs, and calibrate its quantiles on held out simulations.

Earlier editions: Problem 30.1, page 1067 (edition 1).

Source: ch/lo3.tex, label lo:prob:sampler.

Open problem: characterise the level one Thorin densities of functions of the Selberg class without reference to zeros

Volume II, Problem 9.2, page 197 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Characterise, without reference to zeros, the functions \(\frac1\pi[g(v)-\lim_{X\to\infty}(\sum_{n\le X}a_n\cos(v\log n)-\sin(v\log X)/v)]\) with \(a_n\ge0\) supported on prime powers and \(g\) a smooth gamma part that are the Thorin densities at level one of a function \(\Xi(1)/\Xi(1+\sqrt s)\) with \(\Xi\) in the Selberg class. The Thorin conditions constrain the growth and the integrability near \(0\) but say nothing about the arithmetic of the \(a_n\). The Selberg class needs the completed function, the gamma factors, the conductor, the poles and the sign of the coefficients of \(-L'/L\) to be specified.

Earlier editions: Problem 30.2, page 1067 (edition 1).

Source: ch/lo3.tex, label lo:prob:selberg.

Open problem: the rate of convergence of the backward flow and the exponential growth lower bound when a zero lies to the right

Volume II, Problem 9.3, page 197 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) have zeros \(\rho=\sigma_\rho+i\gamma_\rho\), \(\theta_0=\sup_\rho\sigma_\rho\), \(\mathfrak U_\theta=\sum_\rho\mathrm{Cauchy}(\gamma_\rho,\theta-\sigma_\rho)\) its Thorin measure at level \(\theta\), and \(B_{\theta',\Omega}\mathfrak U_1=\mathfrak U_1*K_{\theta',\Omega}\) the band limited backward flow, \(K_{\theta',\Omega}\) the inverse Fourier transform of \((1-|\omega|/\Omega)_+e^{(1-\theta')|\omega|}\). For \(\theta'\in(\theta_0,1)\) give the rate of locally uniform convergence of \(B_{\theta',\Omega}\mathfrak U_1\) to \(\mathfrak U_{\theta'}\) in terms of \(\Omega\) and \(\theta'-\theta_0\), and, when \(\sigma_{\max}=\sup\sigma_\rho>\theta'\) is attained by a zero, show that \(\sup_{|v|\le V}|B_{\theta',\Omega}\mathfrak U_1(v)|\) grows like \(e^{(\sigma_{\max}-\theta')\Omega}\) up to polynomial factors. For one zero with \(d=\sigma-\theta'>0\) the Fejér inverse at distance \(t\) from its ordinate is \((\pi\Omega)^{-1}\operatorname{Re}[(e^{z\Omega}-1-z\Omega)/z^2]\) with \(z=d+it\), asymptotic to \(e^{d\Omega}/(\pi d^2\Omega)\) at \(t=0\); with infinitely many zeros the supremum need not be attained and the cancellations are part of the problem.

Earlier editions: Problem 30.3, page 1067 (edition 1).

Source: ch/lo3.tex, label lo:prob:rate.

Open problem: a prior allowing off line zeros and the posterior of the offset delta at the real level one density

Volume II, Problem 9.4, page 197 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Replace the GUE prior for the zeros in the generative Bayesian computation inversion of the level one Thorin density \(u_1(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(1+iv)\) of \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) by a prior that allows a fraction of the zeros off the critical line, for instance pairs at \(\frac12\pm\delta+i\gamma\) with \(\delta\) drawn from a prior on \([0,\frac12)\), and train the quantile network to report the posterior of \(\delta\) for each detected zero. The posterior of \(\delta\) at the real \(u_1\) is then a window by window numerical diagnostic at the resolution the data allow, conditional on the prior, and not a certificate.

Earlier editions: Problem 30.4, page 1067 (edition 1).

Source: ch/lo3.tex, label lo:prob:delta.

One sided approach in the Dyck chain: the criterion implies RH, and conditionally on the lattice bias conjecture it fails for T above gamma 64

Volume II, Result 10, page 576 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The lattice bias conjecture, that \(F_n(s)=2\xi(s)-\frac{3s}{\sqrt{\pi n}}\xi(s-1)+O(1/n)\) locally uniformly on \(\operatorname{Re}s>0\) (for the failure of the hypothesis).

Statement. Let \(H_n\) be the height of a uniform Dyck path of length \(2n\), \(Y_n=\sqrt{2/\pi}\,H_n/\sqrt{2n}\), \(F_n(s)=\mathbb{E}Y_n^s\) (a Dirichlet polynomial), \(\xi\) the Riemann xi function, and \(\rho_k=\frac12+i\gamma_k\) the zeros of \(\xi\) on the critical line in order of height, with \(\gamma_{64}=169.9119\ldots\). (1) Proved: if \(F_n\to2\xi\) locally uniformly on \(\operatorname{Re}s>0\) and, for every \(T>0\), \(F_n\) has no zero in \(\frac12<\operatorname{Re}s<1\), \(0<\operatorname{Im}s<T\) for all large \(n\), then the Riemann Hypothesis holds: if it fails, the functional equation and the symmetry \(\xi(\bar s)=\overline{\xi(s)}\) give a zero \(\rho\) of \(\xi\) with \(\frac12<\operatorname{Re}\rho<1\) and \(\operatorname{Im}\rho>0\), and by Hurwitz’s theorem, for \(T>\operatorname{Im}\rho\), \(F_n\) has zeros in that rectangle for all large \(n\). (2) Conditional on the conjecture that, uniformly on compact subsets of \(\operatorname{Re}s>0\), \(F_n(s)=2\xi(s)-\frac{3s}{\sqrt{\pi n}}\xi(s-1)+O(1/n)\): the zero free hypothesis holds for \(T<\gamma_{64}\) and fails for every \(T>\gamma_{64}\). The zeros of \(\xi\) below height \(170\) are \(\rho_1,\dots,\rho_{64}\), simple and on the line. For \(T>\gamma_{64}\) the zero \(z_n(\rho_{64})\) of \(F_n\) near \(\rho_{64}\) tends to \(\rho_{64}\) and has real part \(\frac12+\delta_n\operatorname{Re}c_{\rm D}(\rho_{64})+O(1/n)>\frac12\) for large \(n\), with \(\delta_n=3/(2\sqrt{2n})\) and \(c_{\rm D}(\rho)=\sqrt{2/\pi}\,\rho\,\xi(\rho-1)/\xi'(\rho)\), since \(\operatorname{Re}c_{\rm D}(\rho_{64})>53.08\) (computed in ball arithmetic). For \(T<\gamma_{64}\), \(F_n\to2\xi\) uniformly on the closed rectangle, and the only zeros of \(F_n\) near \(\rho_1,\dots,\rho_{63}\) are to the left of the line for large \(n\) because \(\operatorname{Re}c_{\rm D}(\rho_k)<0\) (computed in ball arithmetic). So conditionally on the conjecture the hypothesis fails, and this criterion of one sided approach cannot prove RH along this chain. For \(10^5\le n\le10^8\) the zero near \(\rho_{64}\) lies to the right of the line (computed). Open: prove the conjecture, which would also give the locally uniform convergence of the first hypothesis. The implication and the argument are from notes of Nicholas Polson merged into the book.

Earlier editions: Result 10, page 947 (edition 1).

Source: ch/ev4.tex, label ev:res:dyckside.

Volume II, Chapter 10. Complexity and acceleration of the backward flow

Log concavity of the Poisson likelihood and of the gamma prior posterior in the discretised Cauchy deconvolution

Volume II, Theorem 10.2.1, page 204 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(A\) be a matrix with positive entries, \(y\) a vector of nonnegative data, and \(\ell(x)=\sum_i\bigl(y_i\log(Ax)_i-(Ax)_i\bigr)\) the Poisson log likelihood on the open orthant \(x>0\). Then \(\ell\) is concave. With independent gamma priors \(x_j\sim\mathrm{Gamma}(\alpha_j,\beta_j)\), \(\alpha_j\ge1\), the posterior density \(\pi(x)\propto e^{\ell(x)}\prod_jx_j^{\alpha_j-1}e^{-\beta_jx_j}\) is log concave on the orthant. In the discretised deconvolution of the book, \(A_{ij}=hP_{1-\theta'}(v_i-w_j)\) with \(P_a\) the Cauchy density of scale \(a\). The sampling clause of the chapter (sampling to total variation accuracy \(\varepsilon\) in polynomial time for a log concave density, by a theorem of Applegate, Kannan and Polson) is cited and needs a bound on a ratio of radii that is not proved.

Earlier editions: Theorem 31.2.1, page 1076 (edition 1).

Source: ch/cx1.tex, label cx:thm:logconcave.

The level one data must be supplied to absolute precision e to the minus (1 minus theta’) Omega for the linear inverse

Volume II, Proposition 10.2.2, page 205 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the Riemann xi function, \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\) the level one density, \(\theta'\in[\frac12,1)\) and let the Thorin measure at level \(\theta'\) be recovered from \(u_1\) by inverting the convolution with the Cauchy density of scale \(1-\theta'\), whose Fourier transform is \(e^{-(1-\theta')|\omega|}\). Let \(\Omega\) be the largest frequency to be resolved. Then for the linear inverse the level one data must be supplied to absolute precision \(e^{-(1-\theta')\Omega}\) on the scale of \(u_1\), that is, to \((1-\theta')\Omega/\log 2\) binary digits, and this is sharp: an error of size \(\delta\) at frequency \(\omega\) in the Fourier transform of \(u_1\) produces an error of size \(\delta e^{(1-\theta')|\omega|}\) in the inverted measure.

Earlier editions: Proposition 31.2.2, page 1077 (edition 1).

Source: ch/cx1.tex, label cx:prop:precision.

What EM computes for the backward Cauchy flow: the Thorin measure at level theta’ when zeta has no zeros to the right of theta’

Volume II, Proposition 10.3.4, page 208 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12\le\theta'<1\), \(a=1-\theta'\), \(\xi\) the Riemann xi function and \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\). For a grid of spacing \(h\) with data points \(v_i=ih\), \(|i|\le M\), in the window \([-V,V]\) and unknown points \(w_j=jh\), \(|j|\le N\), in the window \([-W,W]\), \(W>V\), let \(A_{ij}=hP_a(v_i-w_j)\) with \(P_a(x)=(a/\pi)/(a^2+x^2)\), \(y_i=u_1(v_i)\), \(R_h(x)=h\sum_{|i|\le M}\bigl(y_i\log\frac{y_i}{(Ax)_i}-y_i+(Ax)_i\bigr)\), \(R_h^*=\min_{x\ge0}R_h(x)\), let \(x_h\) be any minimiser (for instance the limit of EM from a positive start) and \(\mu_h=\sum_jx_{h,j}h\,\delta_{w_j}\). Consider grids with \(h\to0\) and \(V\to\infty\). (a) If \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta'\) and \((\log W)^2/(W-V)\to0\), \((W-V)^2\log V/V^2\to0\) and \(h^2V\log^2V\to0\), then \(R_h^*\to0\) and \(\mu_h\) converges vaguely to the Thorin measure \(\mathfrak U_{\theta'}\), whose density is \(\sum_\rho P_{\theta'-\operatorname{Re}\rho}(v-\operatorname{Im}\rho)\) over the zeros \(\rho\) of \(\xi\) with multiplicity, with \(P_0\) read as the unit point mass. (b) If \(\zeta\) has a zero of real part greater than \(\theta'\), then \(\liminf R_h^*>0\), whatever the choice of \(W>V\).

Earlier editions: Proposition 31.3.4, page 1080 (edition 1).

Source: ch/cx1.tex, label cx:prop:what.

Curvature bounds for the logarithm of the Cauchy smoothed zero measure (conditional on RH)

Volume II, Proposition 10.3.6, page 210 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis.

Statement. Assume the Riemann Hypothesis. Let \(h>0\), let \(P_h(x)=h/(\pi(h^2+x^2))\) be the Cauchy kernel, and let \(u_{1/2+h}(v)=\sum_\gamma P_h(v-\gamma)\), the sum over the ordinates \(\gamma\) of all nontrivial zeros of \(\zeta\) with multiplicity. Then \(u_{1/2+h}>0\) and, on \(\mathbb{R}\), \(\big|\frac{d}{dv}\log u_{1/2+h}\big|\le\frac1h\) and \(-\frac2{h^2}\le\frac{d^2}{dv^2}\log u_{1/2+h}\le\frac9{8h^2}\), and each bound is sharp over positive Cauchy mixtures of width \(h\) (the upper bound is attained at the midpoint of an isolated pair at distance \(2h\sqrt{5/3}\)). The first bound holds for every positive Cauchy mixture of width \(h\).

Source: ch/cx1.tex, label cx:prop:curv.

Duality for the Poisson Kullback Leibler grid deconvolution problem

Volume II, Theorem 10.3.7, page 211 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(y_j>0\) be data, \(A\ge0\) a matrix, \(b_j>0\) a fixed far field, \(\mu=Ax+b\), and \(\operatorname{KL}_{\min}=\min_{x\ge0}\sum_j\big(y_j\log\frac{y_j}{\mu_j}-y_j+\mu_j\big)\). Then \(\operatorname{KL}_{\min}=\max\{\sum_jy_j\log(1-\varphi_j)+\sum_j\varphi_jb_j:\ \varphi_j<1,\ A^\top\varphi\ge0\}\). The maximum is attained at \(\varphi^*=1-y/\hat\mu\), the relative residual of any minimiser \(\hat x\), \(\hat\mu=A\hat x+b\), and the optimality conditions are \(A^\top\varphi^*\ge0\), \(\hat x\ge0\), \(\hat x^\top A^\top\varphi^*=0\). The last is the fixed point condition of the EM iteration with the far field held fixed: an EM fixed point satisfies it, and is optimal exactly when it also satisfies \(A^\top\varphi^*\ge0\). Finally \(\operatorname{KL}_{\min}>0\) if and only if some \(\varphi\) with \(A^\top\varphi\ge0\) has \(\langle\varphi,y-b\rangle<0\).

Source: ch/cx1.tex, label cx:thm:dual.

Optimality gap bound for the weighted Poisson objective at any iterate

Volume II, Proposition 10.3.8, page 212 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f(x)=\sum_jw_j(\mu_j-y_j\log\mu_j)\) with weights \(w_j>0\), data \(y_j>0\), \(\mu=Ax+b\) with \(b_j>0\) and \(A\) with positive entries, \(s=A^\top w\), and for \(x\ge0\) let \(\delta(x)=\max\big(0,-\min_i\partial_if(x)/s_i\big)\). Then \(0\le f(x)-\min_{x'\ge0}f(x')\le\sum_ix_i\,\partial_if(x)+\delta(x)\sum_jw_jy_j\). The bound needs no knowledge of the minimiser; at an EM fixed point the first term vanishes.

Source: ch/cx1.tex, label cx:prop:cert.

Window certificates for the Cauchy deconvolution misfit

Volume II, Proposition 10.3.9, page 213 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12\le\theta'<1\), \(a=1-\theta'\), \(P_a(x)=a/(\pi(a^2+x^2))\), \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\) with \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(v_1,\dots,v_J\) lie in a window, and let \(\varphi\in\mathbb{R}^J\) satisfy \(\sum_j|\varphi_j|\le1\) and \(\sum_j\varphi_jP_a(v_j-t)\ge0\) for every real \(t\). Then for every positive measure \(\mu\), \(\max_j|(P_a*\mu)(v_j)-u_1(v_j)|\ge-\sum_j\varphi_ju_1(v_j)\). If \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta'\), the right side is at most \(0\) for every such \(\varphi\). The converse, that a zero with real part above \(\theta'\) yields a positive right side for some finite set of points in the window, is open. The certificates in the book are computed in floating point from samples and are not certified.

Source: ch/cx1.tex, label cx:prop:wcert.

The von Neumann and Ulam Monte Carlo estimator for the backward Cauchy flow has infinite variance for small grid spacing

Volume II, Proposition 10.5.1, page 220 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\theta'\in[\frac12,1)\), let \(A\) be the square section of order \(2N+1\) of the Toeplitz matrix \(A_{ij}=hP_{1-\theta'}((i-j)h)\) with \(P_a(x)=(a/\pi)/(a^2+x^2)\), and let \(t=h/(\pi(1-\theta'))\). Let \(t_0=0.52219\ldots\) be the root in \((0,1)\) of \(\coth(1/t)=2t\). If \(t<t_0\), in particular if \(h<\pi(1-\theta')/2\), the bi infinite operator \(|I-A|\) has spectral radius \(1-2t+\coth(1/t)>1\), and so do its sections for all \(N\ge N_0(h,\theta')\). For such a section consider the von Neumann and Ulam estimator of \(\sum_{k\ge0}(I-A)^ky\): at state \(j\) it stops with probability \(q_j\in(0,1)\) and otherwise moves to \(k\) with probability \((1-q_j)p_{jk}\), \(p_{jk}>0\) wherever \((I-A)_{jk}\ne0\), and it returns the product of the weights \((I-A)_{jk}/((1-q_j)p_{jk})\) along the path times \(y_j/q_j\) at the stopping state \(j\). Its second moment is infinite from every starting state. If \(t_0<t\le1\) the spectral radius of \(|I-A|\) is below one for the bi infinite operator and all its sections, and the conclusion fails for the choice \(q_j=1-r_j\), \(p_{jk}=|(I-A)_{jk}|/r_j\) with \(r_j\) the row sums of \(|I-A|\).

Earlier editions: Proposition 31.5.1, page 1087 (edition 1).

Source: ch/cx1.tex, label cx:prop:vnvar.

Subordinated heat trace: the Thorin measure at level theta is the heat flow of the zeros run to first passage times of Brownian motion

Volume II, Proposition 10.6.1, page 221 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(T_a\) be the first passage time to level \(a>0\) of a Brownian motion, with density \(\ell_a(t)=a(2\pi)^{-1/2}t^{-3/2}e^{-a^2/(2t)}\). Then \(\frac{a/\pi}{a^2+(v-\gamma)^2}=\int_0^\infty\frac{e^{-(v-\gamma)^2/(2t)}}{\sqrt{2\pi t}}\ell_a(t)\,dt\), so \(\gamma+B(T_a)\) has the Cauchy law of centre \(\gamma\) and scale \(a\) for an independent Brownian motion \(B\), and \(e^{-a|\omega|}=\int_0^\infty e^{-t\omega^2/2}\ell_a(t)\,dt\). Consequently, if \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta_0\) and \(\theta>\theta_0\), the Thorin measure \(\mathfrak U_\theta\), with density \(\sum_\rho P_{\theta-\sigma_\rho}(v-\gamma_\rho)\) over the nontrivial zeros \(\rho=\sigma_\rho+i\gamma_\rho\) of \(\zeta\) with multiplicity and \(P_a(x)=(a/\pi)/(a^2+x^2)\), equals \(\sum_\rho\int_0^\infty(e^{t\Delta/2}\delta_{\gamma_\rho})\,\ell_{\theta-\sigma_\rho}(t)\,dt\), the sum over the nontrivial zeros of the heat flow of \(\delta_{\gamma_\rho}\) run to the random time \(T_{\theta-\sigma_\rho}\). Under the Riemann hypothesis it is the heat flow of \(\sum_\rho\delta_{\gamma_\rho}\) to the single random time \(T_{\theta-1/2}\).

Earlier editions: Proposition 31.6.1, page 1089 (edition 1).

Source: ch/cx1.tex, label cx:prop:subord.

The backward Cauchy multiplier is not a positive mixture of backward heat multipliers

Volume II, Proposition 10.6.2, page 222 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For every \(a>0\) there is no positive measure \(\mu\) on \([0,\infty)\) with \(\int_0^\infty e^{v\omega^2/2}\mu(dv)=e^{a|\omega|}\) for all real \(\omega\). Consequently the Cauchy deconvolution, whose Fourier multiplier is \(e^{a|\omega|}\), is not a mixture of backward heat steps.

Source: ch/cx1.tex, label cx:prop:noinverse.

EM halves the location error exactly for one Cauchy atom

Volume II, Proposition 10.6.4, page 224 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\epsilon>0\) be known, \(P_\epsilon(x)=\epsilon/(\pi(\epsilon^2+x^2))\), and let the function data be \(y=P_\epsilon(\cdot-\gamma_0)\) on the whole line, fitted by one atom of known scale \(\epsilon\) and fixed mass at location \(\gamma\). The EM map for the location is \(M(\gamma)=\int y\,w_\gamma\,t\,dt\big/\int y\,w_\gamma\,dt\) with \(w_\gamma(t)=1/((t-\gamma)^2+\epsilon^2)\). Then \(M(\gamma)-\gamma_0=\frac12(\gamma-\gamma_0)\), so every step halves the error exactly, for every \(\epsilon\).

Source: ch/cx1.tex, label cx:prop:ratehalf.

Atoms of known count are locally identifiable from 4n minus 1 sample points

Volume II, Proposition 10.6.5, page 225 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=a/(\pi(a^2+x^2))\) with \(a>0\), let \(z_1<\dots<z_n\) be locations, \(m_k>0\) masses, and let \(v_1,\dots,v_J\) be distinct points with \(J\ge4n-1\). The map \(F(z,m)=\big(\sum_km_kP_a(v_j-z_k)\big)_{j\le J}\) has an injective derivative at \((z,m)\). Hence \((z,m)\) is locally determined by the data, Gauss Newton converges locally quadratically for exact data, and the fitted atoms are locally Lipschitz in the data.

Source: ch/cx1.tex, label cx:prop:atoms.

Excess of the conditional precision when one atom fits a pair of Cauchy kernels

Volume II, Proposition 10.6.6, page 226 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0\le\delta<\epsilon\), \(P_a(x)=a/(\pi(a^2+x^2))\), and let the data near an atom be the pair \(P_{\epsilon-\delta}+P_{\epsilon+\delta}\) (each normalised, so the data density is their average), fitted by one atom of scale \(\epsilon\) at the origin. In the Cauchy model the conditional mean of the precision \(\lambda\) of a datum at distance \(t\) from the atom is \(2/(t^2+\epsilon^2)\), and its model value is \(1/\epsilon^2\). The mean \(\mathbb{E}[\lambda]\) over the data exceeds its model value by the factor \(1/(1-(\delta/2\epsilon)^2)\), that is \(\mathbb{E}[\lambda]=4/(4\epsilon^2-\delta^2)\).

Source: ch/cx1.tex, label cx:prop:prec.

The reciprocal xi transform at level theta is a GGC if and only if zeta has no zeros right of theta, if and only if the real part of xi’/xi is positive there

Volume II, Theorem 10.9.1, page 230 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the Riemann xi function and, for \(\frac12\le\theta<1\), \(F_\theta(s)=\xi(\theta)/\xi(\theta+\sqrt s)\) on the plane cut along \((-\infty,0]\) with the principal square root. A generalised gamma convolution (GGC) has Laplace transform \(\exp(-as-\int_0^\infty\log(1+s/u)U(du))\) with \(a\ge0\) and a positive Thorin measure \(U\). For \(\frac12\le\theta<1\) the following are equivalent: (i) \(F_\theta\) is the Laplace transform of a GGC, that is, the measure \(U_\theta\) of the zeros is a positive Thorin measure; (ii) \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta\); (iii) \(\operatorname{Re}(\xi'/\xi)(s)>0\) for \(\operatorname{Re}s>\theta\), that is, \(u_\sigma(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\sigma+iv)>0\) for every \(\sigma>\theta\) and every real \(v\). For \(\theta=\frac12\) the equivalence of (ii) and (iii) is the theorem of Lagarias.

Earlier editions: Theorem 31.9.1, page 1094 (edition 1).

Source: ch/cx2.tex, label cx:thm:positivity.

Mass deficit identity: twice the number of zeros right of the level equals the drop of the integrated density plus a boundary term

Volume II, Proposition 10.9.2, page 230 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the Riemann xi function, \(u_\sigma(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\sigma+iv)\) and \(M_\sigma(T)=\int_0^Tu_\sigma(v)\,dv\). Let \(\frac12<\theta<1\) and \(T>0\), and suppose no zero of \(\xi\) lies on the segment \(\operatorname{Re}s=\theta\), \(0\le\operatorname{Im}s\le T\), nor on the segment \(\operatorname{Im}s=T\), \(\theta\le\operatorname{Re}s\le1\). Let \(N(\theta,T)\) be the number of zeros with \(\operatorname{Re}\rho>\theta\) and \(0<\operatorname{Im}\rho<T\). Then \(2N(\theta,T)=M_1(T)-M_\theta(T)+\pi^{-1}\Delta_{\mathrm{top}}(\theta,T)\), where \(\Delta_{\mathrm{top}}(\theta,T)=-\int_\theta^1\operatorname{Im}(\xi'/\xi)(\sigma+iT)\,d\sigma\) is the change of \(\arg\xi\) along the top side.

Earlier editions: Proposition 31.9.2, page 1095 (edition 1).

Source: ch/cx2.tex, label cx:prop:deficit.

The level mass function theta to M_theta(T) is continuous if and only if every zero below height T is on the critical line

Volume II, Corollary 10.9.3, page 231 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With \(\xi\), \(u_\sigma\), \(M_\sigma(T)=\int_0^Tu_\sigma(v)\,dv\), \(N(\theta,T)\) and \(\Delta_{\mathrm{top}}(\theta,T)=-\int_\theta^1\operatorname{Im}(\xi'/\xi)(\sigma+iT)\,d\sigma\) as in the mass deficit identity, let \(T>0\) not be the ordinate of a zero. For \(\theta\in(\frac12,1]\) not the real part of a zero below height \(T\), \(M_\theta(T)=M_1(T)+\pi^{-1}\Delta_{\mathrm{top}}(\theta,T)-2N(\theta,T)\). The first two terms form a real analytic function of \(\theta\) with derivative \(\pi^{-1}\operatorname{Im}(\xi'/\xi)(\theta+iT)\), and \(2N(\theta,T)\) is a step function that drops by \(2m\) as \(\theta\) decreases through \(\operatorname{Re}\rho\), for each zero \(\rho\) of multiplicity \(m\) with \(\operatorname{Re}\rho>\frac12\) and \(0<\operatorname{Im}\rho<T\). Hence \(\theta\mapsto M_\theta(T)\) extends continuously to \((\frac12,1]\) if and only if every zero below height \(T\) lies on the critical line, and its jumps occur exactly at the real parts of the off line zeros, of size twice the total multiplicity of the zeros with that real part and ordinate below \(T\).

Earlier editions: Corollary 31.9.3, page 1096 (edition 1).

Source: ch/cx2.tex, label cx:cor:levelmass.

Certified positivity of u_{3/4} and u_{7/8} on [0,1420] and 1000 zeros on the critical line below 1419.9, in ball arithmetic

Volume II, Theorem 10.9.6, page 236 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\xi\) be the Riemann xi function, \(u_\theta(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta+iv)\), \(\Xi(v)=\xi(\frac12+iv)\), \(T^*=1419.9\), \(N(T)\) the number of zeros \(\rho\) of \(\zeta\) with \(0<\operatorname{Im}\rho<T\) and \(N(\theta,T)\) the number of those with \(\operatorname{Re}\rho>\theta\). A computation in ball arithmetic (run at \(96\) and at \(160\) bits, with Taylor bounds for \(u_\theta\) on cells and the argument change of \(\xi\) along a path and a rectangle) establishes: (i) \(u_{3/4}(v)>0\) and \(u_{7/8}(v)>0\) for \(0\le v\le1420\); (ii) \(N(T^*)=1000\); (iii) \(N(\frac34,T^*)=0\), with \(\int_0^{T^*}u_{3/4}(v)\,dv=1000.01952570266\ldots\) to within \(10^{-20}\); (iv) \(\Xi\) has \(1000\) certified sign changes in \((0,T^*)\). Consequently every zero of \(\zeta\) with \(0<\operatorname{Im}\rho<T^*\) lies on the critical line and is simple. The statement covers only this range of heights.

Earlier editions: Theorem 31.9.6, page 1102 (edition 1).

Source: ch/cx2.tex, label cx:thm:certified.

Global minimum of the Thorin density u_{7/8} at the origin, given the announced zero free half plane

Volume II, Theorem 10.10.2, page 239 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The zero free half plane \(\operatorname{Re}s>\frac78\) for the Riemann zeta function, announced in the quasi Riemann hypothesis preprints (OpenAI 2026, Liu 2026), unrefereed and not checked for this book.

Statement. Let \(\xi\) be the Riemann xi function and \(u_\theta(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta+iv)\). If \(\zeta\) has no zeros in \(\operatorname{Re}s>\frac78\), then, as measures, \(\mathfrak U_{7/8}\ge u_{7/8}(0)\,dv\), where \(\mathfrak U_{7/8}\) is the Thorin measure with density \(u_{7/8}\) (with atoms at the ordinates of zeros on \(\operatorname{Re}s=\frac78\) if there are any) and \(u_{7/8}(0)=\pi^{-1}(\xi'/\xi)(\frac78)=0.0055146635278\ldots\), and \(u_{7/8}(v)>u_{7/8}(0)\) wherever \(v\ne0\) is not the ordinate of a zero on the line \(\operatorname{Re}s=\frac78\). The proof is a computation, with least certified bound on \([13.5,1420]\) equal to \(0.0272\) at \(v=17.49\) for \(\theta=\frac78\).

Earlier editions: Theorem 31.10.2, page 1105 (edition 1).

Source: ch/cx2.tex, label cx:thm:globalmin.

Global minimum of u_theta at the origin for every level between 7/8 and 1, given the announced zero free half plane

Volume II, Theorem 10.11.1, page 243 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The zero free half plane \(\operatorname{Re}s>\frac78\) for the Riemann zeta function, announced in the quasi Riemann hypothesis preprints (OpenAI 2026, Liu 2026), unrefereed and not checked for this book. The monotonicity of \(\theta\mapsto u_\theta(0)\) is unconditional.

Statement. Let \(\xi\) be the Riemann xi function, \(u_\theta(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta+iv)\) and \(\mathfrak U_\theta\) the Thorin measure at level \(\theta\), with density \(u_\theta\) for \(\theta>\frac78\). Suppose \(\zeta\) has no zeros in \(\operatorname{Re}s>\frac78\). Then for every \(\theta\in[\frac78,1]\) and every real \(v\ne0\) that is not the height of a zero on \(\operatorname{Re}s=\theta\), \(u_\theta(v)>u_\theta(0)=\pi^{-1}(\xi'/\xi)(\theta)\), and the singular part of \(\mathfrak U_\theta\) consists of positive atoms, which occur only if \(\theta=\frac78\) and a zero lies on \(\operatorname{Re}s=\frac78\); so \(\mathfrak U_\theta-u_\theta(0)\,dv\) is a positive measure. Unconditionally, \(\theta\mapsto u_\theta(0)\) is increasing on \([\frac12,\infty)\). The values of \(u_\theta(0)\) run from \(0.0055147\) at \(\theta=\frac78\) to \(0.0073516\) at \(\theta=1\).

Earlier editions: Theorem 31.11.1, page 1110 (edition 1).

Source: ch/cx3.tex, label cx:thm:uniform.

Level by level theorem: a zero free half plane right of theta is equivalent to the global minimum of u_theta’ at the origin for all theta’ in [theta,1], with an explicit excess bound

Volume II, Theorem 10.11.3, page 244 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\xi\) be the Riemann xi function, \(u_\theta(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta+iv)\) and \(\frac12<\theta\le1\). The following are equivalent: (i) \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta\); (ii) for every \(\theta'\in[\theta,1]\), \(u_{\theta'}(v)\ge u_{\theta'}(0)\) for all real \(v\) that are not heights of zeros on \(\operatorname{Re}s=\theta'\). When they hold, \(\mathfrak U_{\theta'}\ge u_{\theta'}(0)\,dv\) as measures for \(\theta'\in[\theta,1]\), and for every \(\theta'\in[\theta,1]\), with \(c=2\theta'-1\) and the sum over all nontrivial zeros \(\rho=\sigma_\rho+i\gamma_\rho\) of \(\zeta\) with multiplicity, \(u_{\theta'}(v)-u_{\theta'}(0)\ge\frac c{2\pi}\bigl(\sum_\rho\frac1{c^2+(v-\gamma_\rho)^2}-\pi w^*\bigr)\) with \(w^*=0.0294450\), where the bracket \(\sum_\rho\frac1{c^2+(v-\gamma_\rho)^2}-\pi w^*\) is at least \(\pi(0.06174-w^*)>0.1014\) for \(13.5\le|v|\le1418.9\) and at least \(7/65-\pi w^*>0.0151\) for \(|v|\ge1418.9\). The numerical constants and brackets are computed, not proved by hand.

Earlier editions: Theorem 31.11.3, page 1112 (edition 1).

Source: ch/cx3.tex, label cx:thm:lbl.

The slope of the excess of the Thorin density: certified infima near 0.1758 to 0.1793, and under RH the minimum 0.183899 in the first gap at v equal to 17.4979

Volume II, Proposition 10.11.4, page 247 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann hypothesis, for the clause on the limiting bracket \(X\) (its minimiser and value in the first gap). The lower bounds for \(\inf B_c\) are a computation in ball arithmetic that needs only the hypothesis stated, no zeros of \(\zeta\) in \(\operatorname{Re}s>\theta\), which holds for \(\theta=1\).

Statement. Let \(\xi\) be the Riemann xi function, \(\theta\in(\frac12,1]\) be such that \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta\), \(\theta'\in[\theta,1]\), \(c=2\theta'-1\), \(\gamma\) run over the ordinates of the zeros with multiplicity (both signs), \(V_c(v)=\sum_{|\gamma|<1419.9}\bigl(\frac{c^2}4+(v-\gamma)^2\bigr)^{-1}\) and \(B_c(v)=V_c(v)-V_c(0)-2\sum_{|\gamma|\ge1419.9}\gamma^{-2}\), the lower bound for \(\frac{2\pi}c(u_{\theta'}(v)-u_{\theta'}(0))\) in which the zeros below height \(1419.9\) are on the critical line by the certified computation of the book. A computation shows \(\inf_{13.5\le|v|\le1418.9}B_c(v)\ge0.17575\) for \(c=1\), \(0.17838\) for \(c=\frac12\), \(0.17904\) for \(c=\frac14\) and \(0.17927\) as \(c\downarrow0\), each infimum approached at \(v=17.49\) to \(17.50\), between the first two zeros. Under the Riemann hypothesis the limiting bracket \(X(v)=\lim_{c\downarrow0}\frac{2\pi}c(u_{\theta'}(v)-u_{\theta'}(0))=\sum_\rho\bigl(\frac1{(v-\gamma_\rho)^2}-\frac1{\gamma_\rho^2}\bigr)\) has in the first gap between the zeros \(14.1347\) and \(21.0220\) a unique minimiser \(v^*\in[17.4978,17.4980]\), and \(0.1838985448\le X(v^*)\le0.1838987563\). The statement that \(17.4979\) minimises \(X\) over all of \(13.5\le|v|\le1418.9\) is the author’s computation, not rerun, and is not claimed here.

Earlier editions: Proposition 31.11.4, page 1114 (edition 1).

Source: ch/cx3.tex, label cx:prop:slope.

A posteriori bound: the error of a window mass of a positive measure against the Thorin measure is bounded by the residual against a weight

Volume II, Proposition 10.11.5, page 248 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\theta'\in[\frac12,1)\), \(a=1-\theta'\), \(P_a(x)=(a/\pi)/(a^2+x^2)\), and suppose \(\zeta\) has no zeros in \(\operatorname{Re}s>\theta'\), so that the Thorin measure \(\mathfrak U_{\theta'}\) exists and \(P_a*\mathfrak U_{\theta'}=u_1\,dv\) with \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\). Let \(\mu\) be a finite positive measure and \(r=P_a*\mu-u_1\) on \(\mathbb{R}\). For a test function \(\varphi\) with \(\int e^{a|\omega|}|\widehat\varphi(\omega)|\,d\omega<\infty\) put \(\psi=\mathcal F^{-1}[e^{a|\omega|}\widehat\varphi]\), and assume that \(\psi\cdot(P_a*\mu+u_1)\) is integrable. Then \(\bigl|\int\varphi\,d\mu-\int\varphi\,d\mathfrak U_{\theta'}\bigr|\le\int_{\mathbb{R}}|r(v)||\psi(v)|\,dv\).

Earlier editions: Proposition 31.11.5, page 1116 (edition 1).

Source: ch/cx3.tex, label cx:prop:apost.

From certified Gaussian window mass and centroid to the location of a zero, under the Riemann hypothesis

Volume II, Corollary 10.11.9, page 254 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann hypothesis, so that \(\mathfrak U_{1/2}=\sum_\rho\delta_{\gamma_\rho}\).

Statement. Assume the Riemann hypothesis, so that the Thorin measure at level \(\frac12\) is \(\mathfrak U_{1/2}=\sum_\rho\delta_{\gamma_\rho}\) over the ordinates with multiplicity. Let \(\varphi(v)=e^{-(v-c)^2/(2\sigma^2)}\), let \(\int\varphi\,d\mathfrak U_{1/2}\in[\widehat M-B_0,\widehat M+B_0]\) with \(\widehat M>B_0\) and \(\int(v-c)\varphi\,d\mathfrak U_{1/2}\in[\widehat m-B_1,\widehat m+B_1]\) be given bounds, let \(\gamma_*\) be an ordinate with \(|\gamma_*-c|\le R\), and let \(\Theta\ge\sum_{\gamma\ne\gamma_*}(|\gamma-c|+R)\varphi(\gamma)\), the sum over the other ordinates with multiplicity. Then \(|\gamma_*-c|\le\frac{|\widehat m|+B_1+\Theta}{\widehat M-B_0}\).

Earlier editions: Corollary 31.11.9, page 1122 (edition 1).

Source: ch/cx3.tex, label cx:cor:zeroloc.

Smoothed prime sum for u_1: exact identity and error bound under a local hypothesis on the zeros

Volume II, Theorem 10.11.10, page 255 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(v\ge2\), \(X\ge2\), \(s=1+iv\), \(L=\zeta'/\zeta\), \(\Lambda\) the von Mangoldt function and \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\), and let \(u_1^X(v)=\frac1\pi\bigl[\frac1{1+v^2}-\frac{\log\pi}2+\frac12\operatorname{Re}\psi(\frac{1+iv}2)-\sum_{n\ge2}\frac{\Lambda(n)}n e^{-n/X}\cos(v\log n)\bigr]\) with \(\psi\) the digamma function. Then \(\pi(u_1(v)-u_1^X(v))=-\operatorname{Re}\sum_\rho\Gamma(\rho-s)X^{\rho-s}+\operatorname{Re}\Gamma(1-s)X^{1-s}+\operatorname{Re}[X^{-1}L(s-1)]+\operatorname{Re}R_X(s)\), with \(R_X(s)=(2\pi i)^{-1}\int_{\operatorname{Re}w=-3/2}(-L(s+w))\Gamma(w)X^w\,dw\) and \(|R_X(s)|\le2.5X^{-3/2}\log(v+2)\). If every zero with \(|\gamma-v|\le Y\), \(Y\ge1\), lies on the critical line, then \(|u_1(v)-u_1^X(v)|\le\frac1\pi\bigl[c_\Gamma n(v)X^{-1/2}+2.6c_\Gamma'n(v+Y+1)e^{-\pi Y/2}+\sqrt{\pi/(v\sinh\pi v)}+\frac{27\log v}X+\frac{2.5\log(v+2)}{X^{3/2}}\bigr]\), where \(n(t)=0.39\log t+7\) bounds the number of ordinates in any interval of length one below height \(t\), \(c_\Gamma\le8.34\) and \(c_\Gamma'\le2.51\) are the explicit gamma function constants of the chapter.

Earlier editions: Theorem 31.11.10, page 1124 (edition 1).

Source: ch/cx3.tex, label cx:thm:fastu1.

Open problem: the two phase rate of EM for the continuous Cauchy deconvolution at an atomic target

Volume II, Problem 10.1, page 260 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For EM applied to the continuous Cauchy deconvolution \(P_a*\mu=u_1\) at an atomic target, with \(R_k\) the residual after \(k\) steps and \(R^*\) its limit, prove the two phase picture of the chapter: a sublinear phase with \(k(R_k-R^*)\) near a constant of order \(50\) (observed \(54.7\), \(51.9\), \(49.5\), \(48.1\) at \(k=500\), \(1000\), \(2000\), \(3000\)), lasting of the order of \(10^4\) steps on the data of the chapter, followed by faster convergence on the support of the atoms with a grid independent rate, and compute that rate from the curvature of the divergence at the atoms. For the accelerated scheme with restart, explain the observed three point exponents, which oscillate between \(0.6\) and \(2.8\) with a running value near \(1.3\). The problem is open, and the numbers are observations.

Earlier editions: Problem 31.1, page 1129 (edition 1).

Source: ch/cx3.tex, label cx:prob:twophase.

Open problem: a closed form for the limiting slope 0.183899 of the excess of the Thorin density under RH

Volume II, Problem 10.2, page 260 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Under the Riemann hypothesis, with \(\gamma\) running over the positive ordinates of the zeros, \(\log|\xi(\frac12+iv)|=\log\xi(\frac12)+\sum_{\gamma>0}\log|1-v^2/\gamma^2|\). Put \(\mathcal C(v)=\partial_v^2\log|\xi(\frac12+iv)|\), so that the limiting bracket of the excess of the Thorin density is \(X(v)=\mathcal C(0)-\mathcal C(v)\), and let \(v^*=17.4979\) be the zero of \(\mathcal C'\) between the first two zeros, where \(X(v^*)=0.183899\) to six digits. Does \(\mathcal C(v^*)\), equivalently \(X(v^*)\), have an expression in known constants? This is not known.

Earlier editions: Problem 31.2, page 1129 (edition 1).

Source: ch/cx3.tex, label cx:prob:closedform.

Open problem: evaluate the level one Thorin density to accuracy epsilon in time polynomial in log v and log(1/epsilon)

Volume II, Problem 10.3, page 260 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\). The smoothed prime sum evaluates \(u_1(v)\) to accuracy \(\varepsilon\) in time polynomial in \(\log v\) and in \(1/\varepsilon\). Is there an algorithm polynomial in \(\log v\) and \(\log(1/\varepsilon)\)? Such an algorithm for \(\zeta\) and \(\zeta'\) on \(\operatorname{Re}s=1\) would give one; the converse is not known, and the question is open.

Earlier editions: Problem 31.3, page 1129 (edition 1).

Source: ch/cx3.tex, label cx:prob:polylog.

Open problem: the best residual after k steps of a first order method for a single Cauchy blurred atom

Volume II, Problem 10.4, page 260 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a single atom at \(\gamma\) blurred by the Cauchy density \(P_a\) and observed exactly on a grid of spacing \(h\), with \(r_i\) the residuals at the grid points \(v_i\) and \(\psi_\gamma\) the weight of the a posteriori window bound for a window centred at \(\gamma\), what is the best achievable value of \(h\sum_i|\psi_\gamma(v_i)||r_i|\) after \(k\) steps of any first order method on the cone of nonnegative measures, as a function of \(k\), \(h\) and \(a\)? The question is open.

Earlier editions: Problem 31.4, page 1129 (edition 1).

Source: ch/cx3.tex, label cx:prob:bestresidual.

Volume II, Chapter 11. The explicit formula as a deconvolution

The explicit formula for every smooth compactly supported test function, read as the deconvolution at level one half

Volume II, Theorem 11.1.1, page 264 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, let \(\rho=\sigma_\rho+i\gamma_\rho\) run over its zeros (the nontrivial zeros of \(\zeta\), with multiplicity), let \(\Lambda\) be the von Mangoldt function and \(\gamma_E\) Euler’s constant. For every \(g\in C_c^\infty(\mathbb{R})\), with the sum over \(\rho\) absolutely convergent, \[\sum_\rho\int_{\mathbb{R}}g(\omega)\,e^{(\sigma_\rho-\frac12)|\omega|}\,e^{-i\omega\gamma_\rho}\,d\omega=\int_{\mathbb{R}}g(\omega)\,e^{|\omega|/2}\,d\omega-\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\bigl(g(\log n)+g(-\log n)\bigr)+(1-\log\pi-\gamma_E)\,g(0)+\sum_{k\ge1}\Bigl[\frac{g(0)}{k+1}-\int_{\mathbb{R}}g(\omega)\,e^{-(2k+\frac12)|\omega|}\,d\omega\Bigr].\] For even \(g\) this is the explicit formula of Guinand and Weil; for odd \(g\) both sides vanish. Under the Riemann hypothesis the left side is \(\sum_\rho\widehat g(\gamma_\rho)\), \(\widehat g(x)=\int_{\mathbb{R}}g(\omega)e^{-i\omega x}\,d\omega\). Under the Riemann hypothesis the theorem is the Fourier form of the deconvolution \(P_{1/2}*\mu=u_1\), \(\mu=\sum_\rho\delta_{\gamma_\rho}\), of the level one density \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)\), with \(P_a\) the Cauchy density of width \(a\): the multiplier \(e^{|\omega|/2}\) that undoes the Cauchy kernel is the factor \(\sqrt n\) that turns the weights \(\Lambda(n)/n\) of the level one spectrum into the weights \(\Lambda(n)/\sqrt n\) of the formula.

Earlier editions: Theorem 32.1.1, page 1135 (edition 1).

Source: ch/es1.tex, label es:thm:explicit.

An unresolvable perturbation of the Cauchy deconvolution at width one half

Volume II, Proposition 11.1.2, page 266 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Omega>0\), \(L\ge1\), \(\chi_L(v)=(2\pi L^2)^{-1/2}e^{-v^2/2L^2}\), \(N=\int_{\mathbb{R}}|\cos(\Omega v)|\,\chi_L(v)\,dv\) and \(\nu=N^{-1}\cos(\Omega v)\,\chi_L(v)\,dv\), and let \(P_{1/2}(x)=(1/2\pi)/(1/4+x^2)\) be the Cauchy density of width \(\frac12\). Then the total variation norm of \(\nu\) is \(1\), the Fourier transform \(\widehat\nu\) is concentrated within \(O(1/L)\) of \(\pm\Omega\), and \(\|P_{1/2}*\nu\|_\infty\le\sqrt{\pi/8}\,L^{-1}e^{1/8L^2}e^{-\Omega/2}\le0.72\,e^{-\Omega/2}\). Consequently, if \(0.72\,e^{-\Omega/2}\le2\varepsilon\) and \(\mu\) is a positive measure with \(\mu\ge N^{-1}\chi_L\,dv\), then \(\mu\) and \(\mu+\nu\) are positive measures at total variation distance \(1\) whose data \(P_{1/2}*\mu\) and \(P_{1/2}*(\mu+\nu)\) lie within \(\varepsilon\) in the supremum norm of one function, so data of accuracy \(\varepsilon\) do not distinguish them. The heuristic reading of this at scale \(\delta=\pi/\Omega\) (primes to \(e^{\pi/\delta}\)) is not a theorem.

Earlier editions: Proposition 32.1.2, page 1137 (edition 1).

Source: ch/es1.tex, label es:prop:resolution.

The Gaussian tapered Fourier transform of the level one density is a sum of damped exponentials, one per zero

Volume II, Theorem 11.2.1, page 267 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\sigma_\rho+i\gamma_\rho\), \(a_\rho=1-\sigma_\rho\in(0,1)\), \(P_a(x)=(a/\pi)/(a^2+x^2)\), and \(u_1(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(1+iv)=\sum_\rho P_{a_\rho}(v-\gamma_\rho)\). Let \(w(v)=e^{-(v-c)^2/2\ell^2}\) with \(\ell>0\) and \(F(\omega)=\int_{\mathbb{R}}u_1(v)\,w(v)\,e^{i\omega(v-c)}\,dv\). Then for \(\omega\ell^2>1\), \[F(\omega)=\sum_\rho w(\gamma_\rho+ia_\rho)\,e^{-a_\rho\omega}e^{i\omega(\gamma_\rho-c)}+R(\omega),\] where \(w(x+iy)=w(x)e^{y^2/2\ell^2}e^{-iy(x-c)/\ell^2}\) is the entire extension of the taper, the series converges absolutely, and with \(Y=\omega\ell^2\) \[|R(\omega)|\le\frac1\pi\,e^{-\ell^2\omega^2/2}\sum_\rho a_\rho\int_{\mathbb{R}}\frac{e^{-(x-c)^2/2\ell^2}\,dx}{(x-\gamma_\rho)^2+(Y-a_\rho)^2},\qquad\int_{\mathbb{R}}\frac{e^{-(x-c)^2/2\ell^2}\,dx}{(x-\gamma)^2+B^2}\le\min\Bigl(\frac\pi B,\frac{\sqrt{2\pi}\,\ell}{B^2}\Bigr).\] In particular for \(\omega_j=\omega_0+j\Delta\) with \(\omega_0\ell^2>1\), \(F(\omega_j)=\sum_\rho A_\rho z_\rho^{\,j}+R(\omega_j)\) with \(|R(\omega_j)|\le C_0e^{-\ell^2\omega_j^2/2}\), \(C_0\) the above bound on \(|R(\omega_0)|\) divided by \(e^{-\ell^2\omega_0^2/2}\), \(A_\rho=w(\gamma_\rho+ia_\rho)e^{-a_\rho\omega_0}e^{i\omega_0(\gamma_\rho-c)}\) and \(z_\rho=e^{\Delta(-a_\rho+i(\gamma_\rho-c))}\): the pole \(z_\rho\) has modulus giving \(a_\rho\) and argument giving \(\gamma_\rho-c\) modulo \(2\pi/\Delta\). Nothing is assumed about the real parts \(\sigma_\rho\).

Earlier editions: Theorem 32.2.1, page 1138 (edition 1).

Source: ch/es1.tex, label es:thm:taper.

Posterior concentration for the atoms of the tapered transform model on a window

Volume II, Theorem 11.4.1, page 271 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Fix a window centre \(c\), a taper width \(\ell\), a step \(\Delta\), samples \(\omega_j=\omega_0+j\Delta\), \(0\le j<J\), with \(\omega_0\ell^2>1\), and a radius \(R\) with \(2R\Delta<2\pi\). The data are \(F=\Phi(\vartheta^0)+b\) with \(\Phi(\vartheta)_j=\sum_{m=1}^MA_mz_m^{\,j}\), \(z_m=e^{\Delta(-a_m+i(\gamma_m-c))}\), where \(\vartheta^0\) collects the zeros \((\gamma_m,a_m,A_m)\) with \(|\gamma_m-c|\le R\) (ordinate, width \(a_m=1-\sigma_m\), amplitude) and \(b\) is the deterministic bias from the zeros beyond \(R\) and the remainder of the tapered transform; the observation is \(Y=F+\tau\xi\) with \(\xi\in\mathbb{C}^J\) having independent standard normal real and imaginary parts. Fix \(s>0\), \(0<a_-<a_+<1\), \(0<\alpha_-<A_+\) and an integer \(M_{\max}\) with \(J\ge2M_{\max}\). For \(0\le M\le M_{\max}\) let \(K_M\) be the compact set of \(\vartheta=(\gamma_m,a_m,A_m)_{m\le M}\) with \(c-R\le\gamma_1<\dots<\gamma_M\le c+R\), \(\gamma_{m+1}-\gamma_m\ge s\), \(a_m\in[a_-,a_+]\), \(\alpha_-\le|A_m|\le A_+\). Let the prior be \(\Pi=\sum_Mp_M\Pi_M\) with \(\Pi_M\) having a density on \(K_M\) between \(\pi_->0\) and \(\pi_+\). Let \(\vartheta^0\) be an interior point of \(K_{M^0}\) with \(p_{M^0}>0\), \(\|b\|_2\le B\tau\), and \(d(\vartheta,\vartheta^0)=\max_m(|\gamma_m-\gamma^0_m|+|a_m-a^0_m|+|A_m-A^0_m|)\) if \(M=M^0\), \(d=\infty\) otherwise. Then there are \(\tau_0,r_0,K,\kappa_0>0\), depending on \(\vartheta^0\), the sets \(K_M\), the prior, \(J\), \(\Delta\) and \(B\), such that for \(\tau\le\tau_0\) and \(K\tau\le\varepsilon\le r_0\), with probability at least \(1-e^{-\kappa_0\varepsilon^2/\tau^2}\), \(\Pi(d(\vartheta,\vartheta^0)\le\varepsilon\mid Y)\ge1-e^{-\kappa_0\varepsilon^2/\tau^2}\). In particular, for fixed \(\varepsilon_1>0\) and \(\|b\|_2\le B\tau\), as \(\tau\to0\) the posterior probability that every component has \(|\sigma_m-\frac12|\le\varepsilon_1\) tends to \(1\) in probability if every zero of \(\vartheta^0\) has \(|\sigma^0_m-\frac12|<\varepsilon_1\), and to \(0\) if one has \(|\sigma^0_m-\frac12|>\varepsilon_1\). The hypotheses (bias below the noise, no aliasing) are incompatible at the settings of the chapter’s computations, and the theorem is indifferent between the Riemann hypothesis and its negation.

Earlier editions: Theorem 32.4.1, page 1142 (edition 1).

Source: ch/es1.tex, label es:thm:concentration.

The real part of an off line pair of zeros is resolved only at the rate tau to the one half

Volume II, Theorem 11.5.2, page 274 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Work in the window model of the tapered transform at a level \(\theta>\frac12\) with \(a_0=\theta-\frac12\), samples \(\omega_j=\omega_0+j\Delta\), \(0\le j<J\), and \(f(a)_j=\exp(-q^2/2\ell^2+i\omega_jq)\), \(q=(\gamma-c)+ia\). One ordinate \(\gamma\) carries either a zero on the line or a pair at real parts \(\frac12\pm\delta\), \(0\le\delta\le\delta_+<a_0\), entering the samples as \(A\,g(\eta;\gamma)\) with \(g(\eta;\gamma)=f(a_0-\sqrt\eta)+f(a_0+\sqrt\eta)\), \(\eta=\delta^2\), and one complex amplitude \(A\) common to the two members (the functional equation puts zeros off the line in such pairs); the other \(M\) zeros enter as \(\sum_{m\le M}A_mz_m^{\,j}\) with \(z_m=e^{\Delta(-a_m+i(\gamma_m-c))}\). Write \(\vartheta'=(\gamma,A,(\gamma_m,a_m,A_m)_{m\le M})\), of real dimension \(D'=3+4M\), and \(\Phi(\eta,\vartheta')\in\mathbb{C}^J=\mathbb{R}^{2J}\) for the noiseless samples. Assume (H1) \(Y=\Phi(\eta^0,\vartheta'^0)+b+\tau\xi\) with \(\xi\) having independent standard normal real and imaginary parts and \(\|b\|_2\le B\tau\); (H2) the parameter set is \([0,\eta_+]\times K'\) with \(\eta_+=\delta_+^2\), \(K'\) compact, \(\eta^0\in[0,\eta_+)\), \(\vartheta'^0\) interior to \(K'\) and \(\Phi\) injective on it; (H3) the Jacobian \(T'\) of \(\vartheta'\mapsto\Phi(\eta^0,\vartheta')\) at \(\vartheta'^0\) has rank \(D'\) and, with \(t=\partial_\eta\Phi(\eta^0,\vartheta'^0)\) and \(t_\perp=(I-P')t\) for the orthogonal projection \(P'\) onto the range of \(T'\), \(\|t_\perp\|>0\); (H4) the prior puts mass \(1-p<1\) at \(\eta=0\) and density \(p\,\pi(\sqrt\eta)/2\sqrt\eta\) on \((0,\eta_+]\) with \(\pi\) continuous and positive on \([0,\delta_+]\), and independently \(\vartheta'\) has a density bounded on \(K'\), continuous and positive at \(\vartheta'^0\). Let \(r_\tau=\tau/\|t_\perp\|\) and \(\tau\to0\) with \(B\) and \(\vartheta'^0\) fixed, and with \(\eta^0\) fixed in (i) and (ii). Then (i) for every \(M_\tau\to\infty\), \(\Pi(|\eta-\eta^0|>M_\tau r_\tau\mid Y)\to0\) in probability, so \(\delta\) is located within \((M_\tau r_\tau)^{1/2}\) if \(\delta^0=0\); (ii) for \(\eta^0=0\) and \(\delta_1=c\,r_\tau^{1/2}\) the two error probabilities of any test of \(\delta=0\) against \(\delta=\delta_1\), whatever the method, sum to at least \(1-c^2/2-O(\tau)\); (iii) for \(\eta^0=v\,r_\tau\), \(v\ge0\) fixed, with \(\|b\|_2=o(\tau)\), the matched filter statistic \(Z=\langle t_\perp,r\rangle/(\tau\|t_\perp\|)\), with \(t_\perp\) taken at \(\eta=0\) and \(r\) the residual of the least squares fit of \(\Phi(0,\cdot)\) over \(K'\), satisfies \(Z=v+\zeta_\perp+o_P(1)\) with \(\zeta_\perp\) standard Gaussian, the one sided test \(Z>z_\alpha\) has asymptotic level \(\alpha\) and power \(1-N(z_\alpha-v)\), \(N\) the standard normal distribution function and \(z_\alpha=N^{-1}(1-\alpha)\), which is \(\frac12\) at \(\delta^*=(z_\alpha r_\tau)^{1/2}\), and twice the log generalised likelihood ratio of \(\eta\ge0\) against \(\eta=0\) is \(\max(Z,0)^2+o_P(1)\). The theorem needs the bias below the noise and no aliasing, so it does not cover the chapter’s runs on the zeros of \(\zeta\).

Earlier editions: Theorem 32.5.2, page 1146 (edition 1).

Source: ch/es2.tex, label es:thm:singular.

Bayes factors for a zero on the line against a pair at one half plus and minus delta

Volume II, Theorem 11.5.4, page 278 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. In the setting of the rate theorem for the real part of a pair (hypotheses H1 to H4, noise level \(\tau\), scale \(s=\|t_\perp\|/\tau\)), put the spike and slab prior on the pair: \(\delta=0\) with probability \(1-p\), and otherwise \(\delta\) with a density \(\pi\) continuous and positive on \([0,\delta_+]\). Let \(G(Z)=\int_0^\infty u^{-1/2}e^{Zu-u^2/2}\,du\) and \(Z\) the matched filter statistic of that theorem. (a) If \(\eta^0=0\), or \(\eta^0=v\,r_\tau\) with \(v\) fixed, \(\|b\|_2=o(\tau)\), and \(\vartheta'\) has the same prior under the line and the pair, the Bayes factor for the line against the pair is \(\mathrm{BF}_{01}=\dfrac{2\,s^{1/2}}{\pi(0)\,G(Z)}(1+o_P(1))\); under the null \(Z\) is asymptotically standard Gaussian and \(\mathrm{BF}_{01}\asymp\tau^{-1/2}\) in probability. (b) Under a pair with fixed \(\delta\in(0,\delta_+)\), if the prior density of \(\vartheta'\) is bounded below on \(K'\) and every point of \(K'\) has prior mass at least \(c\tau^{D'}\) within distance \(\tau\), then \(\log\mathrm{BF}_{01}=-D(\delta)^2(1+o_P(1))/2\tau^2\) with \(D(\delta)=\delta^2\|t_\perp\|+O(\delta^4)\). (c) Against the alternative of a single zero of width \(a_0+\varepsilon\), with \(\varepsilon\) having a density positive and continuous at \(0\) and \(\|(I-P')\partial_\varepsilon\Phi\|>0\), the same data give \(\mathrm{BF}_{01}\asymp\tau^{-1}\) in probability under the null. All of this is in the free amplitude model with a common amplitude for the pair.

Earlier editions: Theorem 32.5.4, page 1150 (edition 1).

Source: ch/es2.tex, label es:thm:bf.

The density at a level between one half and one is a signed sum of Lorentzians, and the Riemann hypothesis fixes every damping rate

Volume II, Proposition 11.8.1, page 285 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12<\theta\le1\) with \(\theta\ne\sigma_\rho\) for every zero \(\rho=\sigma_\rho+i\gamma_\rho\) of the completed zeta function \(\xi\), let \(P_b(x)=(b/\pi)/(b^2+x^2)\) and \(u_\theta(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta+iv)\). Then \(u_\theta(v)=\sum_\rho\operatorname{sgn}(\theta-\sigma_\rho)P_{|\theta-\sigma_\rho|}(v-\gamma_\rho)\) and \(\widehat{u_\theta}(\omega)=\sum_\rho\operatorname{sgn}(\theta-\sigma_\rho)e^{-|\theta-\sigma_\rho||\omega|}e^{-i\omega\gamma_\rho}\). For \(\omega\ell^2>1\) the tapered transform \(F_\theta(\omega)=\int u_\theta(v)w(v)e^{i\omega(v-c)}dv\) with the Gaussian taper \(w(v)=e^{-(v-c)^2/2\ell^2}\) equals \(\sum_\rho\operatorname{sgn}(\theta-\sigma_\rho)\,w(\gamma_\rho+i|\theta-\sigma_\rho|)\,e^{-|\theta-\sigma_\rho|\omega}e^{i\omega(\gamma_\rho-c)}+R_\theta(\omega)\), with \(R_\theta\) bounded as in the tapered transform theorem with \(a_\rho\) replaced by \(|\theta-\sigma_\rho|\). For samples \(\omega_j=\omega_0+j\Delta\), a zero with \(\sigma_\rho>\theta\) gives the pole \(e^{\Delta(-(\sigma_\rho-\theta)+i(\gamma_\rho-c))}\), of modulus \(e^{-\Delta(\sigma_\rho-\theta)}\), with an amplitude of sign opposite to the taper profile, while its partner \(1-\sigma_\rho<\theta\) gives a pole at the same ordinate with the profile’s sign. The Riemann hypothesis holds if and only if every damping rate \(|\theta-\sigma_\rho|\) equals \(\theta-\frac12\).

Earlier editions: Proposition 32.8.1, page 1157 (edition 1).

Source: ch/es2.tex, label es:prop:level.

Matrix pencil on a window at height 3e12, a demonstration inside the verified range

Volume II, Result 11.8.6, page 287 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. A demonstration, computed and not a finding about the zeros: the window centred at \(c=3\times10^{12}+100\) lies inside the range \(0<\gamma\le3\,000\,175\,332\,800\) in which Platt and Trudgian verified the Riemann hypothesis. From \(1001\) values of \(u_{0.6}(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(0.6+iv)\) at spacing \(0.02\) (level \(0.6\), taper \(\ell=2\), ball arithmetic at \(160\) bits), the matrix pencil returned \(52\) components with amplitudes on the taper profile within \(6\) of the centre, the smooth count being \(51.36\), each with real part \(0.5000000\pm1.4\times10^{-8}\) read from the modulus of its pole; this is a floating point estimate, not a certificate, and the selection of components presumes the real part \(\frac12\). The authors’ certificate (not rerun) is a sign change of Hardy’s function \(Z\) in ball arithmetic on \([\gamma-10^{-7},\gamma+10^{-7}]\) for all \(52\) of the \(52\) ordinates, which places a zero on the line within \(10^{-7}\) of each; an independent scan in ball arithmetic at \(160\) bits finds exactly \(52\) certified sign changes in \([c-6,c+6]\), but the comparison with the pencil’s ordinates one by one was not made. Not certified: that these are all the zeros of the window (a sign scan cannot see two zeros inside one step, and no winding number or Turing method was completed) and the real parts.

Earlier editions: Result 32.8.6, page 1159 (edition 1).

Source: ch/es2.tex, label es:res:window3e12.

Matrix pencil on a window at height 1e13 plus 100, above the verified range, computed

Volume II, Result 11.8.7, page 289 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. A computation with scripts written for the book: the window centred at \(c=10^{13}+100\), above the verified range of Platt and Trudgian (\(\gamma\le3\,000\,175\,332\,800\)), level \(0.6\), taper \(\ell=2\), \(1001\) values \(u_{0.6}(v_k)\), \(v_k=c-10+0.02k\), each a ball at \(160\) bits, \(\omega\in[5,85]\) at spacing \(0.2\), a Hankel matrix of order \(200\) in double precision and model order \(132\). The pencil returns \(53\) components with amplitude on the taper profile within \(6\) of the centre, at ordinates from \(c-5.7\) to \(c+5.8\) (the smooth count is \(53.66\)), the ratio of amplitude to profile being \(1\) to within \(1.6\times10^{-7}\); the selection compares the amplitude with the profile at the pole’s own width, so no real part is assumed. The largest \(|\sigma_\rho-\frac12|\) over the \(53\) poles is \(4.7\times10^{-9}\) (median \(7.1\times10^{-11}\)), an estimate from a double precision pencil and not a certificate, and it is not a proof of the Riemann hypothesis in the window.

Earlier editions: Result 32.8.7, page 1161 (edition 1).

Source: ch/es2.tex, label es:res:pencil1e13.

Fifty three certified zeros on the critical line in a window at height 1e13 plus 100, above the verified range

Volume II, Result 11.8.7, page 289 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. In ball arithmetic at \(160\) bits, with each step of \(0.02\) having opposite signs of Hardy’s function \(Z(t)=e^{i\vartheta(t)}\zeta(\frac12+it)\) at its ends, the interval \([c-6,c+6]\) with \(c=10^{13}+100\), above the height \(3\,000\,175\,332\,800\) verified by Platt and Trudgian, contains \(53\) certified sign changes, with bisection brackets of width \(10^{-10}\) certified at both ends again at \(192\) bits; hence \(\zeta\) has at least \(53\) zeros on the critical line in that interval, each located to \(10^{-10}\). The pencil of the same chapter returns the same \(53\) ordinates (the largest difference is \(3.9\times10^{-9}\)). Not certified: the number of zeros in the window (a sign scan cannot see two zeros inside one step of \(0.02\), the smallest gap found is \(0.0326\), and no Turing method or winding number was run) and the real parts; nothing is certified off the line. The certified statement is only that these \(53\) zeros lie on the line, and it is not a proof of the Riemann hypothesis in the window.

Earlier editions: Result 32.8.7, page 1161 (edition 1).

Source: ch/es2.tex, label es:res:zeros1e13.

Open problem, a rigorous matrix pencil with a perturbation bound under the model error

Volume II, Problem 11.1, page 290 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The matrix pencil applied to the tapered transform \(F(\omega_j)=\sum_\rho A_\rho z_\rho^{\,j}+R(\omega_j)\) of the level one density \(u_1\) of the completed zeta function makes the Hankel matrix of exact finite rank only in the limit, because of the far zeros and the remainder \(R\). The problem: state and prove a perturbation bound for the pencil under this model error, which together with verified linear algebra on ball data (in the sense of Rump) would give rigorous enclosures of the zeros from level one data, and compare the width of the resulting enclosures with the winding number route of the argument principle. Status: open.

Earlier editions: Problem 32.1, page 1162 (edition 1).

Source: ch/es2.tex, label es:prob:rigorous_pencil.

Open problem, characterise and remove the spurious pencil components at the edges of windows

Volume II, Problem 11.2, page 290 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. In the many window use of the matrix pencil on the tapered transform of \(u_1\), a few components at the edges of overlapping windows pass the amplitude filter without being zeros. The problem: characterise them (the expectation is that they are the pencil’s representation of the taper’s tail of the next zeros outside the window, with poles close to the genuine ones and amplitudes off the profile), and prove under explicit hypotheses on the separation and the amplitudes that a rule based on the agreement of two windows, or on the phase \(e^{i(\omega_0-a/\ell^2)(\gamma-c)}\) of the amplitude predicted by the tapered transform theorem, removes all of them. Status: open.

Earlier editions: Problem 32.2, page 1162 (edition 1).

Source: ch/es2.tex, label es:prob:spurious.

Open problem, windows at greater height with a rigorous argument principle and the cost of a full certificate

Volume II, Problem 11.3, page 290 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The matrix pencil runs on one window of fifty two or fifty three zeros at heights \(3\times10^{12}\) and \(10^{13}\) at the cost of the Riemann and Siegel sum, \(t^{1/2}\) terms per value, which Hiary’s method would bring to \(t^{1/3+o(1)}\). The problem: make the argument principle rigorous at large height, with an explicit bound for \(|\zeta'|\) on a small disc from the explicit convexity bounds, so that the count of zeros in the window is certified and not only their existence on the line; estimate the height at which a window of the pencil with its full certificate is feasible within a day, and run it; and determine whether the many window scheme at a fixed height, for example \(10^3\) zeros above \(3\,000\,175\,332\,800\), is competitive with Turing’s method for that range when the real parts are wanted as output and not only tested. Status: open.

Earlier editions: Problem 32.3, page 1163 (edition 1).

Source: ch/es2.tex, label es:prob:higher_wider.

Open problem, the adaptive prior in full: sine kernel process with marked ordinates and a spike and slab on the pairs

Volume II, Problem 11.4, page 290 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The problem: place the sine kernel determinantal process on marked distinct ordinates, a pair of zeros at real parts \(\frac12\pm\delta\) counting as one point marked by its displacement, and implement the posterior on the number, the positions and the marks of the zeros in a window, with the spike and slab prior on the marks, by reversible jump or by a determinantal birth and death sampler started at the pencil; measure on the many window run whether it removes the spurious components; state the count with its multiplicity (a double zero counts twice), and report the posterior probability of each count, conditional on the model, separately from the argument principle certificate, which it does not replace. Status: open.

Earlier editions: Problem 32.4, page 1163 (edition 1).

Source: ch/es2.tex, label es:prob:adaptive_prior.

Open problem, the constant in the tau to the one half rate and the design that maximises the information

Volume II, Problem 11.5, page 291 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The detectable displacement of an off line pair of zeros is \(\delta^*=(z_\alpha\tau/\|t_\perp\|)^{1/2}\), with \(\tau\) the standard deviation per real coordinate and \(t_\perp\) the part of the tangent of the pair orthogonal to the tangent space of the other zeros, depending on the taper, the bandwidth and the level. The problem: find the design that maximises the information \(\|t_\perp\|/\tau\), and not \(\|t_\perp\|\) alone, for a given cost, and determine how it scales with the height, through the gaps to the neighbouring zeros, which enter \(t_\perp\) through the projection, and not through the mean density alone. Status: open.

Earlier editions: Problem 32.5, page 1163 (edition 1).

Source: ch/es2.tex, label es:prob:constant_rate.

Volume II, Chapter 12. From level one to the critical line: particle filters and nested sampling

Cauchy transport: the density at level theta is the Cauchy smoothing of the critical zeros plus a signed defect from zeros off the line

Volume II, Proposition 12.1.2, page 294 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\sigma_\rho+i\gamma_\rho\) counted with multiplicity, \(P_a(x)=\frac{a}{\pi(a^2+x^2)}\) for \(a>0\), \(Q_a(x)=\frac{a}{\pi(a^2+x^2)}\) for real \(a\ne0\), \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\) and \(\mathfrak U_{1/2}=\sum_{\sigma_\rho=1/2}\delta_{\gamma_\rho}\). For \(\theta>\frac12\) and \(\theta+iv\) not a zero, \(u_\theta=P_{\theta-1/2}*\mathfrak U_{1/2}+R_\theta\) with \(R_\theta(v)=\sum_{\sigma_\rho>1/2}\bigl[Q_{\theta-\sigma_\rho}+Q_{\theta-1+\sigma_\rho}\bigr](v-\gamma_\rho)\), the sum over the zeros off the critical line each taken together with its reflection \(1-\bar\rho\). In particular \(u_1=P_{1/2}*\mathfrak U_{1/2}+R_1\), the pair of a zero off the line being positive at \(\theta=1\) and signed only for \(\theta<\sigma_\rho\). Under the Riemann hypothesis \(R_\theta=0\).

Earlier editions: Proposition 34.1.2, page 1182 (edition 1).

Source: ch/c34.tex, label cf:prop:transport.

Local positivity of the Thorin density from any zero free region that narrows with the height

Volume II, Corollary 12.1.3, page 295 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\sigma_\rho+i\gamma_\rho\) and \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\). Let \(\eta:[0,\infty)\to(0,\frac12]\) be nonincreasing and suppose every zero satisfies \(\sigma_\rho\le1-\eta(\lvert\gamma_\rho\rvert)\). Then \(u_\theta(v)>0\) whenever \(\theta>\frac12\), \(\theta\ge1-\eta(\lvert v\rvert+\frac12)\) and \(\xi(\theta+iv)\ne0\). The proof groups the zeros of the Hadamard sum into orbits of \(\rho\mapsto1-\bar\rho\); an orbit with \(\sigma_\rho>\theta\) is negative only within distance \(\frac12\) of its ordinate.

Earlier editions: Corollary 34.1.3, page 1182 (edition 1).

Source: ch/c34.tex, label cf:cor:localpos.

The density tends to the critical zero counting measure as the level tends to one half, zeros off the line being invisible

Volume II, Proposition 12.2.1, page 295 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\sigma_\rho+i\gamma_\rho\) and \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)=\sum_\rho Q_{\theta-\sigma_\rho}(v-\gamma_\rho)\) with \(Q_a(x)=\frac{a}{\pi(a^2+x^2)}\). As \(\theta\downarrow\frac12\), in the sense of distributions, \(u_\theta\to\mathfrak U_{1/2}=\sum_{\sigma_\rho=1/2}\delta_{\gamma_\rho}\). A zero \(\rho\) off the line and its reflection \(1-\bar\rho\) contribute \(Q_{\theta-\sigma_\rho}+Q_{\theta-1+\sigma_\rho}\to Q_{1/2-\sigma_\rho}+Q_{\sigma_\rho-1/2}=0\), because \(a\mapsto Q_a\) is continuous from \(\mathbb{R}\setminus\{0\}\) to \(L^1(\mathbb{R})\) and the far terms are bounded by \(\lvert\gamma_\rho\rvert^{-2}\) uniformly in \(\theta\in[\frac12,1]\).

Earlier editions: Proposition 34.2.1, page 1183 (edition 1).

Source: ch/cf1.tex, label cf:prop:ends.

The four zeros of the Davenport and Heilbronn function off the critical line below height 200 and their enclosures

Volume II, Result 12.2, page 295 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\kappa=(\sqrt{10-2\sqrt5}-2)/(\sqrt5-1)\) and \(f(s)=5^{-s}\bigl[\zeta(s,\frac15)+\kappa\zeta(s,\frac25)-\kappa\zeta(s,\frac35)-\zeta(s,\frac45)\bigr]\) the Davenport and Heilbronn function, with \(\zeta(s,a)\) the Hurwitz zeta function, and \(\Lambda(s)=(5/\pi)^{s/2}\Gamma(\frac{s+1}2)f(s)\), which is entire with \(\Lambda(s)=\Lambda(1-s)\). Computed in ball arithmetic: the zeros of \(f\) off the critical line with \(0<\gamma<200\) are \(0.8085171825+85.6993484854i\), \(0.6508300806+114.1633427308i\), \(0.5743560505+166.4793059132i\), \(0.7242576946+176.7024612429i\) together with their reflections in the line \(\operatorname{Re}s=\frac12\), each the only zero in a square of half width \(10^{-16}\) about its centre, so the printed decimals are correct. The count rests on \(130\) zeros of \(\Lambda\) with \(0<\gamma<200\) by the argument principle against \(122\) sign changes of \(\Lambda(\frac12+it)\) on a grid of step \(\frac1{50}\), the difference \(8=2\cdot4\) being the off line zeros and their reflections. This is a computation and not a statement for all heights. The zeros were found by Spira (1994); the ball arithmetic enclosures and the count are the book’s.

Earlier editions: Result 34.2, page 1184 (edition 1).

Source: ch/cf1.tex, label cf:res:dhzeros.

A symmetric pair of zeros off the line exceeds a double zero on the line by a second order term in the deviation, growing like the fifth inverse power of the level

Volume II, Proposition 12.3.1, page 299 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_a(x)=\frac{a}{\pi(a^2+x^2)}\), \(a=\theta-\frac12>\delta>0\). At the centre \(v=\gamma\) the density at level \(\theta\) of a pair of zeros at \(\frac12\pm\delta+i\gamma\) exceeds that of a double zero at \(\frac12+i\gamma\) by \(\Delta(a,\delta)=\frac1\pi\bigl(\frac1{a-\delta}+\frac1{a+\delta}\bigr)-\frac2{\pi a}=\frac{2\delta^2}{\pi a(a^2-\delta^2)}\). In the parameter \(\eta=\delta^2\) the pair differs from the double zero by \(\eta\,\partial_a^2P_a+O(\eta^2)\), and \(\int_{\mathbb{R}}(\partial_a^2P_a(x))^2dx=\frac3{4\pi a^5}\).

Earlier editions: Proposition 34.3.1, page 1187 (edition 1).

Source: ch/cf1.tex, label cf:prop:detect.

Open: the best resolution of the symmetric pair parameter from data at levels above a threshold

Volume II, Problem 12.1, page 300 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Consider a symmetric pair of zeros of the completed Riemann zeta function at \(\frac12\pm\delta+i\gamma\), with parameter \(\eta=\delta^2\), and data given by the density \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\) at levels \(\theta\ge\theta_{\min}\) with observation error. What is the best possible resolution of \(\eta\), and does the particle filter with the pair parametrisation reach it? This is posed as an open problem.

Earlier editions: Problem 34.1, page 1188 (edition 1).

Source: ch/cf1.tex, label cf:op:pairres.

Non asymptotic concentration inequality for a Feynman Kac particle system with multinomial resampling and fixed potentials under a ratio minorisation of the mutation kernels

Volume II, Theorem 12.5.1, page 306 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\eta_0\) be a probability measure on a space \(E\), \(G_0,\dots,G_{n-1}\) positive bounded potentials with \(\sup G_k/\inf G_k<\infty\), \(M_1,\dots,M_n\) Markov kernels, \(\eta_{k+1}=\Psi_{G_k}(\eta_k)M_{k+1}\) with \(\Psi_G(\mu)(dx)=G(x)\mu(dx)/\mu(G)\), and let \(\eta_k^N\) be the empirical measure of the particle system that draws \(N\) independent particles from \(\eta_0\) and then, given generation \(k\), \(N\) independent particles from \(\Psi_{G_k}(\eta_k^N)M_{k+1}\) (multinomial resampling followed by mutation). With \(\mathcal Q_{k+1}(x,dy)=G_k(x)M_{k+1}(x,dy)\), \(G_{k,n}=\mathcal Q_{k+1}\cdots\mathcal Q_n(1)\), \(\mathcal P_{k,n}f=\mathcal Q_{k+1}\cdots\mathcal Q_n(f)/G_{k,n}\), \(g_{k,n}=\sup_{x,y}G_{k,n}(x)/G_{k,n}(y)\), \(\varkappa_{k,n}\) the Dobrushin coefficient of \(\mathcal P_{k,n}\) and \(S_n=\sum_{k=0}^ng_{k,n}\varkappa_{k,n}\), for every measurable \(f\) with oscillation at most \(1\), every \(N\) and every \(x>0\), \(\mathbb P(\lvert\eta_n^N(f)-\eta_n(f)\rvert\ge x)\le2(n+1)\exp(-Nx^2/(2S_n^2))\) and \(\mathbb E[(\eta_n^N(f)-\eta_n(f))^2]^{1/2}\le S_n/\sqrt N\). The theorem covers a fixed schedule and fixed kernels, not samplers with adaptive tempering or random walk Metropolis moves. The proof is given (in the style of Del Moral 2004).

Earlier editions: Theorem 34.5.1, page 1195 (edition 1).

Source: ch/cf2.tex, label cf:thm:conc.

Concentration bound uniform in the number of tempering steps for the Cauchy flow when the parameters stay away from the levels

Volume II, Corollary 12.5.4, page 308 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Consider the Gaussian tolerance posterior of the chapter, in which zeros \(\sigma_j+i\gamma_j\), \(j\le K\), in a window of heights are fitted to the values \(y_l(v_i)\) of the density at levels \(\theta_l\) and heights \(v_i\) with tolerance \(\varepsilon\), and the parameter space is restricted to \(\Theta_\varsigma=\{z:\lvert\theta_l-\sigma_j\rvert\ge\varsigma\text{ for all levels }\theta_l\text{ and zeros }j\}\). Then the total log likelihood \(\ell\) has \(\operatorname{osc}(\ell)\le\Omega_\varsigma=\frac1{2\varepsilon^2}\sum_{l,i}\bigl(\lvert y_l(v_i)\rvert+\frac K{\pi\varsigma}\bigr)^2\), and a tempering schedule with \(n=\max(1,\lceil\Omega_\varsigma/c\rceil)\) equal increments has potentials with \(\sup G/\inf G\le e^c\). If the mutation kernels satisfy the ratio minorisation \(M(x,\cdot)\ge\epsilon M(y,\cdot)\) for all \(x,y\), then for every posterior functional \(f\) of oscillation at most one, for the idealised particle system of the concentration theorem of the chapter, \(\mathbb P(\lvert\eta_n^N(f)-\eta_n(f)\rvert\ge x)\le2\exp(-\frac12Nx^2\epsilon^6e^{-2c})\) for every \(N\) and every \(x>0\). Without the minorisation the concentration theorem holds with its finite constant \(S_n\), which is not uniform in \(n\). Random walk Metropolis kernels do not satisfy the minorisation.

Earlier editions: Corollary 34.5.4, page 1197 (edition 1).

Source: ch/cf2.tex, label cf:cor:cauchyconc.

What the density at level one determines: unique continuation for entire functions of order one, a counterexample without the order condition, and the Poisson inversion

Volume II, Proposition 12.6.1, page 312 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F_1,F_2\) be real entire functions of order at most one with all zeros in \(0<\operatorname{Re}s<1\) and \(u^{(j)}_\theta(v)=\pi^{-1}\operatorname{Re}(F_j'/F_j)(\theta+iv)\). (a) If \(u^{(1)}_1=u^{(2)}_1\) on an interval then \(F_1'/F_1=F_2'/F_2\), so \(u^{(1)}_\theta=u^{(2)}_\theta\) for every \(\theta\). (b) The order hypothesis cannot be dropped: \(F=\xi e^{G}\) with \(\xi\) the completed Riemann zeta function and \(G(s)=-(10\pi)^{-1}\cos(10\pi(s-1))\) is real entire with the zeros of \(\xi\), its densities at the levels \(1\) and \(0.9\) equal those of \(\xi\) on the whole line, and its density at \(\frac34\) is \(u_{3/4}(v)-\pi^{-1}\cosh(10\pi v)\). (c) If no zero of \(\xi\) has real part at least \(\frac34\) (in particular under the Riemann hypothesis) then \(u_1=P_{1/4}*u_{3/4}\) with \(P_a\) the Cauchy density of scale \(a\), whose Fourier multiplier is \(e^{-\lvert\omega\rvert/4}\), so an error at frequency \(\omega\) in \(u_1\) is magnified by \(e^{\lvert\omega\rvert/4}\) in the recovery of \(u_{3/4}\).

Earlier editions: Proposition 34.6.1, page 1202 (edition 1).

Source: ch/cf2.tex, label cf:prop:determine.

The negative part of the Thorin density at level theta is bounded by the number of zeros to the right of theta, and is a fraction T to the minus (theta minus one half) over four of its mass unconditionally

Volume II, Theorem 12.7.1, page 313 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\) the density of \(\mathfrak U_\theta\), \(u_\theta^-=\max(-u_\theta,0)\) and \(N(\theta,T)\) the number of zeros with \(\operatorname{Re}\rho>\theta\) and \(0<\operatorname{Im}\rho\le T\), with multiplicity. Let \(\frac12<\theta<1\) and \(T\ge2\). (a) \(\int_0^Tu_\theta^-(v)\,dv\le N(\theta,T+\frac12)\). (b) If \(T\) is not an ordinate and no zero with \(0<\gamma_\rho<T\) has real part \(\theta\) then \(\mathfrak U_\theta((0,T])=\int_0^Tu_\theta=\frac T{2\pi}\log\frac T{2\pi e}-2N(\theta,T)+O(\log T)\). (c) By Selberg’s density theorem \(N(\theta,T)\ll T^{1-(\theta-1/2)/4}\log T\), for \(T\) as in (b) the negative part of \(\mathfrak U_\theta\) on \((0,T]\) is a fraction \(O(N(\theta,T+\frac12)/(T\log T))=O(T^{-(\theta-1/2)/4})\) of its mass. The Riemann hypothesis is not assumed.

Earlier editions: Theorem 34.7.1, page 1203 (edition 1).

Source: ch/cf3.tex, label cf:thm:negmass.

Open: how negative can the Thorin measure be at a level between one half and one without assuming RH

Volume II, Problem 12.2, page 313 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\mathfrak U_\theta\) be the measure on the line with density \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\), \(\xi\) the completed Riemann zeta function. How negative can \(\mathfrak U_\theta\) be for \(\frac12<\theta<1\) without assuming the Riemann hypothesis? The book proves a bound through the zero density function (negative part on \((0,T]\) at most \(N(\theta,T+\frac12)\)) and leaves the question open.

Earlier editions: Problem 34.2, page 1203 (edition 1).

Source: ch/cf3.tex, label cf:op:negdepth.

Open: does a linear bound on the negative Thorin mass defect imply a zero density bound

Volume II, Problem 12.3, page 314 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(N(\theta,T)\) be the number of zeros of the Riemann zeta function with real part above \(\theta\) and ordinate in \((0,T]\), and \(\mathrm{def}(\theta)\) the defect of the negative Thorin mass at level \(\theta\) as defined in the chapter. Does \(\mathrm{def}(\theta)\le A(1-\theta)\) for all \(\theta\) imply \(N(\theta,T)\ll T^{A(1-\theta)+\epsilon}\) for every \(\epsilon>0\)? The book proves the converse for separated zeros and a weighted bound for all zeros, and leaves the general question open.

Earlier editions: Problem 34.3, page 1204 (edition 1).

Source: ch/cf3.tex, label cf:op:converse.

A separated zero off the line forces negative Thorin mass in a half disc to its left

Volume II, Proposition 12.7.2, page 315 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\), \(u_\theta^-=\max(-u_\theta,0)\), \(k(\rho)\) the multiplicity of a zero \(\rho\), \(d(\rho)\) the distance from \(\rho\) to the nearest other zero and \(\mathrm{cr}(\rho)=\sum k(\rho')/\lvert\rho'-\rho\rvert\) over distinct zeros \(\rho'\ne\rho\) with \(\lvert\gamma_{\rho'}-\gamma_\rho\rvert\le2\), the reflected zero included. Let \(\rho_0=\sigma_0+i\gamma_0\) be a zero of multiplicity \(k\) with \(\sigma_0>\frac12\) and \(\gamma_0\ge3\). There is an absolute constant \(C_1\ge1\) such that if \(r\le\min\bigl(1,\sigma_0-\frac12,\frac12d(\rho_0),k/(\pi(\mathrm{cr}(\rho_0)+C_1\log\gamma_0))\bigr)\) then \(\int_{\sigma_0-r}^{\sigma_0}\int_{\gamma_0-r}^{\gamma_0+r}u_\theta^-(v)\,dv\,d\theta\ge kr/\pi\).

Earlier editions: Proposition 34.7.2, page 1205 (edition 1).

Source: ch/cf3.tex, label cf:prop:converse.

Two sided bound between the negative Thorin mass and the number of separated zeros to the right of a level

Volume II, Corollary 12.7.3, page 316 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With the notation of the converse proposition for a separated zero (the density \(u_\theta\), \(u_\theta^-=\max(-u_\theta,0)\), \(\mathrm{cr}\) and the constant \(C_1\)), let \(\frac12<\theta_0<1\), \(T\ge3\) and \(E(\theta_0,T)=\int_{\theta_0}^1\int_0^{T+1}u_\theta^-(v)\,dv\,d\theta\). Let \(N_{\mathrm{sep}}(\theta_0,T;c,\delta)\) count the zeros with \(\operatorname{Re}\rho>\theta_0\), \(3\le\operatorname{Im}\rho\le T\), \(\mathrm{cr}(\rho)\le c\) and distance at least \(2\delta\) from every other zero, and \(r=\min(\delta,1/(\pi(c+C_1\log T)))\). Then \(\frac r\pi N_{\mathrm{sep}}(\theta_0+r,T;c,\delta)\le E(\theta_0,T)\le\int_{\theta_0}^1N(\theta,T+\frac32)\,d\theta=\sum_{\operatorname{Re}\rho>\theta_0,\,0<\operatorname{Im}\rho\le T+3/2}(\operatorname{Re}\rho-\theta_0)\), where \(N(\theta,T)\) counts zeros with real part above \(\theta\) and imaginary part in \((0,T]\).

Earlier editions: Corollary 34.7.3, page 1206 (edition 1).

Source: ch/cf3.tex, label cf:cor:twosided.

Every zero off the line forces negative Thorin mass at least its isolation distance over the logarithm of the height

Volume II, Theorem 12.7.4, page 316 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With \(u_\theta\), \(k(\rho)\), \(d(\rho)\), \(\mathrm{cr}(\rho)\), \(C_1\) and \(E(\theta_0,T)=\int_{\theta_0}^1\int_0^{T+1}u_\theta^-\,dv\,d\theta\) as in the converse proposition and its corollary, let \(\theta_0>\frac12\) and for a zero \(\rho\) with \(\operatorname{Re}\rho>\theta_0\) put \(r(\rho)=\min\bigl(1,\operatorname{Re}\rho-\theta_0,\frac12d(\rho),k(\rho)/(\pi(\mathrm{cr}(\rho)+C_1\log\gamma_\rho))\bigr)\). Then for \(T\ge3\), \(E(\theta_0,T)\ge\frac1\pi\sum k(\rho)r(\rho)\), the sum over distinct zeros with \(\operatorname{Re}\rho>\theta_0\) and \(3\le\gamma_\rho\le T\). For an absolute \(c>0\), \(r(\rho)\ge c\min(\operatorname{Re}\rho-\theta_0,\min(d(\rho),1)/\log\gamma_\rho)\), and so \(E(\theta_0,T)\gg\frac1{\log T}\sum\min((\operatorname{Re}\rho-\theta_0)\log T,d(\rho),1)\) over the same zeros.

Earlier editions: Theorem 34.7.4, page 1206 (edition 1).

Source: ch/cf3.tex, label cf:thm:converse.

Open: what a zero free half plane at theta_0, for instance 7/8, gives in the Cauchy flow framework

Volume II, Problem 12.4, page 317 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let QRH(\(\theta_0\)) be the statement that the Riemann zeta function has no zero with real part greater than \(\theta_0\), equivalent to positivity of the Thorin measure \(\mathfrak U_\tau\) for every \(\tau>\theta_0\). What does QRH(\(\theta_0\)) give in the framework of the particle filter and the Cauchy flow, for instance at \(\theta_0=\frac78\)? This is posed as an open problem; QRH(7/8) is announced in unrefereed preprints and is not used as a theorem.

Earlier editions: Problem 34.4, page 1207 (edition 1).

Source: ch/cf3.tex, label cf:op:qrh78.

Under a zero free half plane the Thorin density has a positive margin proportional to the distance from the abscissa and to the logarithm of the height

Volume II, Proposition 12.7.5, page 318 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The quasi Riemann hypothesis QRH(theta_0), that is, that the Riemann zeta function has no zero with real part greater than theta_0, for the stated theta_0 with 1/2 <= theta_0 < 1.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\). Assume \(\xi\) has no zero with real part greater than \(\theta_0\), where \(\frac12\le\theta_0<1\). Then there are constants \(c>0\) and \(v_0\) such that \(u_\theta(v)\ge c(\theta-\theta_0)\log v\) for all \(\theta_0<\theta\le1\) and \(v\ge v_0\). The proof combines the positivity of every term of the Hadamard sum, the Riemann and von Mangoldt formula and an explicit bound for the argument of the zeta function on the critical line.

Earlier editions: Proposition 34.7.5, page 1208 (edition 1).

Source: ch/cf3.tex, label cf:prop:margin.

Open: turn the calibrated GBC screen into a proof of the zero count in each window

Volume II, Problem 12.5, page 318 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The GBC screen of the chapter returns a posterior for the real parts of the zeros of \(\xi\) in a window of heights, calibrated on simulations, which is evidence and not proof. Can it be turned into a proof in each window, for instance through the contour moment \(M_0\) computed in interval arithmetic and shown to be less than \(\frac12\) (the argument principle)? This is posed as an open problem.

Earlier editions: Problem 34.5, page 1209 (edition 1).

Source: ch/cf3.tex, label cf:op:certificate.

De la Vallée Poussin inequality for the Thorin density with every nonnegative cosine polynomial (general version)

Volume II, Theorem 12.8.1, page 319 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(u_\theta(v)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\theta+iv)\) and \(g_\theta(v)=\operatorname{Re}\bigl[\frac1s+\frac1{s-1}-\frac12\log\pi+\frac12\psi(\frac s2)\bigr]\) with \(s=\theta+iv\) and \(\psi\) the digamma function. For \(\theta>1\) and every real \(v\), \(3u_\theta(0)+4u_\theta(v)+u_\theta(2v)\le\frac1\pi[3g_\theta(0)+4g_\theta(v)+g_\theta(2v)]\). More generally, for every cosine polynomial \(p(\omega)=\sum_{k=0}^Ka_k\cos k\omega\) with \(p\ge0\) and \(a_k\ge0\), \(\sum_ka_ku_\theta(kv)\le\pi^{-1}\sum_ka_kg_\theta(kv)\). The proof uses \(\pi u_\theta=g_\theta+\operatorname{Re}\frac{\zeta'}{\zeta}(\theta+iv)\) and that \(-\operatorname{Re}\frac{\zeta'}{\zeta}(\theta+iv)=\sum_{n\ge2}\Lambda(n)n^{-\theta}\cos(v\log n)\), with \(\Lambda\) the von Mangoldt function, is a cosine series with nonnegative coefficients.

Earlier editions: Theorem 34.8.1, page 1209 (edition 1).

Source: ch/cf3.tex, label cf:thm:dvp.

Zero free region of width c over log gamma with every c below 14 minus 8 root 3 over 5 from the Thorin form

Volume II, Corollary 12.8.2, page 319 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(c_\ast=(14-8\sqrt3)/5=0.028719\ldots\). For every \(c<c_\ast\) there is \(\gamma_c\) such that every zero \(\rho=\sigma_\rho+i\gamma_\rho\) of the Riemann zeta function with \(\gamma_\rho\ge\gamma_c\) has \(1-\sigma_\rho\ge c/\log\gamma_\rho\). The proof takes \(v=\gamma_0\) and \(\theta=1+a/\log\gamma_0\) in the inequality of the cosine polynomial \(3+4\cos\omega+\cos2\omega\), with optimum \(a=(4\sqrt3-6)/5\). Sharper constants are known from other methods.

Earlier editions: Corollary 34.8.2, page 1210 (edition 1).

Source: ch/cf3.tex, label cf:cor:zfr.

The Davenport and Heilbronn function satisfies the Cauchy flow identities and violates the quasi Riemann hypothesis at every level below one

Volume II, Proposition 12.8.3, page 320 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\kappa=(\sqrt{10-2\sqrt5}-2)/(\sqrt5-1)\) and \(f(s)=5^{-s}\bigl[\zeta(s,\frac15)+\kappa\zeta(s,\frac25)-\kappa\zeta(s,\frac35)-\zeta(s,\frac45)\bigr]\) the Davenport and Heilbronn function, with \(\zeta(s,a)\) the Hurwitz zeta function, and \(\Lambda_{\mathrm{DH}}(s)=(5/\pi)^{s/2}\Gamma(\frac{s+1}2)f(s)\), entire of order one, real on the real axis with \(\Lambda_{\mathrm{DH}}(s)=\Lambda_{\mathrm{DH}}(1-s)\). The Hadamard sum, the Cauchy transport identity, the equivalence of positivity of the density at every level above \(\theta\) with the absence of zeros to the right of \(\theta\), and the detectability proposition hold for \(\Lambda_{\mathrm{DH}}\) with the same proofs; so does the formula of Balazard, Saias and Yor with \(h=f\) and the sum over all zeros of \(f\) to the right of the level, and the deconvolution theorem with the line \(\operatorname{Re}s=1\) replaced by \(\operatorname{Re}s=2\), to the right of all zeros since \(\lvert f(s)-1\rvert\le\zeta(2)-1<1\) there. The function \(f\) has infinitely many zeros with real part above \(1\) (Davenport and Heilbronn), so \(\Lambda_{\mathrm{DH}}\) violates the absence of zeros to the right of \(\theta\) for every \(\theta<1\), and its lowest zeros off the line, \(0.8085\ldots+85.6993\ldots i\) and \(0.6508\ldots+114.1633\ldots i\), already violate it for \(\theta=\frac34\) and \(\theta=\frac58\).

Earlier editions: Proposition 34.8.3, page 1210 (edition 1).

Source: ch/cf3.tex, label cf:prop:barrier.

Multilevel inequality for the Thorin density with a cosine polynomial and a level kernel of nonnegative Laplace transform

Volume II, Theorem 12.8.5, page 321 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Fix \(v\ge2\), \(L=\log v\), and let \(u_\theta\) and \(g_\theta\) be the density of the Thorin measure of the Riemann zeta function and its explicit gamma part. A level kernel is a finite signed measure \(w\) with compact support in \((0,\infty)\) acting by \(w[u](y)=\int u_{1+x/L}(y)\,w(dx)\), and likewise on \(g\), with Laplace transform \(\phi_w(\lambda)=\int e^{-x\lambda}w(dx)\). Let \(p(\omega)=\sum_{k=0}^Ka_k\cos k\omega\ge0\) with \(a_k\ge0\) and let \(\phi_w(\lambda)\ge0\) for \(\lambda>0\). Then \(\sum_{k=0}^Ka_k\,w[u](kv)\le\frac1\pi\sum_{k=0}^Ka_k\,w[g](kv)\).

Earlier editions: Theorem 34.8.4, page 1211 (edition 1).

Source: ch/cf3.tex, label cf:thm:multilevel.

The zero side facts alone allow half of the zeros at real part one half plus a

Volume II, Proposition 12.8.4, page 321 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, let \(0<a<\frac12\), let \(0<\gamma_1\le\gamma_2\le\cdots\) be the ordinates of the zeros of \(\xi\) in the upper half plane, counted with multiplicity, and let \(Z=\{\frac12\pm a\pm i\gamma_{2j}:j\ge1\}\) and \(F(s)=\prod_{\rho\in Z}(1-s/\rho)\), the product taken over conjugate pairs. Let \(N(T)\) and \(N_F(T)\) be the numbers of zeros of \(\xi\) and of \(F\) with ordinate in \([0,T]\), \(N_F(\sigma,T)\) the number of zeros of \(F\) with real part above \(\sigma\) and ordinate in \([0,T]\), and \(N(\sigma,T)\) the number of zeros of \(\xi\) with real part at least \(\sigma\) and ordinate in \((0,T]\). Then \(F\) is entire of order one, real on \(\mathbb{R}\), \(F(0)=1\), \(F(1-s)=F(s)\), and \(N_F(T)=N(T)+O(1)\). Moreover: (i) \(\operatorname{Re}(F'/F)(\sigma+iy)\) is the sum of the Cauchy kernels \(\frac{\sigma-\operatorname{Re}\rho}{(\sigma-\operatorname{Re}\rho)^2+(y-\operatorname{Im}\rho)^2}\) of its zeros; (ii) for \(\sigma>\frac12+a\) the level \(\sigma\) measure of \(F\) (density \(\frac1\pi\operatorname{Re}\frac{F'}{F}(\sigma+iy)\)) is positive and obeys the Cauchy semigroup; (iii) with \(M_\varepsilon(T)=\frac1\pi\int_0^T\operatorname{Re}\frac{F'}{F}(\frac12+\varepsilon+it)\,dt\) and \(\tau_\varepsilon(T)\) equal to \(\frac1\pi\) times the change of \(\arg F\) along the horizontal segment from \(\frac12+\varepsilon+iT\) to \(\frac12+iT\), the zero density identity \(M_\varepsilon(T)+\tau_\varepsilon(T)=N_F(T)-2N_F(\frac12+\varepsilon,T)\), which holds exactly for \(\xi\) (with \(\xi\) in place of \(F\) and \(N\) in place of \(N_F\)) when \(T\) is not an ordinate and no zero lies on the segment from \(\frac12+\varepsilon\) to \(\frac12+\varepsilon+iT\), holds for \(F\) up to \(O(\log T)\); (iv) \(N_F(\sigma,T)=\frac12N_F(T)+O(1)\) for \(\frac12<\sigma<\frac12+a\). Hence no inequality that follows from (i) to (iii), the functional equation and order one improves the trivial bound \(N(\sigma,T)\le\frac12N(T)+O(\log T)\) for \(\frac12<\sigma<\frac12+a\).

Source: ch/cf3.tex, label cf:prop:half.

Zero free constant y star of an admissible pair of a cosine polynomial and a level kernel

Volume II, Corollary 12.8.7, page 322 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(p(\omega)=\sum_{k=0}^Ka_k\cos k\omega\ge0\) with \(a_k\ge0\) and let \(w\) be a level kernel with compact support in \((0,\infty)\) and Laplace transform \(\phi_w\ge0\), such that \(\int w(dx)P_{x+y}(\tau)\ge0\) for all \(y\ge0\) and real \(\tau\), with \(P_a\) the Cauchy density of scale \(a\) (the pair is called admissible, which is equivalent to positive definiteness of the even extension of \(\phi_w\)). Assume \(\phi_w(0)>0\) and put \(I=\int_0^\infty\phi_w\), \(J(y)=\int_0^\infty e^{-y\lambda}\phi_w(\lambda)\,d\lambda\) and \(A=a_1+\dots+a_K\). If \((a_1-a_0)I>\frac A2\phi_w(0)\) the equation \(a_1J(y)=a_0I+\frac A2\phi_w(0)\) has a unique root \(y^\ast>0\) and every zero \(\rho\) of the Riemann zeta function with large ordinate has \((1-\operatorname{Re}\rho)\log\gamma_\rho\ge y^\ast-o(1)\). Otherwise the scheme gives no positive constant.

Earlier editions: Corollary 34.8.6, page 1212 (edition 1).

Source: ch/cf3.tex, label cf:cor:ystar.

No admissible pair of a cosine polynomial and a level kernel gives a zero free region wider than a constant below one over log gamma

Volume II, Proposition 12.8.8, page 323 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(p(\omega)=\sum_{k=0}^Ka_k\cos k\omega\ge0\) with \(a_k\ge0\) and \(a_1>a_0\), \(A=a_1+\dots+a_K\), and let \(w\) be a level kernel with Laplace transform \(\phi_w\), admissible, that is \(\phi_w(\lambda)\ge0\) for \(\lambda>0\) and the even extension of \(\phi_w\) is positive definite, and normalised by \(\phi_w(0)=1\). If \(y>0\) satisfies \(a_1\int_0^\infty e^{-y\lambda}\phi_w(\lambda)\,d\lambda\ge a_0\int_0^\infty\phi_w(\lambda)\,d\lambda+\frac A2\) then \(y\le\frac2A\bigl(a_1-a_0-a_0\log\frac{a_1}{a_0}\bigr)<\frac{2(a_1-a_0)}A\le1\). In particular the zero free constant of the corresponding scheme for the zeros of the Riemann zeta function, \((1-\sigma_\rho)\log\gamma_\rho\ge y^\ast\), obeys these bounds, which are \(0.0548\) for \(3+4\cos\omega+\cos2\omega\) and \(0.1051\) for the cosine polynomial of degree \(4\) with coefficients \(\approx(1,1.705,1.044,0.425,0.089)\).

Earlier editions: Proposition 34.8.7, page 1213 (edition 1).

Source: ch/cf3.tex, label cf:prop:barrier2.

Computed zero free constants of admissible pairs: about 1 over 11.8 log gamma with a level kernel with derivatives

Volume II, Table 12.12, page 324 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed (code cf_arith.py, not certified): in the scheme of the zero free constant of an admissible pair, with \((1-\sigma_\rho)\ge c/\log\gamma_\rho\) for the zeros of the Riemann zeta function, the optimised constants \(c=y^\ast\) over cosine polynomials \(p=\lvert\sum_jb_je^{ij\omega}\rvert^2\) with \(b_j\ge0\) are \(0.0287\) for one level with \(3+4\cos\omega+\cos2\omega\), \(0.0376\), \(0.0542\) and \(0.0573\) for the best single level polynomials of degree \(2\), \(3\) and \(4\) to \(8\), \(0.0377\), \(0.0704\) and \(0.0742\) for two levels in the curvature limit with \(3+4\cos\omega+\cos2\omega\) and the best polynomials of degree \(3\) and \(4\), and \(0.0800\), \(0.0826\), \(0.0841\) and \(0.0849\) for a level with derivatives and a polynomial factor of degree \(2\), \(3\), \(4\) and \(5\) (best level \(0.89\) at degree \(4\)). The rows with derivatives are optima on finite constraint grids and at degree \(4\) and \(5\) the kernels are not certified admissible. All values lie below the proved cap \(0.1051\). The curvature and derivative kernels are limits of level kernels and are not covered by the multilevel inequality itself.

Earlier editions: Table 34.12, page 1214 (edition 1).

Source: ch/cf3.tex, label cf:tab:arith.

Computed ceiling of about 1 over 11.3 log gamma for smoothings of nonnegative positive definite type, by linear programming

Volume II, Table 12.13, page 326 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed by discretised linear programming (not certified: positive definiteness is imposed at finitely many frequencies, the support is truncated at \(\lambda_{\max}\) and the polynomial and the smoothing are optimised alternately, so the rows with an optimised polynomial are lower estimates): the best zero free constant \(c\) in \((1-\sigma_\rho)\log\gamma_\rho\ge c\) of the scheme of the zero free constant of an admissible pair, over every smoothing \(\phi\) with \(0\le\phi\le1\), \(\phi(0)=1\) and positive definite even extension, is \(0.0460\) and \(0.0450\) for \(p=3+4\cos\omega+\cos2\omega\) on a uniform and a low frequency grid, \(0.0874\) for the degree \(4\) polynomial of the chapter and for degree at most \(4\) with both optimised, and \(0.0881\) for degree at most \(8\), that is about \(1/(11.3\log\gamma)\), against \(0.0849\) for the best level kernel and the proved cap \(0.1051\).

Earlier editions: Table 34.13, page 1216 (edition 1).

Source: ch/cf3.tex, label cf:tab:lp.

Stechkin pairing with one far level gives a zero free constant outside the admissible class

Volume II, Proposition 12.8.11, page 327 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(p(\omega)=\sum_{k=0}^Ka_k\cos k\omega\ge0\) with \(a_k\ge0\) and \(a_1>a_0\), \(A=a_1+\dots+a_K\) and \(\kappa_\ast=1/\sqrt5\). Every zero \(\rho_0=\sigma_0+i\gamma_0\) of the Riemann zeta function with \(\gamma_0\) large has \((1-\sigma_0)\log\gamma_0\ge\frac{2(\sqrt{a_1}-\sqrt{a_0})^2}{(1-\kappa_\ast)A}-o(1)\). For \(p=3+4\cos\omega+\cos2\omega\) the constant is \(0.05195\) and for the degree \(4\) polynomial with coefficients \(\approx(1,1.705,1.044,0.425,0.089)\) it is \(0.10367=1/9.646\) (computed values; the inequality is proved). The scheme pairs each zero with its reflection through a two level kernel \(\delta_x-\kappa\delta_{x_1}\) with \(x_1\) of order \(\log\gamma_0\), which is outside the admissible class of the barrier proposition. The pairing is the device of Stechkin (1970), and the degree \(4\) constant \(9.646\) is the constant of Rosser and Schoenfeld (1975), an attribution the book recalls without verifying it; the book adds a proof of the pair inequality by direct algebra and the form for every nonnegative cosine polynomial with \(a_1>a_0\).

Earlier editions: Proposition 34.8.10, page 1217 (edition 1).

Source: ch/cf3.tex, label cf:prop:stechkin.

Nested sampling of the normalised Thorin density finds both zeros of the Davenport and Heilbronn function off the line in heights 1000 to 1020

Volume II, Result 12.9.2, page 329 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed in ball arithmetic: for the Davenport and Heilbronn function \(f\), nested sampling over heights \(t\in[1000,1100]\) of the score \(s(t)=u_{0.6}(t)/\varrho(t)\), with \(u_{0.6}\) the density of the Thorin measure at level \(0.6\) of \(\Lambda'/\Lambda\) and \(\varrho(t)=\frac1{2\pi}\log\frac{5t}{2\pi}\), with \(50\) live heights, \(4\) moves and \(300\) iterations (\(1250\) evaluations of \(\Lambda'/\Lambda\)), found negative values in two clusters near \(1012.0\) and \(1018.8\) (no point of the initial uniform sample had \(s<0\); the sampler crossed \(s=0\) at iteration \(239\), with estimated prior volume \(e^{-4.78\pm0.31}\approx0.008\) for \(\{s<0\}\)). Newton’s method gives the zeros \(0.691404808984+1012.019981489088i\) and \(0.698097971267+1018.795762313197i\) of \(f\) with their reflections, each the only zero in a square of half width \(10^{-18}\) (at \(256\) and \(384\) bits). Both zeros are in the table of Balanzario and Sánchez-Ortiz (2007). The argument principle in interval arithmetic gives \(0\) zeros with real part at least \(0.6\) in \([0.6,2]\times[1000,1010]\) and \(2\) in \([0.6,2]\times[1010,1020]\), so nested sampling found every zero with real part above \(0.6\) in \([1000,1020]\), and the two it found are all of them there. The rest of \([1020,1100]\) was not counted. This is a computation for one function and one band of heights, and none of it is a statement about the Riemann zeta function.

Earlier editions: Result 34.9.2, page 1219 (edition 1).

Source: ch/cf4.tex, label cf:res:nsdh.

Volume II, Chapter 13. Balazard, Saias and Yor: Monte Carlo along vertical lines

The Balazard, Saias and Yor criterion at every level theta and every height: the Cauchy average of log of zeta minus log of h equals the sum of Green potentials of the zeros right of the level

Volume II, Theorem 13.2.1, page 336 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\beta+i\gamma\) its nontrivial zeros counted with multiplicity, \(P_a(x)=\frac{a}{\pi(a^2+x^2)}\) for \(a>0\), \(\frac12\le\theta<1\), \(H=\{\operatorname{Re}s>\theta\}\) and \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\), which is analytic in \(H\) and has \(|h_\theta|=|\zeta|\) on the line \(\operatorname{Re}s=\theta\). For a zero \(\rho\) with \(\beta>\theta\) let \(G_\theta(s,\rho)=\log\bigl|\frac{s-2\theta+\bar\rho}{s-\rho}\bigr|\ge0\) be the Green function of \(H\) with pole \(\rho\). For every \(s=\sigma+it\in H\) with \(h_\theta(s)\ne0\), \(\int_{-\infty}^{\infty}\log|\zeta(\theta+i\tau)|\,P_{\sigma-\theta}(t-\tau)\,d\tau-\log|h_\theta(s)|=\sum_{\beta>\theta}G_\theta(s,\rho)\ge0\), the integral absolutely convergent and the sum convergent. In particular the defect \(D_\theta(t_0)=\int_{-\infty}^\infty\log|\zeta(\theta+i\tau)|\,P_{1-\theta}(t_0-\tau)\,d\tau-\log|h_\theta(1+it_0)|\) equals \(\sum_{\beta>\theta}G_\theta(1+it_0,\rho)\), and \(\zeta\) has no zero with \(\beta>\theta\) (the quasi Riemann hypothesis \(\mathrm{QRH}(\theta)\)) if and only if \(D_\theta(t_0)=0\) for one, equivalently every, real \(t_0\). At \(\theta=\frac12\), \(t_0=0\) this is the theorem of Balazard, Saias and Yor (1999): \(\frac1{2\pi}\int_{-\infty}^\infty\frac{\log|\zeta(\frac12+it)|}{\frac14+t^2}\,dt=\sum_{\beta>1/2}\log\bigl|\frac{\rho}{1-\rho}\bigr|\).

Earlier editions: Theorem 35.2.1, page 1228 (edition 1).

Source: ch/bsy1.tex, label bsy:thm:bsy.

Computed bound: zeros above height 3 times 10 to the 12 can change the global defect D_theta(0) by at most about 10 to the minus 12

Volume II, Remark 13.2.2, page 337 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. A zero \(\rho=\beta+i\gamma\) with \(\beta>\theta\) contributes about \(2(\beta-\theta)(1-\theta)/\gamma^2\) to the defect \(D_\theta(0)\), the Cauchy average of \(\log|\zeta(\theta+i\tau)|\) with centre \(0\) and scale \(1-\theta\) minus \(\log\frac1{2-2\theta}\), which is zero exactly when no zero has real part above \(\theta\). Using the verification of the Riemann hypothesis to height \(3\times10^{12}\) (Platt and Trudgian 2021) and the explicit zero counting bound of Hasanalizade, Shen and Wong (2022), the computation in floating point of the book gives that the sum over all possible zeros above that height is below \(7.4\times10^{-13}\) at \(\theta=\frac12\), \(1.85\times10^{-13}\) at \(\theta=\frac34\) and \(4.7\times10^{-14}\) at \(\theta=\frac78\). So the global criterion at \(t_0=0\) cannot detect a zero above the verified range, and the criterion is useful only locally, centred where the zeros are. This is a computation from the cited bounds and not a certified interval bound.

Earlier editions: Remark 35.2.2, page 1229 (edition 1).

Source: ch/bsy1.tex, label bsy:rem:local.

Equivalent forms of the Riemann hypothesis and of QRH(theta): zero defect, positivity of the Thorin measures, and Cauchy deconvolution from level one

Volume II, Corollary 13.3.2, page 338 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\xi\) the completed zeta function, \(\rho=\beta+i\gamma\) its zeros, \(P_a(x)=\frac{a}{\pi(a^2+x^2)}\) and \(\mathfrak U_\sigma\) the measure on \(\mathbb{R}\) with density \(m_\sigma(t)=\pi^{-1}\operatorname{Re}\frac{\xi'}{\xi}(\sigma+it)\) for \(\sigma>\frac12\) not the real part of a zero. Let \(D_\theta(t_0)\) be the Cauchy defect at level \(\theta\) and height \(t_0\) (the Cauchy average of \(\log|\zeta(\theta+i\tau)|\) against \(P_{1-\theta}(t_0-\tau)\) minus \(\log|h_\theta(1+it_0)|\), \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\)). The following are equivalent: (i) the Riemann hypothesis; (ii) \(D_{1/2}(t_0)=0\) for some \(t_0\); (iii) \(\mathfrak U_\sigma\) is a positive measure for every \(\sigma\in(\frac12,1)\); (iv) \(\mathfrak U_1=P_{1/2}*\nu\) for a positive measure \(\nu\) on \(\mathbb{R}\) with \(\int(1+t^2)^{-1}\nu(dt)<\infty\), and then \(\nu=\sum_{\beta=1/2}\delta_\gamma\). More generally, for \(\theta\in[\frac12,1)\), \(\mathrm{QRH}(\theta)\) (no zero with \(\beta>\theta\)) holds if and only if \(D_\theta\equiv0\), if and only if \(\mathfrak U_\sigma\ge0\) for all \(\sigma>\theta\). The equivalence of (i) and (ii) is the criterion of Balazard, Saias and Yor and the equivalence with (iii) is the argument of Lagarias; the book gives the proofs and the form for \(\mathrm{QRH}(\theta)\).

Earlier editions: Corollary 35.3.3, page 1231 (edition 1).

Source: ch/bsy1.tex, label bsy:cor:equiv.

The Levy density of the reciprocal Xi function is positive with no hypothesis, and is an extended generalised gamma convolution under the Riemann hypothesis

Volume II, Proposition 13.4.1, page 339 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function with zeros \(\rho=\beta+i\gamma\) counted with multiplicity, and \(K(u)=\sum_{\gamma>0}e^{-\gamma|u|}\cos\bigl((\beta-\tfrac12)u\bigr)\). Then \(K(u)>0\) for every \(u\ne0\), with no hypothesis. Hence \(x\mapsto\xi(\frac12)/\xi(\frac12+x)\) is the characteristic function of an infinitely divisible law with Levy density \(K(u)/|u|\), since \(\log\frac{\xi(1/2+x)}{\xi(1/2)}=\int_{-\infty}^{\infty}2\sin^2(\frac{xu}2)\frac{K(u)}{|u|}du\). Under the Riemann hypothesis it is moreover an extended generalised gamma convolution (EGGC) \(\exp\bigl(-\int_{(0,\infty)}\log(1+x^2/y^2)\,U(dy)\bigr)\) with \(U=\sum_{\gamma>0}\delta_\gamma\). The proof uses that the zeros with \(0<\gamma<100\) lie on the line, the lowest at \(\gamma_1=14.1347\ldots\), and the bound \(\sum_\rho(1+\gamma^2)^{-1}\le2+\gamma_E-\log4\pi<0.05\) (\(\gamma_E\) Euler’s constant).

Earlier editions: Proposition 35.4.1, page 1232 (edition 1).

Source: ch/bsy1.tex, label bsy:prop:kpos.

The Riemann hypothesis holds if and only if the reciprocal of xi is the characteristic function of an extended generalised gamma convolution

Volume II, Proposition 13.4.2, page 340 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function. The Riemann hypothesis holds if and only if \(x\mapsto\xi(\frac12)/\xi(\frac12+x)\) is the characteristic function of an extended generalised gamma convolution, that is of the form \(\exp\bigl(-\int\log(1+x^2/y^2)\,U(dy)\bigr)\) for a measure \(U\) on \((0,\infty)\) (a drift or a Gaussian part changes the exponent by an entire function and does not affect the argument). When it is, \(U=\sum_{\gamma>0}\delta_\gamma\), the sum over the ordinates of the zeros \(\frac12+i\gamma\) with \(\gamma>0\) counted with multiplicity. The proof uses that a zero \(\frac12+\delta+i\gamma\) with \(\delta\ne0\) produces a pole of \(1/\xi(\frac12+x)\) off the imaginary axis, where an extended generalised gamma convolution is analytic.

Earlier editions: Proposition 35.4.2, page 1232 (edition 1).

Source: ch/bsy1.tex, label bsy:prop:eggc.

The reciprocal of the completed Davenport and Heilbronn function is infinitely divisible with positive Levy density but is not an extended generalised gamma convolution

Volume II, Theorem 13.4.3, page 341 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\kappa=(\sqrt{10-2\sqrt5}-2)/(\sqrt5-1)\), \(f(s)=5^{-s}\bigl[\zeta(s,\frac15)+\kappa\zeta(s,\frac25)-\kappa\zeta(s,\frac35)-\zeta(s,\frac45)\bigr]\) the Davenport and Heilbronn function (\(\zeta(s,a)\) the Hurwitz zeta function) and \(\Lambda(s)=(5/\pi)^{s/2}\Gamma(\frac{s+1}2)f(s)\), so that \(\Lambda(s)=\Lambda(1-s)\). Let \(K_{\mathrm{DH}}(u)=\sum e^{-\gamma|u|}\cos(\delta u)\), the sum over the zeros \(\frac12+\delta+i\gamma\) of \(\Lambda\) with \(\gamma>0\) counted with multiplicity. Then \(K_{\mathrm{DH}}(u)>0\) for every \(u\ne0\), and \(\Lambda(\frac12)/\Lambda(\frac12+x)\) is the characteristic function of an infinitely divisible law. That law is not an extended generalised gamma convolution, because \(\Lambda\) has a zero off the line near \(0.8085+85.6993\,i\) (known to Spira 1994 and Balanzario and Sanchez-Ortiz 2007). The proof rests on three finite computations certified in ball arithmetic at two precisions: (a) \(\Lambda\) has a zero \(\frac12+ig\) with \(5.09<g<5.10\); (b) \(\Lambda\) has no zero \(\beta+i\gamma\) with \(\beta\ne\frac12\) and \(0<\gamma<85.69\) (the argument principle gives \(86\) zeros in \([-2,3]\times[-85.69,85.69]\) against \(43\) certified sign changes of \(\Lambda(\frac12+it)\) on \((0,85.69)\)); (c) \(\Lambda\) has exactly one zero in a square of half width \(10^{-16}\) about \(0.8085171824566374+85.6993484853775922\,i\). The floating point zero list to height \(300\) is evidence only and is not used in the proof.

Earlier editions: Theorem 35.4.3, page 1233 (edition 1).

Source: ch/bsy1.tex, label bsy:thm:dhid.

A combination of L functions of characters modulo 4, 4p, 4q and 4pq with a functional equation whose reciprocal is not infinitely divisible

Volume II, Remark 13.4.5, page 342 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(B(s)=L(s,\chi_4)\) with \(\chi_4\) the character modulo \(4\), \(\Lambda_B(s)=(4/\pi)^{s/2}\Gamma(\frac{s+1}2)B(s)\), \(p<q<2p\) odd primes, \(a=\frac12\log(pq)\), \(b=\frac12\log(p/q)\), \(P(s)=1+2\sqrt p\,p^{-s}+2\sqrt q\,q^{-s}+\sqrt{pq}\,(pq)^{-s}\). Then \(F(s)=(4pq/\pi)^{s/2}\Gamma(\frac{s+1}2)P(s)B(s)=2(pq)^{1/4}\Lambda_B(s)\bigl[\cosh(a(s-\frac12))+2\cosh(b(s-\frac12))\bigr]\) is entire of order one, real and positive on \(\mathbb{R}\) and invariant under \(s\mapsto1-s\), and \(P B\) is a real linear combination of Dirichlet \(L\) functions of characters modulo \(4\), \(4p\), \(4q\) and \(4pq\) (those induced by \(\chi_4\)). For \(p\) large, \(F\) has a lowest pair of zeros above the real axis of the form \(\frac12\pm\delta_0+i\gamma_0\) with \(\delta_0>0\) and \(\gamma_0\) below the first zero of \(\Lambda_B\), the trace \(K(u)=\sum_{\gamma>0}e^{-\gamma|u|}\cos((\beta-\frac12)u)\) over the zeros \(\beta+i\gamma\) of \(F\) satisfies \(K(u)=2e^{-\gamma_0u}\cos(\delta_0u)+O(e^{-(\gamma_0+\eta)u})\) for some \(\eta>0\), and \(K((2k+1)\pi/\delta_0)<0\) for all large \(k\). So \(F(\frac12)/F(\frac12+x)\) is not the characteristic function of an infinitely divisible law. For \(p=101\), \(q=103\) the computation in floating point gives \(\delta_0=0.28475\), \(\gamma_0=0.67928\) and \(K(\pi/\delta_0)=-1.11\times10^{-3}\).

Earlier editions: Remark 35.4.5, page 1236 (edition 1).

Source: ch/bsy1.tex, label bsy:rem:negW.

Level series of the alternating sum of Thorin densities at levels one half plus odd multiples of a

Volume II, Proposition 13.4.7, page 344 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the Riemann Hypothesis (part (a) only; parts (b) and (c) are unconditional).

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(\rho=\beta+i\gamma\) run over its zeros, and for \(\sigma>\frac12\) let \(\mathfrak U_\sigma\) be the measure on \(\mathbb{R}\) with density \(\pi^{-1}\operatorname{Re}(\xi'/\xi)(\sigma+it)\) in \(t\), so that \(\mathfrak U_\sigma=\sum_\rho P_{\sigma-\beta}(\cdot-\gamma)\) with the Cauchy kernel \(P_b(x)=b/(\pi(b^2+x^2))\) (signed when \(\sigma<\sup\beta\)). Let \(\chi_4\) be the nontrivial character modulo \(4\), let \(\mathrm{HS}_a\) be the hyperbolic secant law with density \((2a)^{-1}\operatorname{sech}(\pi x/2a)\), sums over odd \(j\) run over the positive odd integers, and let \(W_a=\frac12\sum_{j\ \mathrm{odd}}\chi_4(j)\mathfrak U_{1/2+ja}\) (Abel sum). (a) Assume RH. For every \(a>0\) and every \(t\), with \(r=e^{-\varepsilon a}\), the series \(\frac12\sum_{j\ \mathrm{odd}}\chi_4(j)r^j\mathfrak U_{1/2+ja}(t)\) converges absolutely for \(\varepsilon>0\) and tends as \(\varepsilon\downarrow0\) to \(W_a(t)=\frac14\sum_\gamma\mathrm{HS}_a(t-\gamma)\), the sum over the ordinates of the zeros with multiplicity. In particular \(W_{1/2}=\frac12(\mathfrak U_1-\mathfrak U_2+\mathfrak U_3-\cdots)\) is a positive measure. (b) For \(\alpha\ge1\) every \(\mathfrak U_\alpha\) is a positive measure with no hypothesis, given for \(\alpha>1\) by the absolutely convergent prime side; the terms \(\mathfrak U_{1/2+ja}\) with \(\frac12+ja<1\) are known to be positive only under RH. (c) At \(a=\frac12\) the partial sums \(\frac12\sum_{j\ \mathrm{odd},\,j\le4N+1}\chi_4(j)\mathfrak U_{j/2+1/2}\) with \(2N+1\) terms are positive with no hypothesis, because each zero contributes a positive kernel; this positivity is the nonvanishing of \(\zeta\) on \(\operatorname{Re}s=1\) and carries no further information.

Source: ch/bsy1.tex, label bsy:prop:levelser.

Closed form kernel of a mirror pair of zeros in the hyperbolic secant level series

Volume II, Proposition 13.4.8, page 345 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0\le c<a\), let \(\mathrm{HS}_a\) be the hyperbolic secant law with density \((2a)^{-1}\operatorname{sech}(\pi x/2a)\), and let \(\rho=\frac12\pm c+i\gamma\) be a mirror pair of zeros of \(\xi\) (a zero and its image under \(s\mapsto1-\bar s\)). In the level series \(W_a=\frac12\sum_{j\ \mathrm{odd}}\chi_4(j)\mathfrak U_{1/2+ja}\) built from the Cauchy kernels \(P_b(x)=b/(\pi(b^2+x^2))\) of the zeros, the pair contributes \(\frac12M_{a,c}(\cdot-\gamma)\) with \(M_{a,c}(x)=\frac{\cos(\pi c/2a)\cosh(\pi x/2a)}{a(\cosh(\pi x/a)+\cos(\pi c/a))}\), a probability density with \(\int M_{a,c}=1\) and Fourier transform \(\widehat M_{a,c}(\omega)=\cosh(c\omega)/\cosh(a\omega)\), equal to \(\mathrm{HS}_a\) for \(c=0\) and to \(\frac12[\mathrm{HS}_a(x+ic)+\mathrm{HS}_a(x-ic)]\). Each zero of the pair alone contributes a kernel with transform \(\frac14e^{\pm c|\omega|}\operatorname{sech}(a\omega)\); the one with the plus sign has positive derivative at \(0^+\) and so is not a positive kernel, and positivity is restored by the mirror partner. The closed form was also checked numerically to twelve digits.

Source: ch/bsy1.tex, label bsy:prop:meixner2.

A mirror pair kernel is in the Thorin class if and only if a over c is an odd integer

Volume II, Theorem 13.4.9, page 345 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0<c<a\) and let \(M_{a,c}\) be the probability density on \(\mathbb{R}\) with Fourier transform \(\cosh(c\omega)/\cosh(a\omega)\) (the law of Brownian motion run to the exit time from \((-a,a)\) started at \(c\)). Then \(M_{a,c}\) is infinitely divisible with the positive Levy density \(\nu_{a,c}(x)=\frac1{|x|}\sum_{k\ge1}\big[e^{-(2k-1)\pi|x|/2a}-e^{-(2k-1)\pi|x|/2c}\big]>0\), and \(|x|\nu_{a,c}(x)\) is the Laplace transform in \(|x|\) of the signed measure \(\sum_k\delta_{(2k-1)\pi/2a}-\sum_k\delta_{(2k-1)\pi/2c}\). The kernel is in the Thorin class on the line if and only if \(a/c\) is an odd integer, and then the negative atoms cancel exactly, for example \(\cosh(c\omega)/\cosh(3c\omega)=1/(2\cosh(2c\omega)-1)\). For \(c=0\) the kernel is the hyperbolic secant law \(\mathrm{HS}_a\), which is in the class.

Source: ch/bsy1.tex, label bsy:thm:dichotomy.

Reformulation of RH by the Thorin class of the mirror pair kernels at two dampings

Volume II, Corollary 13.4.10, page 346 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a_1,a_2\ge\frac12\) have irrational ratio \(a_1/a_2\). Then the Riemann Hypothesis holds if and only if, at both dampings \(a_1\) and \(a_2\), every kernel \(M_{a_i,c}\) (the probability density with Fourier transform \(\cosh(c\omega)/\cosh(a_i\omega)\)) belonging to a mirror pair of zeros \(\frac12\pm c+i\gamma\) is in the Thorin class. The proof is that if RH holds every \(c=0\), and conversely a pair with \(c>0\) passing at both dampings makes \(a_1/c\) and \(a_2/c\) odd integers, so \(a_1/a_2\) would be rational. This is a reformulation, not a route: a single damping does not suffice since \(c=a/(2n+1)\) passes, and the statement says that every \(c\) is zero.

Source: ch/bsy1.tex, label bsy:cor:reform.

The criterion as optional stopping: log of h at the hitting point of the line for planar Brownian motion, a submartingale that is a martingale exactly when QRH(theta) holds

Volume II, Proposition 13.5.1, page 347 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(B\) be planar Brownian motion started at \(s_0=1+it_0\), \(\tau\) its hitting time of the line \(\{\operatorname{Re}s=\theta\}\), \(\frac12\le\theta<1\), \(H=\{\operatorname{Re}s>\theta\}\), \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\) and \(G_\theta(s,\rho)=\log\bigl|\frac{s-2\theta+\bar\rho}{s-\rho}\bigr|\). Then \(\log|h_\theta(B_\tau)|\) is integrable and \(\mathbb{E}\log|h_\theta(B_\tau)|=\log|h_\theta(s_0)|+\sum_{\beta>\theta}G_\theta(s_0,\rho)\), the sum over the zeros \(\rho=\beta+i\gamma\) of \(\zeta\) with \(\beta>\theta\) with multiplicity. The process \(\log|h_\theta(B_{t\wedge\tau})|\) is a local martingale and a submartingale, and a uniformly integrable martingale if and only if \(\zeta\) has no zero with \(\beta>\theta\). Each zero contributes its Riesz mass \(2\pi\), times \(\frac12\), times the value at \(\rho\) of \(\pi^{-1}G_\theta(s_0,\cdot)\), the density of the expected occupation measure of \(B\) killed at \(\tau\).

Earlier editions: Proposition 35.5.1, page 1236 (edition 1).

Source: ch/bsy1.tex, label bsy:prop:yor.

Under QRH(theta) the complex logarithm of h at the line has all moments equal to powers of its value at the starting point, equal variances of log of zeta and its argument, zero covariance, and a Littlewood Paley variance formula

Volume II, Theorem 13.5.2, page 348 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on QRH(theta): the Riemann zeta function has no zero with real part greater than theta, for the stated theta in [1/2, 1) (for theta = 1/2 this is the Riemann hypothesis).

Statement. Assume \(\mathrm{QRH}(\theta)\), \(\frac12\le\theta<1\): \(\zeta\) has no zero with real part greater than \(\theta\). Let \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\), \(H=\{\operatorname{Re}s>\theta\}\), \(L\) the analytic logarithm of \(h_\theta\) on \(H\) with \(L(\sigma+it)\to0\) as \(\sigma\to+\infty\), and \(G_\theta(1+it_0,w)=\log\bigl|\frac{w-2\theta+1-it_0}{w-1-it_0}\bigr|\) the Green function of \(H\) with pole \(1+it_0\). For almost every \(\tau\) the limit \(Z(\tau)=X(\tau)+iY(\tau)=\lim_{\sigma\downarrow\theta}L(\sigma+i\tau)\) exists, \(X(\tau)=\log|\zeta(\theta+i\tau)|\) and \(Y(\tau)=\arg h_\theta(\theta+i\tau)\) continued along the horizontal path from \(+\infty\). Let \(\tau\) have the Cauchy law with density \(\frac{1-\theta}{\pi((1-\theta)^2+(t_0-\tau)^2)}\) and \(z_0=x_0+iy_0=L(1+it_0)\). Then \(\mathbb{E}|Z|^p<\infty\) for every \(p<\infty\), \(\mathbb{E}[Z^k]=z_0^k\) for every \(k\ge1\), and in particular \(\mathbb{E}X=x_0\), \(\mathbb{E}Y=y_0\), \(\operatorname{Var}X=\operatorname{Var}Y\), \(\operatorname{Cov}(X,Y)=0\), with \(\operatorname{Var}X=\operatorname{Var}Y=\frac1\pi\iint_HG_\theta(1+it_0,w)\bigl|\frac{h_\theta'}{h_\theta}(w)\bigr|^2dA(w)\). The raw argument of \(\zeta\) does not satisfy these identities: at \(\theta=\frac12\), \(t_0=0\) under the Riemann hypothesis the variance of \(\arg\zeta\) exceeds that of \(\log|\zeta|\) by more than \(\pi^2/3\).

Earlier editions: Theorem 35.5.2, page 1237 (edition 1).

Source: ch/bsy1.tex, label bsy:thm:cmart.

Computed moments of the complex logarithm of h at the line for zeta and the Davenport and Heilbronn function: the identities hold for zeta and fail for Davenport and Heilbronn

Volume II, Table 13.2, page 349 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\) for \(\zeta\), \(Z=X+iY=\log h_\theta(\theta+i\tau)\) (continued horizontally from \(+\infty\)) with \(\tau\) Cauchy distributed with centre \(t_0\) and scale \(1-\theta\), \(h=f\) for the Davenport and Heilbronn function, and \(x_0+iy_0=\log h(1+it_0)\). Computed in floating point (Poisson weighted panels between the zeros, with a window \(|\tau-t_0|\le13000\) and a tail correction; the larger of the difference between two meshes and half the tail correction is at most \(2.1\times10^{-5}\) in each entry): for \(\zeta\), \(\mathbb{E}X-x_0=\mathbb{E}Y-y_0=\operatorname{Cov}(X,Y)=0\) and \(\operatorname{Var}X=\operatorname{Var}Y\) equal to \(0.1415\) at \((\theta,t_0)=(\frac12,0)\), \(0.7056\) at \((\frac12,85.7)\), \(1.0539\) at \((\frac12,1000)\), \(1.1298\) at \((\frac12,5000)\) and \(0.1341\) at \((\frac34,85.7)\) to that accuracy; for the Davenport and Heilbronn function at \((\frac12,85.7)\), \(\mathbb{E}X-x_0=1.4407\), \(\mathbb{E}Y-y_0=-0.0016\), \(\operatorname{Var}X=0.6410\), \(\operatorname{Var}Y=4.4414\), \(\operatorname{Cov}(X,Y)=-0.0727\), and at \((\frac34,85.7)\), \(\mathbb{E}X-x_0=0.4770\), \(\mathbb{E}Y-y_0=-0.0003\), \(\operatorname{Var}X=1.0124\), \(\operatorname{Var}Y=3.2682\), \(\operatorname{Cov}(X,Y)=-0.0162\), so the identities fail there. The agreement for \(\zeta\) is a computation and not a proof for any \(\theta\).

Earlier editions: Table 35.2, page 1239 (edition 1).

Source: ch/bsy1.tex, label bsy:tab:cm.

Without the hypothesis that no zero lies right of the line Re s = theta: the defects of the complex moments of log h as integrals along horizontal cuts from the zeros to the line

Volume II, Proposition 13.5.4, page 350 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12\le\theta<1\), \(z=1+it_0\), \(z^*=2\theta-1+it_0\), \(K(s)=\frac1{s-z^*}-\frac1{s-z}\) and \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\). For each ordinate \(\gamma\) of a zero with \(\beta>\theta\) let \(b_\gamma\) be the largest real part of the zeros with that ordinate, \(C_\gamma=[\theta+i\gamma,b_\gamma+i\gamma]\), and for \(x+i\gamma\in C_\gamma\) let \(M_\gamma(x)\) be the number of zeros \(\beta'+i\gamma\) with \(\beta'>x\), with multiplicity. Let \(L\) be the logarithm of \(h_\theta\) on \(\{\operatorname{Re}s>\theta\}\) minus the cuts, continued along horizontal paths from \(+\infty\) where it tends to \(0\), \(z_0=L(z)\), \(L_-\) its boundary value from below on \(C_\gamma\) (the value above minus the value below is \(2\pi iM_\gamma\)), \(Z(\tau)=\lim_{\sigma\downarrow\theta}L(\sigma+i\tau)\) and \(\tau\) Cauchy distributed with centre \(t_0\) and scale \(1-\theta\). Then for every polynomial \(F\), \(\mathbb{E}F(Z)-F(z_0)=\sum_\gamma\frac1{2\pi i}\int_{C_\gamma}[F(L_-+2\pi iM_\gamma)-F(L_-)]K(s)\,ds\), each integral taken from left to right and the sum absolutely convergent. In particular \(D:=\mathbb{E}Z-z_0=\sum_{\beta>\theta}[\log\frac{\rho-z^*}{\rho-z}-\log\frac{\theta+i\gamma-z^*}{\theta+i\gamma-z}]\) over the zeros \(\rho=\beta+i\gamma\) with multiplicity, with the logarithms continued along the cut, and \(\mathbb{E}Z^2-z_0^2=2\sum_\gamma\int_{C_\gamma}M_\gamma(L_-+\pi iM_\gamma)K\,ds\). The real part of the formula for \(D\) is the Balazard, Saias and Yor identity at \(1+it_0\). The second order balance \(\operatorname{Var}X=\operatorname{Var}Y\), \(\operatorname{Cov}(X,Y)=0\) for \(Z=X+iY\) holds exactly when \(2\sum_\gamma\int_{C_\gamma}M_\gamma(L_-+\pi iM_\gamma-z_0)K\,ds=D^2\). The same holds for the Davenport and Heilbronn function with \(h=f\) when no zero has \(\gamma=t_0\) and \(\beta\ge1\). A floating point check on the Davenport and Heilbronn function at \(t_0=85.7\) with the zeros below height \(700\) gives at \(\theta=\frac12\) the defect \(\mathbb{E}Z-z_0=1.4407-0.0016\,i\) and the balance defect \(\operatorname{Var}X-\operatorname{Var}Y+2i\operatorname{Cov}(X,Y)=-3.8003-0.1455\,i\), against \(1.4407-0.0016\,i\) and \(-3.8004-0.1455\,i\) from the computed moments, and at \(\theta=\frac34\) \(0.4770-0.0003\,i\) and \(-2.2558-0.0323\,i\) for both; the jump across every cut is \(2\pi i\) to \(7\) digits (computed). Heuristic and not claimed: that the balance fails for most \(t_0\) when a zero lies right of the line.

Earlier editions: Proposition 35.5.4, page 1239 (edition 1).

Source: ch/bsy1.tex, label bsy:prop:cuts.

Finite moments of log of zeta under the Cauchy law on a line at every level and height

Volume II, Lemma 13.6.1, page 352 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12\le\theta<1\), \(t_0\in\mathbb{R}\), \(f(\tau)=\log|\zeta(\theta+i\tau)|\) and \(P\) the Cauchy law with density \(\frac{1-\theta}{\pi((1-\theta)^2+(t_0-\tau)^2)}\). Then \(\mathbb{E}_P|f|^p<\infty\) for every \(p<\infty\). Hence the plain Monte Carlo estimator of \(\mathbb{E}_Pf\) that averages \(f\) at \(n\) independent exact Cauchy draws is unbiased and has root mean square error \((\operatorname{Var}_Pf/n)^{1/2}\) with no truncation of the heavy tail. The proof bounds \(\int_T^{T+1}|f|^p\,dt\ll_p\log^pT\) using the convexity bound and the local formula of Titchmarsh for \(\log|\zeta|\).

Earlier editions: Lemma 35.6.1, page 1241 (edition 1).

Source: ch/bsy2.tex, label bsy:lem:var.

Computed bound on the gain of any weight over plain Monte Carlo for independent draws: the factor Var f over the squared mean absolute deviation is between 1.35 and 2.46

Volume II, Remark 13.6.2, page 353 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\mu=\mathbb{E}_Pf\), \(w>0\), \(Z=\mathbb{E}_Pw\) and \(q=Pw/Z\). For independent draws from \(q\) the self normalised estimator \(\hat\mu=\sum_if(\tau_i)/w(\tau_i)\big/\sum_i1/w(\tau_i)\) has \(n\operatorname{Var}\hat\mu\to Z\,\mathbb{E}_P[(f-\mu)^2/w]\), at least \((\mathbb{E}_P|f-\mu|)^2\) with equality exactly for \(w\propto|f-\mu|\), and \((\mathbb{E}_P|f-\mu|)^2\le\operatorname{Var}_Pf\). For the nine integrands \(f=\log|h|\) of the book’s table of local defects (zeta and Davenport and Heilbronn at several levels and heights), the factor \(\operatorname{Var}_Pf/(\mathbb{E}_P|f-\mu|)^2\) ranges from \(1.35\) to \(2.46\) with an error up to \(0.03\) (computed by quadrature), so for independent draws no weight gains more than that over plain Monte Carlo. For a Markov chain negative autocorrelation can bring the limit below the value for independent draws, so the factor does not bound Markov samplers. The optimal weight itself is the standard one of self normalised importance sampling.

Earlier editions: Remark 35.6.2, page 1242 (edition 1).

Source: ch/bsy2.tex, label bsy:rem:weight.

Computed local defects D_theta(t_0) for zeta and the Davenport and Heilbronn function by quadrature, plain Monte Carlo and vertical Monte Carlo

Volume II, Table 13.3, page 354 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(D_\theta(t_0)\) be the defect of the Cauchy average of \(\log|h|\) on the line \(\operatorname{Re}s=\theta\) against the Cauchy law of centre \(t_0\) and scale \(1-\theta\), minus \(\log|h(1+it_0)|\), with \(h=h_\theta\), \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\), for \(\zeta\) and \(h=f\) for the Davenport and Heilbronn function, whose zero \(0.808517182457+85.699348485378\,i\) lies off the critical line. Computed in floating point (adaptive quadrature with the zeros as breakpoints on \(|\tau-t_0|\le W\), plain Monte Carlo with \(20{,}000\) exact Cauchy draws for \(\zeta\) and \(4{,}000\) for the Davenport and Heilbronn function, and a vertical likelihood Metropolis sampler): for \(\zeta\) at \((\theta,t_0)=(\frac12,0),(\frac78,0),(\frac12,85.7),(\frac34,85.7),(\frac78,85.7)\) the quadrature at \(W=200\) gives values of absolute value at most \(5.0\times10^{-6}\), and both Monte Carlo columns agree with \(0\) within \(2.4\) standard errors; for the Davenport and Heilbronn function the quadrature at \(W=200\) gives \(1.4406\) at \((\frac12,85.7)\), \(0.47698\) at \((\frac34,85.7)\), \(1.47\times10^{-4}\) at \((\frac34,100)\) and \(-3.6\times10^{-7}\) at \((\frac78,85.7)\), against the predictions \(1.4407\), \(0.4770\), \(1.49\times10^{-4}\) and \(0\) (the sum of the Green potentials of the zeros found below height \(700\); the exact defect at \(\theta=\frac78\) is positive, since \(f\) has zeros with real part above \(1\)). The neglected Cauchy mass outside the window makes the realised truncation errors larger than the printed digits suggest. This is a computation at these points and not a statement for all heights.

Earlier editions: Table 35.3, page 1244 (edition 1).

Source: ch/bsy2.tex, label bsy:tab:defects.

Computed errors of plain Monte Carlo, the Yakowitz, Krimmel and Szidarovszky trapezoid rule and its Dirichlet polynomial control for the Cauchy average of log of zeta

Volume II, Table 13.4, page 356 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed in floating point over \(400\) replications at \(t_0=0\) for the Cauchy average of \(\log|\zeta(\theta+i\tau)|\), with \(N\) between \(500\) and \(16{,}000\) nodes: at \(\theta=\frac12\) the root mean square error of plain Monte Carlo, of the sorted trapezoid rule (weights of Yakowitz, Krimmel and Szidarovszky, 1978) and of the trapezoid rule with the exact control by the Selberg smoothed truncation of \(\log\zeta\) (\(M=100\)) is \(1.7\times10^{-2}\), \(9.6\times10^{-3}\) and \(5.3\times10^{-3}\) at \(N=500\) and \(3.1\times10^{-3}\), \(9.7\times10^{-4}\) and \(7.2\times10^{-4}\) at \(N=16{,}000\), a gain of \(3.2\) to \(4.3\); at \(\theta=\frac78\) (\(M=1000\)) it is \(3.0\times10^{-2}\), \(2.3\times10^{-3}\) and \(4.0\times10^{-4}\) at \(N=500\) and \(5.1\times10^{-3}\), \(2.9\times10^{-4}\) and \(6.9\times10^{-5}\) at \(N=16{,}000\), a gain of \(65\) to \(75\). The fitted slopes over \(500\le N\le16{,}000\) at \(\theta=\frac12\) and \(\frac78\) are \(-0.49\) and \(-0.50\) (plain), \(-0.66\) and \(-0.60\) (trapezoid) and \(-0.58\) and \(-0.52\) (trapezoid with control), not the \(-2\) of the bound for \(g\in C^2\). The sampling error of each entry is about \(3.5\%\). The reference values \(0\) and \(\log4\) are those the criterion gives under the Riemann hypothesis and QRH(7/8); with no hypothesis they hold to within \(7.4\times10^{-13}\) and \(4.7\times10^{-14}\), a floating point computation. This is a computation and not a proof of a rate.

Earlier editions: Table 35.4, page 1246 (edition 1).

Source: ch/bsy2.tex, label bsy:tab:yks.

The integrand of the Cauchy average of log of zeta is not C^1 on the unit interval, so the N to the minus four bound of Yakowitz, Krimmel and Szidarovszky does not apply

Volume II, Proposition 13.6.4, page 357 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12\le\theta<1\), \(f(\tau)=\log|\zeta(\theta+i\tau)|\) and \(g=f\circ\tau\) with \(\tau(u)=t_0+(1-\theta)\tan(\pi(u-\frac12))\), \(u\in(0,1)\). If \(\theta=\frac12\), \(g\) is unbounded near \(\tau^{-1}(\gamma)\) for every ordinate \(\gamma\) of a zero on the line. If \(\frac12<\theta<1\), \(g'\) is unbounded near \(u=0\) and near \(u=1\). In both cases \(g\notin C^1[0,1]\), so the bound \(\mathbb{E}(T_N-\mathbb{E}_Pf)^2=O(N^{-4})\) of Yakowitz, Krimmel and Szidarovszky (1978) for the sorted trapezoid rule \(T_N\) with \(N\) uniform nodes, which assumes \(g\in C^2[0,1]\), does not apply. The proof for \(\frac12<\theta<1\) uses the mean value theorems \(T^{-1}\int_1^T|\zeta(\theta+it)|^2dt\to\zeta(2\theta)\) and \(T^{-1}\int_1^T|\zeta(\theta+it)|^4dt\to\zeta(2\theta)^4/\zeta(4\theta)\) and the inequality \(\zeta(4\theta)<\zeta(2\theta)^2\).

Earlier editions: Proposition 35.6.4, page 1247 (edition 1).

Source: ch/bsy2.tex, label bsy:prop:rate.

The Balazard, Saias and Yor identity on a half disc: exact local form with the Green function of the half disc and zeros on the diameter subtracted

Volume II, Theorem 13.6.6, page 359 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(c=\theta+it_0\), \(R>0\), \(H=\{|s-c|<R,\operatorname{Re}s>\theta\}\), \(z\in H\) and \(\omega_z\) the harmonic measure of \(\partial H\) seen from \(z\). Let \(h\not\equiv0\) be analytic on a neighbourhood of \(\overline H\), with no zero on the closed arc \(\{|s-c|=R,\operatorname{Re}s\ge\theta\}\), with zeros \(\rho_1,\dots,\rho_n\) of multiplicities \(m_1,\dots,m_n\) on the open diameter, and \(h(z)\ne0\). Then \(\int_{\partial H}(\log|h|-\sum_jm_j\log|s-\rho_j|)\,d\omega_z=\log|h(z)|-\sum_jm_j\log|z-\rho_j|+\sum_{\rho\in H}G_H(z,\rho)\), the last sum over the zeros of \(h\) in \(H\) with multiplicity, and the integrand on the left is real analytic on each of the two closed pieces of \(\partial H\). In the coordinate \(w=(s-c)/R\) with \(w_z=(z-c)/R\), \(w^*=-\overline{w_z}\) and \(G_D(a,w)=\log|(1-\bar aw)/(w-a)|\), the Green function is \(G_H(z,\cdot)=G_D(w_z,\cdot)-G_D(w^*,\cdot)\); on the arc \(w=e^{i\phi}\), \(|\phi|\le\frac\pi2\), the harmonic measure has density \(\frac{1-|w_z|^2}{2\pi|e^{i\phi}-w_z|^2}-\frac{1-|w^*|^2}{2\pi|e^{i\phi}-w^*|^2}\) with respect to \(d\phi\), and on the diameter \(w=iy\) the density with respect to \(dy\) is \(\frac1{2\pi}\partial_x[G_D(w_z,x+iy)-G_D(w^*,x+iy)]\) at \(x=0\). Both densities are real analytic up to the corners \(w=\pm i\), where they vanish to first order. In particular \(h\) has no zero in \(H\) if and only if the identity holds without the Green sum.

Earlier editions: Theorem 35.6.6, page 1249 (edition 1).

Source: ch/bsy2.tex, label bsy:thm:halfdisk.

The half disc identity becomes the Balazard, Saias and Yor identity on the half plane as the radius tends to infinity

Volume II, Remark 13.6.7, page 359 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\) and \(z=1+it_0\). Along radii \(R_k\to\infty\) with \(|t_0\pm R_k-\gamma|\ge c/\log R_k\) for every zero ordinate \(\gamma\), the arc term of the half disc identity tends to \(0\) (on the arc the density is at most \(CR^{-1}\cos\phi\), so the part of the arc with \(\theta\le\operatorname{Re}s\le2\) has harmonic measure \(O(R_k^{-3})\), and there \(|\log|h_\theta||=O(\log^2R_k)\), while \(|\log|h_\theta(s)||=O(2^{-\operatorname{Re}s}+R_k^{-1})\) on the rest of the arc), the density on the diameter increases to the Cauchy density of scale \(1-\theta\) and centre \(t_0\), and the Green sum increases to the sum of the Green potentials of the zeros right of the line. So the half disc identity becomes the Balazard, Saias and Yor identity at \(1+it_0\). That the arc carries vanishing harmonic measure would not suffice by itself: for \(h(s)=e^{s-\theta}\) the arc contributes \(1-\theta\) for every \(R\).

Earlier editions: Remark 35.6.7, page 1250 (edition 1).

Source: ch/bsy2.tex, label bsy:rem:halfplane.

The sorted trapezoid rule at independent uniform nodes: exact leading bias and variance of order N to the minus five, so mean square N to the minus four, and N to the minus five when the end slopes agree

Volume II, Lemma 13.6.8, page 360 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g\in C^4[a,b]\) and let \(T_N\) be the trapezoid rule on the sorted \(N\) independent uniform nodes on \([a,b]\) with the end points \(a\) and \(b\) added. Then \(\mathbb{E}T_N-\int_a^bg=\frac{(b-a)^2(g'(b)-g'(a))}{2(N+1)(N+2)}+O(N^{-3})\) and \(\operatorname{Var}T_N=O(N^{-5})\). So the mean square error is \(O(N^{-4})\), and \(O(N^{-5})\) when \(g'(a)=g'(b)\).

Earlier editions: Lemma 35.6.8, page 1250 (edition 1).

Source: ch/bsy2.tex, label bsy:lem:trap.

Half disc identity: the sorted trapezoid rule on the two smooth pieces of the boundary has mean square error O(N^-4), and O(N^-5) after a change of variable

Volume II, Result 13.6.5, page 361 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. In the half disc identity, each of the two pieces of the boundary integral is a smooth integrand on a closed interval vanishing at the ends. By the sorted trapezoid rule lemma the rule with \(N/2\) nodes on each piece has mean square error \(O(N^{-4})\), its bias \(\frac{2(b-a)^2(g'(b)-g'(a))}{(N+2)(N+4)}+O(N^{-3})\) vanishing to that order only if the end slopes agree. The change of variable \(y=\Phi(v)=v+\sin(\pi v)/\pi\) on each piece, written on \([-1,1]\), makes the transformed integrand \(g(\Phi(v))\Phi'(v)\) vanish to at least fifth order at the ends, since \(\Phi'(\pm(1-r))=\frac{\pi^2}2r^2+O(r^4)\) and \(1-|\Phi(\pm(1-r))|=\frac{\pi^2}6r^3+O(r^5)\); both end slopes are then zero and the mean square error is \(O(N^{-5})\). These are upper bounds.

Earlier editions: Result 35.6.5, page 1251 (edition 1).

Source: ch/bsy2.tex, label bsy:res:halfdisk_rates.

Computed errors of plain Monte Carlo and the sorted trapezoid rule, with and without the change of variable, on the half disc identity for zeta and for Davenport and Heilbronn

Volume II, Table 13.5, page 361 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed in floating point on the half disc identity with \(R=1\), \(t_0=85.7\), \(z=c+\frac12R\), over \(200\) replications with \(N/2\) nodes on each boundary piece, slopes fitted for \(N\ge1024\) (in brackets). For \(\zeta\) at \(\theta=\frac34\) (target \(0.028322\)): plain \(1.2\times10^{-3}\) (\(-0.49\)), sorted trapezoid \(3.4\times10^{-9}\) (\(-2.07\)), with the change of variable \(2.7\times10^{-9}\) (\(-2.49\)). For \(\zeta\) at \(\theta=\frac12\) with the zero \(\frac12+84.7354\ldots i\) on the diameter subtracted (target \(0.048774\)): \(2.7\times10^{-3}\) (\(-0.52\)), \(1.9\times10^{-9}\) (\(-2.28\)), \(5.1\times10^{-9}\) (\(-2.49\)). For the Davenport and Heilbronn function at \(\theta=\frac12\) with its zero \(0.8085\ldots+85.6993\ldots i\) in the half disc, so that the target \(-0.365254\) includes \(G_H(z,\rho)=1.12940\): \(2.1\times10^{-3}\) (\(-0.50\)), \(7.8\times10^{-9}\) (\(-2.16\)), \(1.4\times10^{-8}\) (\(-2.51\)). Gauss and Legendre quadrature of the boundary integral agrees with the right side to \(5\times10^{-15}\). These are computations and not proofs; at these \(N\) the change of variable is not the more accurate rule for \(\zeta\) and the Davenport and Heilbronn function at \(\theta=\frac12\), and whether the rules cross beyond \(N=32{,}768\) was not computed.

Earlier editions: Table 35.5, page 1252 (edition 1).

Source: ch/bsy2.tex, label bsy:tab:halfdisk.

Under RH, the Cauchy tail of log of zeta beyond distance T from the centre is O(log T / (T^2 (log log T)^2)) at level one half and O(T^-2 log T) at levels above

Volume II, Result 13.6.6, page 362 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann hypothesis: every nontrivial zero of the Riemann zeta function has real part 1/2.

Statement. Assume the Riemann hypothesis. Let \(t_0\in\mathbb{R}\) and \(P_{1/2}(t_0-\tau)\) the Cauchy density of centre \(t_0\) and scale \(\frac12\). Using the estimate \(\int_T^t\log|\zeta(\frac12+iu)|\,du=O(\log t/(\log\log t)^2)\) for \(3\le T\le t\) (Bui, Lester and Milinovich 2014) and integration by parts, \(\int_{|\tau-t_0|>T}\log|\zeta(\frac12+i\tau)|\,P_{1/2}(t_0-\tau)\,d\tau=O_{t_0}(\log T/(T^2(\log\log T)^2))\). At a level \(\theta\in(\frac12,1)\), under the same hypothesis, the integral of \(\log\zeta\) around the rectangle with vertices \(\theta+iT\), \(\theta+it\) and \(+\infty\) bounds the primitive of \(\log|\zeta(\theta+iu)|\) by \(O_\theta(\log t)\), and the tail of the Cauchy average (scale \(1-\theta\)) beyond \(T\) by \(O_\theta(T^{-2}\log T)\). The case \(t_0=0\), \(\theta=\frac12\) is a theorem of Bui, Lester and Milinovich; the extension to every centre and to the levels above one half is the book’s.

Earlier editions: Result 35.6.6, page 1252 (edition 1).

Source: ch/bsy2.tex, label bsy:res:tail.

Computed errors of the truncated shifted trapezoid rule for the Cauchy average of log of zeta over a window at height 85.7, with zeros deflated at level one half

Volume II, Table 13.6, page 363 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed in floating point at \(t_0=85.7\): the error of the shifted trapezoid rule for the Cauchy average of \(\log|\zeta(\theta+i\tau)|\) over \(|\tau-t_0|\le T\), minus \(\log|h_\theta(1+it_0)|\), mean over \(10\) random shifts. At \(\theta=\frac34\) (step \(0.02\)): errors \(-9.47\times10^{-4}\), \(-3.40\times10^{-5}\), \(+4.36\times10^{-6}\), \(+6.47\times10^{-8}\), \(-1.36\times10^{-7}\) at \(T=10,40,160,640,1280\). At \(\theta=\frac12\) (step \(0.01\), the zeros with \(|\gamma-t_0|\le T+1\) deflated and their window means added back): \(-2.09\times10^{-3}\), \(-5.37\times10^{-5}\), \(+1.05\times10^{-5}\), \(+7.84\times10^{-8}\) at \(T=10,40,160,640\), with \(10,33,123,742\) zeros deflated. In this range \(T^2\cdot|\mathrm{error}|\) stays below \(0.27\), and the spread over the shifts is at most \(8.1\times10^{-7}\). The shifted grid is unbiased for the window integral, so the mean error is minus the Cauchy tail. This is a computation and not a theorem about the rule: at a fixed step the discretisation error of one shift need not tend to \(0\) as \(T\) grows, and the bounds on the tail are conditional on the Riemann hypothesis. Adding back the whole line means of the deflating functions instead would add their tails, and for that variant \(T^2\cdot|\mathrm{error}|\) grows from \(0.15\) at \(T=10\) to \(1.24\) at \(T=640\) (computed).

Earlier editions: Table 35.6, page 1253 (edition 1).

Source: ch/bsy2.tex, label bsy:tab:trunc.

Rates of the randomly shifted product trapezoid rule on the torus (every power for a smooth periodic integrand, order N to the minus one with a logarithmic singularity) and N to the minus three halves for Voronoi and Delaunay weights at independent uniform nodes

Volume II, Proposition 13.7.2, page 365 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g\) be square integrable on the torus \([0,1)^2\) and \(Q(\delta)=m^{-2}\sum_{j\in\{0,\dots,m-1\}^2}g(j/m+\delta)\), with \(N=m^2\) nodes and \(\delta\) uniform on \([0,1/m)^2\). Then \(\mathbb{E}_\delta Q=\int g\) and \(\mathbb{E}_\delta|Q-\int g|^2=\sum_{\ell\ne0}|\hat g(m\ell)|^2\). (i) If \(g\) has a smooth periodic extension, for example \(g=\log|f|\,\Delta\varphi\) with \(\varphi\) the flat bump \(b(x)b(y)\), \(b(x)=e^{-1/(1-x^2)}\), and \(f\) without zeros in the box, then \(Q(\delta)-\int g=O(N^{-k})\) for every \(k\), uniformly in \(\delta\). (ii) If \(g=\psi\log|s-\rho|\) with \(\psi\in C_c^\infty((0,1)^2)\) and \(\rho\in(0,1)^2\), then \((\mathbb{E}_\delta|Q-\int g|^2)^{1/2}=O(N^{-1})\); for a fixed shift whose nodes all lie at distance at least \(c/m\) from \(\rho\), \(|Q-\int g|=O_c(N^{-1}\log N)\); and if \(\psi(\rho)\ne0\) no bound uniform in the shift holds. (iii) On the box, for \(g\in C^3\) with \(g=0\) near the boundary, or \(g\) smooth with every derivative vanishing on the boundary (for example \(g=\log|f|\,\Delta\varphi\) when \(f\) has no zero in the closed box), Voronoi weights (cells clipped to the box) and Delaunay weights (triangulation of the nodes and the four corners of the box) at independent uniform nodes have root mean square error \(O(N^{-3/2})\), against \(N^{-1/2}\) for the sample mean; this is proved by the weighted Monte Carlo theorem with \(d=2\). At jittered nodes the same rate is computed, not proved. The variance identity and part (i) are standard for randomly shifted lattice rules; parts (ii) and (iii) are the book’s.

Earlier editions: Proposition 35.7.2, page 1256 (edition 1).

Source: ch/bsy3.tex, label bsy:prop:rates2d.

Voronoi and Delaunay weighted Monte Carlo at independent uniform nodes for integrands flat at the boundary: root mean square error O(N^(-1/2-2/d)) in every dimension d

Volume II, Theorem 13.7.3, page 366 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Q=[-1,1]^d\), \(X_1,\dots,X_N\) independent and uniform on \(Q\), and \(g\in C^3(Q)\) with (a) \(g=0\) within distance \(\delta>0\) of \(\partial Q\), or (b) \(g\in C^\infty(Q)\) with every derivative vanishing on \(\partial Q\). Let \(V_N=\sum_i|V_i\cap Q|\,g(X_i)\) be the Voronoi rule with the cells clipped to \(Q\), \(D_N=\int_QI_Ng\) the Delaunay rule, where \(I_Ng\) interpolates \(g\) linearly on the Delaunay triangulation of the nodes and the \(2^d\) corners of \(Q\). With \(d(x)\) the distance from \(x\) to the nearest node and \(v(x)\) the vector from \(x\) to it, \(V_N-\int_Qg=\frac12\int_Qd^2\,\Delta g\,dx+\frac12\int_Qv^{\mathsf T}D^2g\,v\,dx+\int_Qr\,dx\) with \(|r(x)|\le\frac16\|D^3g\|_\infty|v|^3\). For \(A_N=V_N\) and for \(A_N=D_N\), \(|\mathbb{E}A_N-\int_Qg|=O(Ne^{-cN\delta^d})\) in case (a) and \(O(N^{-k})\) for every \(k\) in case (b), and \(\operatorname{Var}A_N=O(N^{-1-4/d})\). So the root mean square error is \(O(N^{-1/2-2/d})\) in every dimension, with constants depending on \(g\), \(d\) and, in case (a), \(\delta\). Case (b) includes \(g=\log|f|\,\Delta\varphi\) for the flat bump \(\varphi\) of the smoothed Riesz identity mapped to \(Q\), when \(f\) is analytic and zero free on a neighbourhood of the closed box.

Earlier editions: Theorem 35.7.3, page 1257 (edition 1).

Source: ch/bsy3.tex, label bsy:thm:wmc.

Computed root mean square errors of the Voronoi, Delaunay and trapezoid rules at independent uniform nodes for flat and non flat integrands in one and two dimensions

Volume II, Table 13.7, page 369 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Computed in floating point, root mean square error at independent uniform nodes on \([-1,1]^d\) at \(N=1000\) and at the largest \(N=1{,}024{,}000\), slopes fitted for \(N\ge4000\), over \(200\) replications in one dimension and \(30\) in two (about \(5\) percent noise in one dimension and about \(13\) percent in two). Bump \(\prod_ib(x_i/0.9)\), \(d=1\), Voronoi: \(1.2\times10^{-6}\) and \(3.5\times10^{-14}\), slope \(-2.51\). \(e^x\), \(d=1\), sorted trapezoid: \(4.8\times10^{-6}\) and \(4.5\times10^{-12}\), slope \(-2.00\). Bump, \(d=2\), Voronoi: \(1.4\times10^{-4}\) and \(5.1\times10^{-9}\), slope \(-1.46\). Bump, \(d=2\), Delaunay: \(1.1\times10^{-4}\) and \(4.2\times10^{-9}\), slope \(-1.42\). \(e^{x+y/2}\), \(d=2\), Delaunay: \(1.0\times10^{-2}\) and \(1.2\times10^{-5}\), slope \(-0.98\). The slopes are consistent with the exponents \(-\frac52\) and \(-\frac32\) of the weighted Monte Carlo theorem for the bump, with \(-2\) of the sorted trapezoid rule lemma for \(e^x\), and with a bias of order \(N^{-2/d}\) for the exponentials, which do not vanish at the boundary. This is a computation and not a proof.

Earlier editions: Table 35.7, page 1259 (edition 1).

Source: ch/bsy3.tex, label bsy:tab:wmc.

Computed errors of plain, Voronoi, Delaunay, jittered, shifted grid and deflated rules for the smoothed Riesz integral of zeta and of the Davenport and Heilbronn function

Volume II, Table 13.8, page 370 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\varphi(x,y)=b(x)b(y)\) with \(b(x)=e^{-1/(1-x^2)}\) on the box \(0.6\le\sigma\le1.4\), \(83.7\le t\le87.7\) mapped to \([-1,1]^2\), and \(I(\varphi)=\iint\log|f|\,\Delta\varphi\). For \(\zeta\) there is no zero in the box and \(I(\varphi)=0\); for the Davenport and Heilbronn function the box contains the zero \(\rho=0.808517182457+85.699348485378\,i\) and \(I(\varphi)=2\pi\varphi(\rho)=0.63165631\). Computed in floating point over \(20\) replicates, the root mean square error at \(N=1024\), \(16{,}384\), \(262{,}144\) nodes is, for \(\zeta\): plain \(1.2\times10^{-1}\), \(3.9\times10^{-2}\), \(9.0\times10^{-3}\); Voronoi \(4.3\times10^{-2}\), \(1.7\times10^{-3}\), \(3.2\times10^{-5}\); Delaunay \(4.5\times10^{-2}\), \(1.5\times10^{-3}\), \(2.4\times10^{-5}\); jittered \(2.0\times10^{-2}\), \(4.4\times10^{-4}\), \(5.6\times10^{-6}\); shifted grid \(3.0\times10^{-3}\), \(1.9\times10^{-6}\), \(2.2\times10^{-14}\); and for the Davenport and Heilbronn function: plain \(2.3\times10^{-1}\), \(6.7\times10^{-2}\), \(1.2\times10^{-2}\); Voronoi \(5.8\times10^{-2}\), \(3.0\times10^{-3}\), \(9.5\times10^{-5}\); Delaunay \(5.3\times10^{-2}\), \(2.5\times10^{-3}\), \(7.4\times10^{-5}\); jittered \(3.3\times10^{-2}\), \(1.0\times10^{-3}\), \(6.1\times10^{-5}\); shifted grid \(1.5\times10^{-2}\), \(6.9\times10^{-4}\), \(4.8\times10^{-5}\); deflated grid with an estimate of the zero off by \(1.4\times10^{-4}\) \(2.7\times10^{-2}\), \(1.5\times10^{-5}\), \(3.5\times10^{-6}\). The fitted slopes for \(N\ge4096\) for \(\zeta\) are \(-0.51\) (plain), \(-1.43\) (Voronoi), \(-1.40\) (Delaunay) and \(-1.54\) (jittered); the relative standard error of an entry is \(16\%\). This is a computation in floating point and not a proof of any rate.

Earlier editions: Table 35.8, page 1260 (edition 1).

Source: ch/bsy3.tex, label bsy:tab:2d.

Approximate deflation: the shifted trapezoid grid error with a zero estimate at distance epsilon has two regimes and no floor

Volume II, Proposition 13.7.5, page 371 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\psi\in C_c^\infty((0,1)^2)\), \(\rho,\hat\rho\in(0,1)^2\) with \(0<\varepsilon=|\hat\rho-\rho|\le\frac12\) and \(r=\psi\,(\log|s-\rho|-\log|s-\hat\rho|)\). The randomly shifted product trapezoid rule with \(N\) nodes is unbiased for \(\int r\), and its root mean square error over the shift is at most \(C\bigl(\varepsilon N^{-1/2}(1+\log^+\frac1{\varepsilon\sqrt N})^{1/2}+\varepsilon\log\frac2\varepsilon\,N^{-1}\bigr)\) if \(\varepsilon\sqrt N\le1\) and \(C\,N^{-1}\) if \(\varepsilon\sqrt N\ge1\), with \(C\) depending only on \(\psi\). So for every fixed \(\varepsilon\) the error tends to zero.

Earlier editions: Proposition 35.7.5, page 1262 (edition 1).

Source: ch/bsy3.tex, label bsy:prop:defl.

The zero count weighted by a bump is a linear functional of the horizontal logarithmic derivative on a half strip

Volume II, Theorem 13.7.6, page 372 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\not\equiv0\) be meromorphic on a connected neighbourhood of \(\{\operatorname{Re}s\ge\sigma_0\}\) and equal, for \(\operatorname{Re}s\) large, to an absolutely convergent Dirichlet series \(1+\sum_{n\ge2}a_nn^{-s}\), such as \(\zeta\) or the Davenport and Heilbronn function. Let \(d_f(u,t)=\pi^{-1}\operatorname{Re}(f'/f)(u+it)\), \(\varphi\in C_c^\infty(\{\operatorname{Re}s>\sigma_0\})\), \(\Phi(u,t)=\int_{-\infty}^u\varphi(\sigma,t)\,d\sigma\) and \(K_\varphi=\partial_u\varphi+\partial_t^2\Phi\). Then \(\sum_{f(\rho)=0}\varphi(\rho)-\sum_{f(p)=\infty}\varphi(p)=-\frac12\int_{-\infty}^\infty\int_{\sigma_0}^\infty d_f(u,t)\,K_\varphi(u,t)\,du\,dt\), with zeros and poles counted with multiplicity and the double integral absolutely convergent. \(K_\varphi\) is smooth, and to the right of the support of \(\varphi\) it equals \(\partial_t^2\Phi(\infty,t)\) and does not depend on \(u\).

Earlier editions: Theorem 35.7.6, page 1262 (edition 1).

Source: ch/bsy3.tex, label bsy:thm:thorincount.

Computed error of the deflated shifted grid for the Davenport and Heilbronn function as a function of the error in the deflating zero

Volume II, Table 13.9, page 372 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. For the Davenport and Heilbronn function on the box of the smoothed Riesz identity, the randomly shifted product trapezoid rule with the zero deflated by an estimate at distance \(\varepsilon\) along \((1+i)/\sqrt2\) has, computed in floating point as the root mean square error over \(10\) shifts at \(N=262{,}144\) nodes, the values \(5.1\times10^{-14}\) (\(\varepsilon=0\)), \(3.1\times10^{-10}\) (\(10^{-8}\)), \(3.1\times10^{-8}\) (\(10^{-6}\)), \(3.0\times10^{-6}\) (\(10^{-4}\)), \(2.1\times10^{-5}\) (\(10^{-3}\)), \(3.0\times10^{-5}\) (\(10^{-2}\)) and \(4.2\times10^{-5}\) without deflation; the relative standard error of an entry is \(22\%\). A synthetic test shows the error falling to \(N=2^{22}\) for every \(\varepsilon\in\{10^{-2},10^{-3},10^{-4}\}\) with slope near \(-0.6\) while \(\varepsilon\sqrt N\le1\) and near \(-1.1\) afterwards. This is a computation and not a proof.

Earlier editions: Table 35.9, page 1263 (edition 1).

Source: ch/bsy3.tex, label bsy:tab:defl.

The explicit formula at a Cauchy smoothed level: the primes compute the Thorin integral against a band limited test function only when no zero lies to the right of the level

Volume II, Remark 13.8.1, page 374 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\ge0\) have a Fourier transform supported in \([-L,L]\), Lipschitz with derivative of bounded variation, \(F_c=F(\cdot-c)\), and \(\frac12<\sigma<1\) not the real part of a zero (otherwise take right limits in \(\sigma\)). The explicit formula of Weil applied to the even part of \(P_{\sigma-1/2}*F_c\), with \(P\) the Cauchy density, has on its prime side only the prime powers \(n\le e^L\), besides the archimedean terms. On its zero side a zero \(\beta+i\gamma\) with \(\beta<\sigma\) contributes \(\int F_c\,P_{\sigma-\beta}(\cdot-\gamma)\), its term in \(\int F_c\,d\mathfrak U_\sigma\), where \(\mathfrak U_\sigma\) is the Thorin measure at level \(\sigma\). So if no zero has \(\beta>\sigma\) the formula gives \(\int F_c\,d\mathfrak U_\sigma\ge0\). A zero with \(\beta>\sigma\) contributes the continuation in \(\beta\) of that term, which exceeds its signed term \(-\int F_c\,P_{\beta-\sigma}(\cdot-\gamma)\) in \(\int F_c\,d\mathfrak U_\sigma\) by \(F_c(\gamma+i(\beta-\sigma))+F_c(\gamma-i(\beta-\sigma))\); these corrections need not be positive. So the primes compute the continued Weil functional, which equals the Thorin integral in general only when no zero lies to the right of \(\sigma\).

Earlier editions: Remark 35.8.1, page 1265 (edition 1).

Source: ch/bsy3.tex, label bsy:rem:weil.

Open problem: an implementable sampler whose variance per unit cost for the defect is below plain Monte Carlo

Volume II, Problem 13.1, page 374 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\frac12\le\theta<1\), \(t_0\in\mathbb{R}\), \(f(\tau)=\log|\zeta(\theta+i\tau)|\), \(P\) the Cauchy law with density \(\frac{1-\theta}{\pi((1-\theta)^2+(t_0-\tau)^2)}\), \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\) and \(D_\theta(t_0)=\mathbb{E}_Pf-\log|h_\theta(1+it_0)|\). Find an implementable sampler for the law \(P_w\propto P\,w(f)\) with \(w\) close to \(|f-\mathbb{E}_Pf|\), or another weight, whose variance per unit of computing cost (pilot, burn in, autocorrelation and evaluations of \(f\) included) is below that of plain Monte Carlo for the defect \(D_\theta(t_0)\), or show that the Metropolis chains of the chapter cannot reach it. For independent draws no weight gains more than the factor \(\mathrm{Var}_Pf/(\mathbb{E}_P|f-\mathbb{E}_Pf|)^2\), which is between \(1.35\) and \(2.46\) for the nine integrands of the book’s table of defects (computed); a Markov chain is not bound by it.

Earlier editions: Problem 35.1, page 1265 (edition 1).

Source: ch/bsy3.tex, label bsy:op:weight.

Open problem: the root mean square rate of Voronoi and Delaunay rules for the negative Thorin mass on a region free of zeros

Volume II, Problem 13.2, page 374 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(m_\sigma(t)=\pi^{-1}\operatorname{Re}\frac{\xi'}{\xi}(\sigma+it)\) be the Thorin density, \(\xi\) the completed zeta function, and \(m_\sigma^-\) its negative part, a nonlinear functional with a kink along the curve \(m_\sigma=0\), integrable but not square integrable near an undeflated zero, so that independent uniform nodes give infinite variance there. Determine the root mean square rate of the Voronoi and Delaunay rules for \(\iint_Rm_\sigma^-\,d\sigma\,dt\) on a bounded region \(R\) free of zeros, or after exact deflation of the zeros in \(R\). The weighted Monte Carlo theorem for smooth integrands does not apply to this nonsmooth functional.

Earlier editions: Problem 35.2, page 1265 (edition 1).

Source: ch/bsy3.tex, label bsy:op:2d.

Open problem: a rate in N for the truncated shifted trapezoid rule on the whole line under RH, and a certified tail correction

Volume II, Problem 13.3, page 375 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. On a half disc the sorted trapezoid rule reaches the mean square rates \(N^{-4}\) and \(N^{-5}\) for an exact local form of the criterion. For the Cauchy average of \(\log|\zeta|\) over the whole line \(\operatorname{Re}s=\theta\): prove, under the Riemann hypothesis, a rate in \(N\) for the shifted trapezoid rule truncated to \(|\tau-t_0|\le T\), with a step that shrinks with \(N\) and with the discretisation and deflation errors bounded; find a correction for the Cauchy tail beyond \(T\) that is finite, efficiently computable and certified. Are the empirical slopes \(-0.60\) and \(-0.66\) of the sorted trapezoid rule for \(\zeta\) a preasymptotic effect of the zeros, or the true rate?

Earlier editions: Problem 35.3, page 1266 (edition 1).

Source: ch/bsy3.tex, label bsy:op:line.

Open problem: extend the rate of Voronoi and Delaunay weights to dependent nodes and find the rate for a box with an undeflated zero

Volume II, Problem 13.4, page 375 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The rate \(N^{-1/2-2/d}\) for the root mean square error of Voronoi and Delaunay weights at independent uniform nodes on a box in dimension \(d\) is proved for smooth integrands flat at the boundary. Extend it to dependent node sets, such as those of the particle filter along the Cauchy flow or of nested sampling, under explicit coverage and mixing hypotheses on the nodes (some hypothesis is needed, since nodes that all coincide give one evaluation); for deterministic quasi uniform meshes the rate can be \(N^{-1}\). For a box containing an undeflated zero, where plain Monte Carlo keeps the rate \(N^{-1/2}\), is the root mean square error of the Voronoi and Delaunay rules at independent uniform nodes of order \(N^{-1}\) up to logarithms, as the observed slopes between \(-1.0\) and \(-1.3\) suggest?

Earlier editions: Problem 35.4, page 1266 (edition 1).

Source: ch/bsy3.tex, label bsy:op:rates.

Open problem: do the moment identities for all heights imply QRH(theta), and a central limit theorem for the logarithm of h under the Cauchy law

Volume II, Problem 13.5, page 375 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Fix \(\theta\in[\frac12,1)\) and let \(Z_{t_0}\) be the boundary value of the branch of \(\log h_\theta\), \(h_\theta(s)=\zeta(s)\frac{s-1}{s-2\theta+1}\), continued horizontally from \(+\infty\), at a height with the Cauchy law of centre \(t_0\) and scale \(1-\theta\). If \(\mathbb{E}[Z_{t_0}^2]=(\mathbb{E}Z_{t_0})^2\) for every \(t_0\), does \(\mathrm{QRH}(\theta)\) follow? By the cut formula for complex moments this is equivalent to the identity \(2\sum_\gamma\int_{C_\gamma}M_\gamma(L_-+\pi iM_\gamma-z_0)K\,ds=D^2\) for every \(t_0\), over the horizontal cuts \(C_\gamma\) from the zeros right of the line to the line, with \(M_\gamma\) the number of zeros on the cut to the right of the point, \(L_-\) the boundary value of the logarithm from below, \(z_0=\log h_\theta(1+it_0)\), \(K\) the kernel \(\frac1{s-z^*}-\frac1{s-z}\) with \(z=1+it_0\), \(z^*=2\theta-1+it_0\), and \(D=\mathbb{E}Z_{t_0}-z_0\). Under the Riemann hypothesis, is \(\operatorname{Var}\log|\zeta(\frac12+i\tau)|=\frac12\log\log t_0+O(1)\) under the Cauchy law of centre \(t_0\) and scale \(\frac12\), for every \(t_0\) or for \(t_0\) outside a set of density zero; and does \((Z_{t_0}-\mathbb{E}Z_{t_0})/(\frac12\log\log t_0)^{1/2}\) converge in law to the standard normal law of \(\mathbb{R}^2\) when \(t_0\) is uniform on \([T,2T]\)?

Earlier editions: Problem 35.5, page 1266 (edition 1).

Source: ch/bsy3.tex, label bsy:op:cmart.

Open problem: a combination of L functions of primitive characters of one modulus, with a functional equation and zeros off the line, whose trace is negative somewhere

Volume II, Problem 13.6, page 375 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is there a linear combination of \(L(s,\chi)\) over primitive characters \(\chi\) of one modulus, with a functional equation and zeros off the line, whose trace \(K(u)=\sum_{\gamma>0}e^{-\gamma|u|}\cos((\beta-\frac12)u)\) over its zeros \(\beta+i\gamma\) is negative for some \(u\)? With imprimitive characters allowed, a real combination of \(L\) functions of characters modulo \(4\), \(4p\), \(4q\) and \(4pq\) gives one. If the lowest zero \(\frac12+i\gamma_1\) of such a function is on the line, the argument of the book’s proof for the Davenport and Heilbronn function gives a positive trace whenever \(\sum e^{-(\gamma-\gamma_1)\pi/(2\delta_{\max})}<1\), the sum over the upper zeros off the line with multiplicity and \(\delta_{\max}\) the largest \(|\beta-\frac12|\); so a negative trace needs the opposite inequality, and if there are finitely many, \(M\), upper zeros off the line, one of them lies within \(\frac{2\delta_{\max}}\pi\log M\) of \(\gamma_1\).

Earlier editions: Problem 35.6, page 1266 (edition 1).

Source: ch/bsy3.tex, label bsy:op:dh.

Volume II, Chapter 14. Weil positivity and Yoshida’s Hermitian forms: certificates and barriers

The heat trace computed from Gamma and the primes agrees with the sum over the first 1700 zeros

Volume II, Result 14.1, page 382 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\rho_j=\beta_j+i\tau_j\) with \(\tau_j>0\) be the nontrivial zeros of the Riemann zeta function \(\zeta\) in the upper half plane, counted with multiplicity (the zeros are closed under \(\rho\mapsto1-\bar\rho\)), let \(\gamma_j=\tau_j+i(\frac12-\beta_j)\), so that \(\rho_j=\frac12+i\gamma_j\) and \(\gamma_j\) is real exactly when \(\rho_j\) is on the critical line, let \(\lambda_j=\gamma_j^2\), and let \(W(t)=\sum_je^{-\lambda_jt}\) for \(t>0\) be the heat trace of the zeros (absolutely convergent because the number of zeros of height at most \(T\) is \(O(T\log T)\)). Let \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\psi=\Gamma'/\Gamma\), \(\Lambda\) the von Mangoldt function, \(\mathcal A(t)=e^{t/4}-\frac12g_t(0)\log\pi+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\) and \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\), so that \(W=\mathcal A-\mathcal P\) for every \(t>0\) by the Guinand and Weil explicit formula. Computed at 110 digits: at each of \(t=0.005,0.02,0.05,0.1,0.2\) the value \(\mathcal A(t)-\mathcal P(t)\) equals \(W(t)\) summed over the first 1700 zeros to \(10^{-70}\) or better, with \(W(0.005)=0.5371919640910408957\), \(W(0.02)=0.018541291731319482254\), \(W(0.05)=4.5878350485735742685\times10^{-5}\), \(W(0.1)=2.1047997958223512845\times10^{-9}\) and \(W(0.2)=4.4301821802236448843\times10^{-18}\). This is a computation, not an interval enclosure.

Earlier editions: Result 26.1, page 952 (edition 1).

Source: ch/wt1.tex, label wt:res:check.

RH holds if and only if the heat trace of the zeros is completely monotone, if and only if the Thorin measure lies on the positive axis

Volume II, Theorem 14.2.1, page 383 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_j=\beta_j+i\tau_j\) with \(\tau_j>0\) be the nontrivial zeros of the Riemann zeta function \(\zeta\) in the upper half plane, counted with multiplicity (the zeros are closed under \(\rho\mapsto1-\bar\rho\)), let \(\gamma_j=\tau_j+i(\frac12-\beta_j)\), so that \(\rho_j=\frac12+i\gamma_j\) and \(\gamma_j\) is real exactly when \(\rho_j\) is on the critical line, let \(\lambda_j=\gamma_j^2\), and let \(W(t)=\sum_je^{-\lambda_jt}\) for \(t>0\) be the heat trace of the zeros (absolutely convergent because the number of zeros of height at most \(T\) is \(O(T\log T)\)). Let \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\psi=\Gamma'/\Gamma\), \(\Lambda\) the von Mangoldt function, \(\mathcal A(t)=e^{t/4}-\frac12g_t(0)\log\pi+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\) and \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\), so that \(W=\mathcal A-\mathcal P\) for every \(t>0\) by the Guinand and Weil explicit formula. Let \(\xi\) be the Riemann xi function, \(U=\sum_j\delta_{\lambda_j}\) and \(T\) the reciprocal Xi variable, the law (when it exists) with Laplace transform \(\xi(\frac12)/\xi(\frac12+\sqrt s)=\prod_j(1+s/\lambda_j)^{-1}\). The following are equivalent: (i) the Riemann Hypothesis (RH); (ii) every \(\lambda_j\) lies in \((0,\infty)\), that is, \(U\) is supported on \((0,\infty)\) and \(T\) is a generalised gamma convolution; (iii) \(W\) is completely monotone on \((0,\infty)\), \((-1)^nW^{(n)}(t)\ge0\) for all \(n\ge0\) and \(t>0\); (iv) \(\mathcal A-\mathcal P\) is completely monotone on \((0,\infty)\). The proof of (iii) implies (i) uses \(S(s)=\sum_j(s+\lambda_j)^{-1}=\int_0^\infty e^{-sx}W(x)\,dx\) for real \(s>\frac14\): if \(W=\int e^{-ux}\mu(du)\) with \(\mu\ge0\) (Bernstein), then \(S(s)=\int(s+u)^{-1}\mu(du)\) is holomorphic off \((-\infty,0]\), while a pole \(-\lambda_j\) outside \((-\infty,0]\) has residue the positive integer multiplicity of \(\lambda_j\), so it cannot be removable; hence every \(\lambda_j\ge0\), and \(\operatorname{Im}\lambda_j=2\tau_j(\frac12-\beta_j)\) gives \(\beta_j=\frac12\).

Earlier editions: Theorem 26.2.1, page 953 (edition 1).

Source: ch/wt1.tex, label wt:thm:main.

The archimedean part of the heat trace is a compensated integral against u times the Levy density of minus half the logarithm of a Gamma(1/4) variable

Volume II, Result 14.3, page 384 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\psi=\Gamma'/\Gamma\), \(\Lambda\) the von Mangoldt function, \(\mathcal A(t)=e^{t/4}-\frac12g_t(0)\log\pi+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\) and \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\), so that \(\mathcal A-\mathcal P\) equals the heat trace of the zeros \(W(t)=\sum_je^{-\lambda_jt}\) with \(\lambda_j=-(\rho_j-\frac12)^2\) and \(\rho_j\) the nontrivial zeros of \(\zeta\) in the upper half plane counted with multiplicity, for every \(t>0\) by the Guinand and Weil explicit formula. Let \(k(u)=e^{-u/2}/(1-e^{-2u})\) for \(u>0\). Then \(\mathcal A(t)=e^{t/4}+\frac12[\psi(\frac14)-\log\pi]\,g_t(0)+\int_0^\infty[g_t(0)-g_t(u)]\,k(u)\,du\), where the compensation \(g_t(0)-g_t(u)=O(u^2)\) is essential because \(k(u)\sim1/2u\) at the origin and \(\int g_tk\) and \(\int g_t(0)k\) diverge separately. Moreover \(k(u)\) is \(u\) times the Levy density \(e^{-u/2}/(u(1-e^{-2u}))\) of \(-\frac12\log G_{1/4}\) for \(G_{1/4}\) with the Gamma\((\frac14)\) law, and more generally \(-\frac12\log G_a\) has Levy density \(e^{-2au}/(u(1-e^{-2u}))\) on \(u>0\), with compensation. The prime part is \(\mathcal P(t)=\int_0^\infty e^{-u/2}g_t(u)\,d\psi_{\mathrm C}(e^u)\) with the atom at \(u=0\) left out, where \(\psi_{\mathrm C}(y)=\sum_{n\le y}\Lambda(n)\).

Earlier editions: Result 26.3, page 954 (edition 1).

Source: ch/wt1.tex, label wt:res:acomp.

The prime part is a positive weighted series of Frechet distribution functions and for small t is governed by the prime 2

Volume II, Result 14.3, page 385 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\Lambda\) the von Mangoldt function and \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\). With \(\Phi_1(y)=e^{-1/y}\) the Frechet distribution function of index one, \(\mathcal P(t)=(4\pi t)^{-1/2}\sum_{n\ge2}\Lambda(n)n^{-1/2}\Phi_1(4t/(\log n)^2)\), a positive weighted series of Frechet distribution functions with scales \((\log n)^2/4\); the weights are not summable (\(\sum_{n\le N}\Lambda(n)n^{-1/2}\sim2\sqrt N\)), so it is not a probability mixture. As \(t\downarrow0\), \(\mathcal P(t)=\frac{\log2}{\sqrt{8\pi t}}\exp[-(\log2)^2/4t]\{1+O(\exp[-((\log3)^2-(\log2)^2)/4t])\}\), by isolating \(n=2\) and factoring out the exponential of the term \(n=3\).

Earlier editions: Result 26.3, page 955 (edition 1).

Source: ch/wt1.tex, label wt:res:frechet.

The spectral density of the archimedean part has exactly one sign change, at the square of the critical point of the Riemann Siegel theta function

Volume II, Lemma 14.4.1, page 386 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\psi=\Gamma'/\Gamma\) and \(w(\lambda)=\big(\operatorname{Re}\psi(\frac14+\frac i2\sqrt\lambda)-\log\pi\big)/(4\pi\sqrt\lambda)\) for \(\lambda>0\). Then \(w\) has exactly one sign change on \((0,\infty)\), from negative to positive, at \(\lambda_*=r_*^2\) with \(r_*=6.28984\) the unique positive root of \(\operatorname{Re}\psi(\frac14+\frac{ir}2)=\log\pi\), and \(w'(\lambda_*)>0\). The reason is that \(r\mapsto\operatorname{Re}\psi(\frac14+\frac{ir}2)\) increases strictly from \(\psi(\frac14)=-4.227\) to \(+\infty\), since \(\operatorname{Im}\psi'(\frac14+\frac{ir}2)=-\sum_{k\ge0}(\frac14+k)r|z+k|^{-4}<0\) for \(r>0\) with \(z=\frac14+\frac{ir}2\).

Earlier editions: Lemma 26.8.1, page 960 (edition 1).

Source: ch/wt1.tex, label wt:lem:w.

Sign structure of the derivatives of the archimedean part: one sign change for odd order, at most two for even order, and a first change near n over lambda star

Volume II, Proposition 14.4.2, page 387 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\psi=\Gamma'/\Gamma\) and \(w(\lambda)=\big(\operatorname{Re}\psi(\frac14+\frac i2\sqrt\lambda)-\log\pi\big)/(4\pi\sqrt\lambda)\) for \(\lambda>0\), so that \(\mathcal A(t)=e^{t/4}+\int_0^\infty e^{-\lambda t}w(\lambda)\,d\lambda\), let \(\lambda_*=r_*^2\) where \(r_*=6.28984\) is the unique positive root of \(\operatorname{Re}\psi(\frac14+\frac{ir}2)=\log\pi\) (the critical point of the Riemann Siegel theta function), and let \(D_n(t)=(-1)^n\mathcal A^{(n)}(t)=(-\frac14)^ne^{t/4}+\int_0^\infty\lambda^ne^{-\lambda t}w(\lambda)\,d\lambda\). For every \(n\ge1\), \(D_n(t)>0\) for small \(t>0\). For odd \(n\), \(D_n\) has exactly one sign change on \((0,\infty)\), at a point \(t_n\), and is negative after it. For even \(n\), \(D_n\) has at most two sign changes and is positive for large \(t\); if it has two, the first is \(t_n\) and the second lies where the atom \(4^{-n}e^{t/4}\) overtakes the negative part of the density. As \(n\to\infty\) a first sign change \(t_n\) exists and \(t_n=n/\lambda_*+O(1)\). The existence of a first sign change for even \(n\) is proved only for \(n\) large; it is computed for \(n=2,4,6\).

Earlier editions: Proposition 26.8.2, page 960 (edition 1).

Source: ch/wt1.tex, label wt:prop:window.

Computed first sign changes of the derivatives of the archimedean part for orders one to six

Volume II, Result 14.4.1, page 387 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\psi=\Gamma'/\Gamma\) and \(w(\lambda)=\big(\operatorname{Re}\psi(\frac14+\frac i2\sqrt\lambda)-\log\pi\big)/(4\pi\sqrt\lambda)\) for \(\lambda>0\), so that \(\mathcal A(t)=e^{t/4}+\int_0^\infty e^{-\lambda t}w(\lambda)\,d\lambda\), let \(\lambda_*=r_*^2\) where \(r_*=6.28984\) is the unique positive root of \(\operatorname{Re}\psi(\frac14+\frac{ir}2)=\log\pi\) (the critical point of the Riemann Siegel theta function), and let \(D_n(t)=(-1)^n\mathcal A^{(n)}(t)=(-\frac14)^ne^{t/4}+\int_0^\infty\lambda^ne^{-\lambda t}w(\lambda)\,d\lambda\). Computed by bisection at 35 and 45 digits (the two runs agree): the first sign change \(t_n\) of \(D_n=(-1)^n\mathcal A^{(n)}\) is \(0.025332\), \(0.051040\), \(0.076135\), \(0.101334\), \(0.126561\), \(0.151805\) for \(n=1,\dots,6\), against \(n/\lambda_*=0.025277\), \(0.050554\), \(0.075830\), \(0.101107\), \(0.126384\), \(0.151661\) with \(\lambda_*=39.5620\), a relative deviation of \(0.22\%,0.96\%,0.40\%,0.22\%,0.14\%,0.095\%\). Since \(t_1<\dots<t_6\), the conditions of all orders up to \(n\) hold together for \(\mathcal A\) only on \((0,t_1]\). For \(n=2,4,6\) a scan on \([0.005,7]\) finds a second sign change at \(1.3612\), \(3.2456\) and \(5.2669\).

Earlier editions: Result 26.4.1, page 957 (edition 1).

Source: ch/wt1.tex, label wt:res:window.

Window law: the first sign change of the derivatives of the archimedean part is n over lambda star plus beta plus delta over n plus O of n to the minus two

Volume II, Proposition 14.4.3, page 388 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\psi=\Gamma'/\Gamma\) and \(w(\lambda)=\big(\operatorname{Re}\psi(\frac14+\frac i2\sqrt\lambda)-\log\pi\big)/(4\pi\sqrt\lambda)\) for \(\lambda>0\), so that \(\mathcal A(t)=e^{t/4}+\int_0^\infty e^{-\lambda t}w(\lambda)\,d\lambda\), let \(\lambda_*=r_*^2\) where \(r_*=6.28984\) is the unique positive root of \(\operatorname{Re}\psi(\frac14+\frac{ir}2)=\log\pi\) (the critical point of the Riemann Siegel theta function), and let \(D_n(t)=(-1)^n\mathcal A^{(n)}(t)=(-\frac14)^ne^{t/4}+\int_0^\infty\lambda^ne^{-\lambda t}w(\lambda)\,d\lambda\). Put \(A_1(c)=c\,w'(c)+\frac12c^2w''(c)\), \(A_2(c)=c^2w''(c)+\frac56c^3w'''(c)+\frac18c^4w^{(4)}(c)\), \(\alpha=-A_1(\lambda_*)/w'(\lambda_*)\) and \(\eta=-[\frac12w''(\lambda_*)\alpha^2+A_1'(\lambda_*)\alpha+A_2(\lambda_*)]/w'(\lambda_*)\). The first sign change \(t_n\) of \(D_n\) satisfies \(t_n=n/\lambda_*+\beta+\delta/n+O(n^{-2})\) with \(\beta=\lambda_*^{-1}\big(1+\lambda_*w''(\lambda_*)/(2w'(\lambda_*))\big)\) and \(\delta=\alpha^2/\lambda_*^3-\eta/\lambda_*^2\). The proof expands \(\mathbb E\,w(cT_n)\), with \(T_n\) of gamma law of shape \(n+1\) and rate \(n\), in powers of \(1/n\) and shows that the atom term is exponentially small near \(\lambda_*\). The numerical value of \(\delta\) is not entered here because no script of the book evaluated it.

Earlier editions: Proposition 26.9.1, page 965 (edition 1).

Source: ch/wt1.tex, label wt:prop:expansion.

Computed short interval variance of the prime sum follows the GUE prediction and not the Poisson level

Volume II, Result 14.6.1, page 391 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\psi_{\mathrm C}(y)=\sum_{n\le y}\Lambda(n)\) and \(V_\delta=\frac1{b-a}\int_a^b[\psi_{\mathrm C}((1+\delta)y)-\psi_{\mathrm C}(y)-\delta y]^2/(\delta y)\,dy\) with \([a,b]=[10^6,10^7]\). The GUE prediction transported to the primes is \(\log(1/\delta)+1-\gamma_E-\log2\pi\), with \(\gamma_E\) Euler’s constant, and a Poisson process of the same intensity such as the Cram'er model gives about \(\log y-1\), averaged over the range \(14.37\). Computed (random points and a sieve to \(2.1\times10^7\), standard errors \(0.003\) to \(0.03\)): for \(\delta=10^{-5},10^{-4},10^{-3},3\times10^{-3},10^{-2},10^{-1}\) the primes give \(10.13,7.71,5.32,4.04,3.22,1.10\) against the GUE values \(10.10,7.80,5.49,4.39,3.19,0.89\), while one realisation of the Cram'er model gives \(14.40,14.53,14.60,14.81,14.76,14.55\). This compares finite numerical results with a conjectural variance prediction (Goldston and Montgomery, under RH); it does not establish GUE statistics for the primes.

Earlier editions: Result 26.8.3, page 963 (edition 1).

Source: ch/wt1.tex, label wt:res:rig.

Thorin covariance of the prime side and its exact zero side form

Volume II, Theorem 14.6.1, page 392 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed zeta function, \(\Lambda\) the von Mangoldt function, \(\psi\) the digamma function and \(H_\alpha(t)=\operatorname{Re}(\xi'/\xi)(\alpha+it)\). For \(\alpha>1\), \(H_\alpha(t)=\operatorname{Re}\big[\frac1{\alpha+it}+\frac1{\alpha-1+it}\big]-\frac12\log\pi+\frac12\operatorname{Re}\psi\big(\frac{\alpha+it}2\big)-\sum_{n\ge2}\Lambda(n)n^{-\alpha}\cos(t\log n)\); let \(h_\alpha\) be the smooth part, so that \(H_\alpha-h_\alpha=-\sum_n\Lambda(n)n^{-\alpha}\cos(t\log n)\). Then \(C_\alpha(u):=\lim_{T\to\infty}\frac1{2T}\int_{-T}^T(H_\alpha-h_\alpha)(t)(H_\alpha-h_\alpha)(t+u)\,dt=\frac12\sum_{n\ge2}\Lambda(n)^2n^{-2\alpha}\cos(u\log n)\), and \(C_\alpha(u)=\frac12\operatorname{Re}\big[\frac1{(2\alpha-1+iu)^2}+\frac1{(2\alpha+iu)^2}-\frac14\psi'(\alpha+\frac{iu}2)-\sum_\rho\frac1{(2\alpha+iu-\rho)^2}-D(2\alpha+iu)\big]\) with \(D(s)=\sum_p(\log p)^2p^{-2s}(1-p^{-s})^{-2}\) over primes \(p\) and \(\rho\) over the nontrivial zeros. This needs only absolute convergence and the Hadamard product. At \(\alpha=1\) the right side of the exact formula continues analytically and is evaluated numerically in the book, but the left side at \(\alpha=1\), a mean value of \(\operatorname{Re}\zeta'/\zeta(1+it)\), is not proved.

Source: ch/wt1.tex, label wt:thm:cov.

Prime by prime window law: omitting the prime powers from q on ends the order one window near log q over twice the first ordinate

Volume II, Result 14.7, page 394 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\gamma_1=14.134725\ldots\) be the ordinate of the first nontrivial zero of \(\zeta\), \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\psi=\Gamma'/\Gamma\), \(\Lambda\) the von Mangoldt function, \(\mathcal A(t)=e^{t/4}-\frac12g_t(0)\log\pi+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\) and \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\), so that \(\mathcal A-\mathcal P\) is the heat trace of the zeros \(W(t)=\sum_je^{-\lambda_jt}\) with \(\lambda_j=-(\rho_j-\frac12)^2\) and \(\rho_j\) the nontrivial zeros of \(\zeta\) in the upper half plane counted with multiplicity, by the Guinand and Weil explicit formula, \(\mathcal P_S\) the prime part restricted to all powers of the primes in a finite set \(S\), \(q\) the smallest prime power not generated by \(S\), and \(t_n(S)\) the first point where \((-1)^n(\mathcal A-\mathcal P_S)^{(n)}\) fails to be positive. A heuristic balance of the exponents \(e^{-(\log q)^2/4t}\) and \(e^{-\gamma_1^2t}\) gives \(t(S)\approx\log q/2\gamma_1\). Computed (scan to \(0.8\)): \(t_1(S)=0.0253,0.0394,0.0575,0.0696,0.0856,0.1010,0.1289,0.1635\) for \(S=\emptyset,\{2\},\{2,3\}\), primes up to \(5,7,13,31,97\), against \(\log q/2\gamma_1=0.0245,0.0389,0.0569,0.0688,0.0848,0.1002,0.1277,0.1633\) for \(q=2,3,5,7,11,17,37,101\). This is an empirical law for this truncation scheme, not a theorem: coefficients, derivative polynomials, the other zeros and the other omitted prime powers contribute corrections for which no uniform bound is given.

Earlier editions: Result 26.8.2, page 961 (edition 1).

Source: ch/wt2.tex, label wt:res:law.

One sided certificates: a first derivative bound with no tail estimate and a second derivative bound with the Rosser and Schoenfeld tail

Volume II, Proposition 14.7.1, page 395 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\psi=\Gamma'/\Gamma\), \(\Lambda\) the von Mangoldt function, \(\mathcal A(t)=e^{t/4}-\frac12g_t(0)\log\pi+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\), \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\) and \(W=\mathcal A-\mathcal P\), which by the Guinand and Weil explicit formula is the heat trace \(W(t)=\sum_je^{-\lambda_jt}\) with \(\lambda_j=-(\rho_j-\frac12)^2\) and \(\rho_j\) the nontrivial zeros of \(\zeta\) in the upper half plane counted with multiplicity; let \(\psi_{\mathrm C}(y)=\sum_{n\le y}\Lambda(n)\), \(\mathcal P_S\) the prime part over the prime powers generated by a finite set \(S\) of primes, and \(q\) the smallest omitted prime power, so that \(W=(\mathcal A-\mathcal P_S)-\mathcal P_{\mathrm{rest}}\) with \(\mathcal P_{\mathrm{rest}}\) the sum over the omitted prime powers, all at least \(q\). (1) For \(t<(\log q)^2/2\), \(-W'(t)\ge-(\mathcal A-\mathcal P_S)'(t)\); hence if \(\mathcal A-\mathcal P_S\) is decreasing on \((0,X]\) with \(X<(\log q)^2/2\), so is \(W\), with no tail estimate. (2) Let \(S\) be all prime powers up to \(N_0\), so that \(\mathcal P_S=\mathcal P_{\le N_0}=\sum_{2\le n\le N_0}\Lambda(n)n^{-1/2}g_t(\log n)\), and let \(f(y)=y^{-1/2}\partial_t^2g_t(\log y)\) be nonnegative and decreasing on \((N_0,\infty)\). Then, by the Rosser and Schoenfeld bound \(\psi_{\mathrm C}(y)<1.03883y\), \(W''(t)\ge(\mathcal A-\mathcal P_{\le N_0})''(t)-1.03883\big(N_0f(N_0)+\int_{\log N_0}^\infty e^{u/2}\partial_t^2g_t(u)\,du\big)\), and the hypotheses on \(f\) hold for every \(0<t\le X\) as soon as \(X\le(\log N_0)^2/[2(5+\sqrt{10})]\).

Earlier editions: Proposition 26.9.2, page 966 (edition 1).

Source: ch/wt2.tex, label wt:prop:cert.

Windows for the positivity of the second derivative of the heat trace suggested by the certificate bound (not certified)

Volume II, Result 14.7, page 396 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(W''\) be the second derivative of the heat trace of the zeros and let the lower bound for \(W''(t)\) be \((\mathcal A-\mathcal P_{\le N_0})''(t)-1.03883\big(N_0f(N_0)+\int_{\log N_0}^\infty e^{u/2}\partial_t^2g_t(u)\,du\big)\), where \(W=\mathcal A-\mathcal P\) is the explicit formula for the heat trace with archimedean part \(\mathcal A\), \(\mathcal P_{\le N_0}(t)=\sum_{2\le n\le N_0}\Lambda(n)n^{-1/2}g_t(\log n)\) is the prime part up to \(N_0\), \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(f(y)=y^{-1/2}\partial_t^2g_t(\log y)\), and \(1.03883\) is the Rosser and Schoenfeld constant in \(\sum_{n\le y}\Lambda(n)<1.03883y\). Computed at 30 digits on the geometric grid \(t=0.01\cdot1.08^k\), the bound stays positive up to \(t=0.0740\) for \(N_0=10\), \(0.1479\) for \(N_0=100\) and \(0.2346\) for \(N_0=1000\), which falls short of the law \(\log q/2\gamma_1\), with \(\gamma_1=14.134725\ldots\) the ordinate of the first nontrivial zero of \(\zeta\) and \(q\) the first omitted prime power (\(0.0848,0.1633,0.2447\) for \(q=11,101,1009\)) by \(12.8\%\), \(9.4\%\) and \(4.1\%\). These are not certified windows: floating point positivity at grid points proves neither the signs of the computed quantities nor positivity between the points, and the range \(0<t<0.01\) is not covered. The first failing grid points, where the bound is negative, are \(0.0799\), \(0.1597\) and \(0.2534\).

Earlier editions: Result 26.9.2, page 966 (edition 1).

Source: ch/wt2.tex, label wt:res:cert.

The heat trace is the sum of an explicit archimedean quantity and a Gaussian derivative smoothing of the normalised prime error

Volume II, Proposition 14.8.1, page 397 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g_t(u)=(4\pi t)^{-1/2}e^{-u^2/4t}\), \(\psi=\Gamma'/\Gamma\), \(\Lambda\) the von Mangoldt function, \(\mathcal A(t)=e^{t/4}-\frac12g_t(0)\log\pi+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\), \(\mathcal P(t)=\sum_{n\ge2}\Lambda(n)n^{-1/2}g_t(\log n)\) and \(W=\mathcal A-\mathcal P\), which by the Guinand and Weil explicit formula is the heat trace \(W(t)=\sum_je^{-\lambda_jt}\) with \(\lambda_j=-(\rho_j-\frac12)^2\) and \(\rho_j\) the nontrivial zeros of \(\zeta\) in the upper half plane counted with multiplicity; let \(\psi_{\mathrm C}(y)=\sum_{n\le y}\Lambda(n)\) and \(D(u)=(\psi_{\mathrm C}(e^u)-e^u)e^{-u/2}\), so that \(D(0)=-1\). For every \(t>0\), without RH, \(W(t)=\Sigma_\infty(t)+\int_0^\infty D(u)k_t(u)\,du\) with \(k_t=\partial_ug_t-\frac12g_t\) and \(\Sigma_\infty(t)=\int_0^\infty g_t(u)e^{-u/2}du-g_t(0)(1+\frac12\log\pi)+\frac1{4\pi}\int_{\mathbb R}e^{-tr^2}\operatorname{Re}\psi(\frac14+\frac{ir}2)\,dr\). The proof writes \(d\psi_{\mathrm C}(e^u)=e^u\,du+d(De^{u/2})\) in the prime part and integrates by parts.

Earlier editions: Proposition 26.9.3, page 967 (edition 1).

Source: ch/wt3.tex, label wt:prop:smooth.

Energy form of RH: the normalised prime error has finite discounted energy if and only if RH holds

Volume II, Proposition 14.8.2, page 397 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\psi_{\mathrm C}(y)=\sum_{n\le y}\Lambda(n)\) and \(D(u)=(\psi_{\mathrm C}(e^u)-e^u)e^{-u/2}\), and let \(E_\epsilon=\int_0^\infty D(u)^2e^{-2\epsilon u}\,du\). RH holds if and only if \(E_\epsilon<\infty\) for every \(\epsilon>0\). Under RH, with \(m_\gamma\) the multiplicity of the zero \(\frac12+i\gamma\) of \(\zeta\) and the sum over the distinct positive ordinates \(\gamma\) of the nontrivial zeros, \(\lim_{U\to\infty}U^{-1}\int_0^UD(u)^2du=2\sum_{\gamma>0}m_\gamma^2/(\frac14+\gamma^2)\). This equals \(\sum_\rho|\rho|^{-2}\), the sum over the nontrivial zeros \(\rho\) of \(\zeta\), \(=2+\gamma_E-\log4\pi=0.0461914179\) if all zeros are simple, with \(\gamma_E\) Euler’s constant, and \(2+\gamma_E-\log4\pi+2\sum_{\gamma>0}m_\gamma(m_\gamma-1)/(\frac14+\gamma^2)\) in general. The equivalence is proved from \(\int_0^\infty D(u)e^{-zu}du=-\zeta'(s)/(s\zeta(s))-1/(s-1)\) with \(s=\frac12+z\) and von Koch’s bound; the formula for the limit follows the mean square theorem of Cram'er, and the multiplicity sensitive form is quoted from the literature.

Earlier editions: Proposition 26.9.4, page 968 (edition 1).

Source: ch/wt3.tex, label wt:prop:energy.

Computed mean square of the normalised prime error on four windows of the log scale

Volume II, Result 14.8.2, page 398 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(D(u)=(\psi_{\mathrm C}(e^u)-e^u)e^{-u/2}\) with \(\psi_{\mathrm C}(y)=\sum_{n\le y}\Lambda(n)\). Computed from the primes up to \(2\times10^7\) by integrating \(D^2\) exactly on each interval between integers: the mean of \(D^2\) over \(u\in[5,10]\), \([8,12]\), \([10,14]\), \([12,16.8]\) is \(0.0509\), \(0.0442\), \(0.0465\), \(0.0468\), against \(2+\gamma_E-\log4\pi=0.046191\) (\(\gamma_E\) Euler’s constant), the limit of the mean that RH with simple zeros gives, where it equals \(\sum_\rho|\rho|^{-2}\) over the nontrivial zeros \(\rho\). Finite windows can be consistent with this value without proving the limit or the simplicity of the zeros.

Earlier editions: Result 26.9.4, page 968 (edition 1).

Source: ch/wt3.tex, label wt:res:energy.

Hankel form of RH: all zeros are on the line if and only if two Hankel families of normalised cumulants are positive semidefinite

Volume II, Proposition 14.8.3, page 400 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_j=\beta_j+i\tau_j\) with \(\tau_j>0\) be the nontrivial zeros of the Riemann zeta function \(\zeta\) in the upper half plane, counted with multiplicity (the zeros are closed under \(\rho\mapsto1-\bar\rho\)), let \(\gamma_j=\tau_j+i(\frac12-\beta_j)\), so that \(\rho_j=\frac12+i\gamma_j\) and \(\gamma_j\) is real exactly when \(\rho_j\) is on the critical line, let \(\lambda_j=\gamma_j^2\), and let \(W(t)=\sum_je^{-\lambda_jt}\) for \(t>0\) be the heat trace of the zeros (absolutely convergent because the number of zeros of height at most \(T\) is \(O(T\log T)\)). Let \(s_j=\sum_k\lambda_k^{-j}\) and \(m_j=s_{j+1}\) for \(j\ge0\) (the normalised cumulants \(\kappa_{j+1}/j!\) of the reciprocal Xi law, whose first two values can be read off the primes). RH holds if and only if the Hankel matrices \(H_d=(m_{i+j})_{0\le i,j\le d}\) and \(H_d^{(1)}=(m_{i+j+1})_{0\le i,j\le d}\) are positive semidefinite for every \(d\ge0\). The converse uses Stieltjes’ moment theorem to get a positive measure with these moments, supported on \([0,R]\) with \(R=\sup|\lambda_k|^{-1}\), and then analytic continuation of \(\sum_jm_jz^j=\sum_k1/(\lambda_k-z)\), whose poles have residues minus the positive integer multiplicities, to force every \(\lambda_k>0\).

Earlier editions: Proposition 26.9.5, page 970 (edition 1).

Source: ch/wt3.tex, label wt:prop:hankel.

One heat time suffices: RH holds if and only if the sine squared functional is nonnegative for every shift at a single heat time

Volume II, Theorem 14.9.1, page 401 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_j=\beta_j+i\tau_j\) with \(\tau_j>0\) be the nontrivial zeros of the Riemann zeta function \(\zeta\) in the upper half plane, counted with multiplicity (the zeros are closed under \(\rho\mapsto1-\bar\rho\)), let \(\gamma_j=\tau_j+i(\frac12-\beta_j)\), so that \(\rho_j=\frac12+i\gamma_j\) and \(\gamma_j\) is real exactly when \(\rho_j\) is on the critical line, let \(\lambda_j=\gamma_j^2\), and let \(W(t)=\sum_je^{-\lambda_jt}\) for \(t>0\) be the heat trace of the zeros (absolutely convergent because the number of zeros of height at most \(T\) is \(O(T\log T)\)). For \(t>0\) and \(a\ge0\) let \(f_t(a)=\sum_je^{-\lambda_jt}\cos(a\gamma_j)\) and \(V(t,a)=W(t)-f_t(a)=\sum_j2\sin^2(a\gamma_j/2)e^{-\lambda_jt}\), the Weil functional of the even part of the test function \(FF^*\) with \(F(z)=(1-e^{iaz})e^{-tz^2/2}/\sqrt2\). Fix \(t>0\). Then the Riemann Hypothesis holds if and only if \(V(t,a)\ge0\) for every \(a\ge0\). Under RH every term is nonnegative. Conversely, a zero off the line with \(\operatorname{Im}\gamma_j=b>0\) gives a genuine pole at \(s=-i\gamma_j\) of the Laplace transform \(\int_0^\infty e^{-sa}V(t,a)\,da=W(t)/s-\sum_je^{-\gamma_j^2t}s/(s^2+\gamma_j^2)\), so its abscissa of convergence is at least \(b>0\), and Landau’s theorem for the nonnegative function \(V(t,\cdot)\) forces a real singularity there, which no pole of the right side can supply: the poles \(\pm i\gamma_j\) are not real because \(\operatorname{Re}\gamma_j>0\) (the function \(\xi\) has no real zeros), and the possible pole at \(s=0\) lies left of \(\sigma_c\ge b>0\).

Earlier editions: Theorem 26.10.1, page 971 (edition 1).

Source: ch/wt3.tex, label wt:thm:sin2.

Ladder rungs are diagonal values of Weil’s form on Hermite Gaussians

Volume II, Proposition 14.11.1, page 406 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho\) run over the nontrivial zeros of the Riemann zeta function, with multiplicity, put \(\gamma_\rho=-i(\rho-\frac12)\) (so \(\gamma_\rho=\gamma-i\delta\) for \(\rho=\frac12+\delta+i\gamma\)), \(\widehat F(t)=\int_{\mathbb{R}}F(y)e^{ity}\,dy\), and let \(\mathcal{Y}(F)=\sum_\rho\widehat F(\gamma_\rho)\) be the functional of Weil’s explicit formula. Let \(\langle\varphi_1,\varphi_2\rangle=\mathcal{Y}(\varphi_1*\widetilde{\varphi_2})\) be the Hermitian form of Weil and Yoshida, with \(\widetilde\varphi(y)=\overline{\varphi(-y)}\). Let \(W(x)=\sum_{\gamma>0}e^{(\rho-1/2)^2x}\) be the heat trace of the zeros (the sum over the zeros with positive ordinate \(\gamma\), one from each pair \(\{\rho,1-\rho\}\)) and \(T_k=(-1)^kW^{(k)}\) the rung of order \(k\) of the ladder. Let \(h_x(y)=(4\pi x)^{-1/2}e^{-y^2/(4x)}\), so that \(\widehat{h_x}(t)=e^{-t^2x}\), and for \(k\ge0\) and \(x>0\) let \(g_{k,x}=(i\,d/dy)^kh_{x/2}\), the Hermite Gaussian vector with \(\widehat{g_{k,x}}(t)=t^ke^{-t^2x/2}\). Then, with no hypothesis on the zeros, \(\langle g_{k,x},g_{k,x}\rangle=\sum_\rho\gamma_\rho^{2k}e^{-\gamma_\rho^2x}=2\,T_k(x)\) and \(W(x)=\frac12\langle h_{x/2},h_{x/2}\rangle\).

Earlier editions: Proposition 17.2.1, page 681 (edition 1).

Source: ch/yo1.tex, label yo:prop:diag.

If RH fails every large rung of the ladder fails: RH is positivity of the odd rungs

Volume II, Theorem 14.12.1, page 406 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho=\frac12+\delta+i\gamma\) run over the nontrivial zeros of the Riemann zeta function, put \(\gamma_\rho=-i(\rho-\frac12)\), let \(W(x)=\sum_{\gamma>0}e^{(\rho-1/2)^2x}\) be the heat trace of the zeros (the sum over the zeros with positive ordinate, with multiplicity) and \(T_k=(-1)^kW^{(k)}\) the rung of order \(k\ge0\) of the ladder. Let \(\langle\varphi_1,\varphi_2\rangle=\mathcal{Y}(\varphi_1*\widetilde{\varphi_2})\) be Weil’s Hermitian form, with \(\mathcal{Y}(F)=\sum_\rho\widehat F(\gamma_\rho)\), \(\widehat F(t)=\int F(y)e^{ity}\,dy\) and \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), and let \(g_{k,x}\) be the Hermite Gaussian vector with Fourier transform \(t^ke^{-t^2x/2}\), so that \(\langle g_{k,x},g_{k,x}\rangle=2T_k(x)\). (i) If \(T_k\ge0\) on \((0,\infty)\) then \(T_j\ge0\) on \((0,\infty)\) for every \(j\le k\), so the set of orders \(k\) with \(T_k\ge0\) on \((0,\infty)\) is an initial segment of the nonnegative integers, and it is all of them if and only if the Riemann Hypothesis (RH) holds. (ii) If RH fails there is \(K_0\) such that for every \(k\ge K_0\) there is \(x>0\) with \(T_k(x)<0\). Consequently, for any infinite set \(\mathcal{K}\) of orders, RH holds if and only if \(T_k\ge0\) on \((0,\infty)\) for every \(k\in\mathcal{K}\). In particular RH is equivalent to \(\langle g_{k,x},g_{k,x}\rangle\ge0\) for all odd \(k\) and all \(x>0\), a positivity statement on odd Hermite Gaussians only.

Earlier editions: Theorem 17.3.1, page 682 (edition 1).

Source: ch/yo1.tex, label yo:thm:rungs.

Where the rungs fail: saddle point location for an off line zero

Volume II, Proposition 14.12.2, page 407 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho=\frac12+\delta+i\gamma\) with \(\gamma>0\) run over the nontrivial zeros of the Riemann zeta function with positive ordinate (one from each pair \(\{\rho,1-\rho\}\), with multiplicity), put \(w_\rho=\gamma_\rho^2=(\gamma-i\delta)^2=r_\rho e^{i\theta_\rho}\), so that \(\theta_\rho=-2\arctan(\delta/\gamma)\) and \(\sec\theta_\rho=(\gamma^2+\delta^2)/(\gamma^2-\delta^2)\), and let \(T_k(x)=\sum_{\gamma>0}w_\rho^ke^{-w_\rho x}\) be the rung of order \(k\) of the ladder (the rung \(T_k\) is \((-1)^k\) times the derivative of order \(k\) of the heat trace \(\sum_{\gamma>0}e^{(\rho-1/2)^2x}\)). Suppose the Riemann Hypothesis fails. Then \(S=\max_\rho\log\sec\theta_\rho>0\) is attained, by finitely many zeros. Let \(w_*=r_*e^{i\theta_*}\) be one maximiser, of multiplicity \(m\), belonging to the zero \(\frac12+\delta_*+i\gamma_*\), and put \(y_*=r_*\cos\theta_*=\gamma_*^2-\delta_*^2\). There are \(\eta>0\) and \(K_1\) and, for every \(k\ge K_1\), a point \(y_k\) with \(|y_k-y_*|=O(1/k)\) such that \(e^{-k(\log y_k-1)}T_k(k/y_k)=-2m\,e^{kS}\bigl(1+O(1/k)\bigr)+O\bigl(e^{k(S-\eta)}\bigr)<0\), so that \(T_k(k/y_k)<0\). The constants \(\eta\) and \(K_1\) depend on the zeros and are not explicit.

Earlier editions: Proposition 17.3.2, page 683 (edition 1).

Source: ch/yo1.tex, label yo:prop:saddle.

The tent coefficient of Yoshida’s matrix: zero side, prime side and residue series

Volume II, Proposition 14.15.1, page 411 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho\) run over the nontrivial zeros of the Riemann zeta function with multiplicity, \(\gamma_\rho=-i(\rho-\frac12)\) (complex for an off line zero), \(\widehat F(t)=\int F(y)e^{ity}\,dy\), \(\mathcal{Y}(F)=\sum_\rho\widehat F(\gamma_\rho)\) the functional of Weil’s explicit formula, \(\Delta_x(y)=(x-|y|)_+\) the tent of height \(x>0\) on \([-x,x]\), and \(D(x)=\mathcal{Y}(\Delta_x)\) the tent coefficient (it equals \(x\langle\chi_0,\chi_0\rangle\) for Weil’s form \(\langle\varphi_1,\varphi_2\rangle=\mathcal{Y}(\varphi_1*\widetilde{\varphi_2})\), with \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), and \(\chi_0=(2a)^{-1/2}1_{[-a,a]}\) with \(a=x/2\)). Let \(k_x(w)=e^{-wx}+wx-1\), let \(\Lambda\) be the von Mangoldt function, \(\gamma_E\) Euler’s constant and \(\psi=\Gamma'/\Gamma\). (a) For complex \(w\) and real \(t\), \(\int_{\mathbb{R}}\Delta_x(y)e^{wy}\,dy=\frac{k_x(w)+k_x(-w)}{w^2}=\frac{4\sinh^2(wx/2)}{w^2}\) and \(\widehat{\Delta_x}(t)=\frac{4\sin^2(xt/2)}{t^2}\ge0\). (b) Unconditionally \(D(x)=\sum_\rho4\sin^2(\gamma_\rho x/2)/\gamma_\rho^2\), the sum over all nontrivial zeros. Under RH, with the Thorin measure \(U=\sum_{\gamma>0}\delta_{\gamma^2}\) (one atom at \(\gamma^2\) for each ordinate \(\gamma>0\) of a zero \(\frac12+i\gamma\), with multiplicity), \(D(x)=8\int\frac{\sin^2(x\sqrt t/2)}{t}\,U(dt)=\frac{8\pi}{x}\int\frac{x\sin^2(x\sqrt t/2)}{\pi\sqrt t}\,\frac{U(dt)}{\sqrt t}\). (c) Unconditionally \(D(x)=32\sinh^2\frac x4-x\log\pi-2\sum_{n\le e^x}\frac{\Lambda(n)}{\sqrt n}\log\frac{e^x}{n}+V_1(\Delta_x)\), where \(V_1(\Delta_x)=\frac1{2\pi}\int\widehat{\Delta_x}(t)\operatorname{Re}\psi(\frac14+\frac{it}2)\,dt\) is the archimedean term of the explicit formula and equals \(x-x\log x-(\gamma_E+\log2)x-2\int_0^x\bigl(\frac{e^{y/2}}{2\sinh y}-\frac1{2y}\bigr)(x-y)\,dy\); the prime term is a Riesz mean of order one of \(\Lambda(n)n^{-1/2}\). (d) \(V_1(\Delta_x)=x\,\psi(\frac14)+\frac12\psi'(\frac14)-2\sum_{k\ge0}\frac{e^{-(2k+1/2)x}}{(2k+\frac12)^2}\).

Earlier editions: Proposition 17.6.1, page 687 (edition 1).

Source: ch/yo2.tex, label yo:prop:tent.

Gaussian field with variogram Psi built from the zeta ordinates

Volume II, Proposition 14.15.3, page 414 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis, for part (b) only; parts (a) and (c) are unconditional.

Statement. Let \(\xi\) be the completed zeta function and \(\Psi(t)=\sum_\rho(1-\cos\gamma_\rho t)/\gamma_\rho^2\) over all nontrivial zeros \(\rho=\beta+i\gamma_\rho\) of \(\zeta\) (the variogram function; \(\Psi=D/2\) for the book’s function \(D\)). (a) Unconditionally, for \(\operatorname{Re}z>\frac12\), \(\int_0^\infty e^{-zt}\Psi(t)\,dt=z^{-2}\,\frac{\xi'}{\xi}(\frac12+z)\). (b) Assume the Riemann Hypothesis. Let \(a_\gamma,b_\gamma\) be independent standard normals, one pair for each zero with \(\gamma>0\), counted with multiplicity, and \(X_t=\sum_{\gamma>0}(a_\gamma\cos\gamma t+b_\gamma\sin\gamma t)/\gamma\). Then the series converges almost surely, uniformly on compact sets, to a stationary centred Gaussian process with continuous paths and \(\mathbb{E}(X_t-X_s)^2=\Psi(|t-s|)\). If moreover the zeros are simple, then almost surely \(\lim_{T\to\infty}\frac1T\int_0^TX_t^2\,dt=\sum_{\gamma>0}E_\gamma/\gamma^2\) with \(E_\gamma=(a_\gamma^2+b_\gamma^2)/2\) independent standard exponentials, a variable with the law of the book’s clock \(\sum_{\gamma>0}E_\gamma/\gamma^2\). (c) Unconditionally, for \(\ell>0\) the following are equivalent: a Gaussian process \((Y_t)_{0\le t\le\ell}\) with \(\mathbb{E}(Y_t-Y_s)^2=\Psi(|t-s|)\) exists; \(G(u,v)=\Psi(u)+\Psi(v)-\Psi(|u-v|)\) is positive semidefinite on \([0,\ell]\); and \(Q(f)=\sum_\rho\hat f(\gamma_\rho)\hat f(-\gamma_\rho)\ge0\) for all real \(f\in C_c^\infty(0,\ell)\).

Source: ch/yo2.tex, label yo:prop:field.

Yoshida’s matrix is a Löwner matrix, and RH is the Pick property of one function

Volume II, Theorem 14.16.1, page 418 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho\) run over the nontrivial zeros of the Riemann zeta function with multiplicity, \(\gamma_\rho=-i(\rho-\frac12)\), \(\widehat F(t)=\int F(y)e^{ity}\,dy\), \(\mathcal{Y}(F)=\sum_\rho\widehat F(\gamma_\rho)\) the functional of Weil’s explicit formula, and \(\langle\varphi_1,\varphi_2\rangle=\mathcal{Y}(\varphi_1*\widetilde{\varphi_2})\) with \(\widetilde\varphi(y)=\overline{\varphi(-y)}\). For \(a>0\) let \(\chi_n(y)=(2a)^{-1/2}e^{\pi iny/a}1_{[-a,a]}(y)\), \(n\in\mathbb{Z}\) (an orthonormal basis of \(L^2[-a,a]\)), and \(\nu_n=\pi n/a\). The Löwner matrix of a function \(g\) on the nodes \(\nu_n\) has the entries \((g(\nu_n)-g(\nu_m))/(\nu_n-\nu_m)\) off the diagonal and \(g'(\nu_n)\) on it, and a Pick function is a function holomorphic on the upper half plane with nonnegative imaginary part there. (i) Unconditionally, \(\langle\chi_n,\chi_m\rangle=(-1)^{n+m}\frac2a\sum_\rho\frac{\sin^2(a\gamma_\rho)}{(\gamma_\rho+\nu_n)(\gamma_\rho+\nu_m)}\), an absolutely convergent series. (ii) Let \(g_a(\nu)=\sum_\rho\frac{\sin^2(a\gamma_\rho)}{\gamma_\rho-\nu}\), the sum over all zeros taken in symmetric pairs; unconditionally it is an odd meromorphic function of \(\nu\). Then for \(n\ne m\), \(\langle\chi_n,\chi_m\rangle=(-1)^{n+m}\frac2a\,\frac{g_a(\nu_n)-g_a(\nu_m)}{\nu_n-\nu_m}\), and \(\langle\chi_n,\chi_n\rangle=\frac2a\,g_a'(\nu_n)\), so that the matrix \([\langle\chi_n,\chi_m\rangle]\) is, up to the sign change \(\operatorname{diag}((-1)^n)\), \(\frac2a\) times the Löwner matrix of \(g_a\) on the lattice \(\frac\pi a\mathbb{Z}\). (iii) Under RH, with the Thorin measure \(U=\sum_{\gamma>0}\delta_{\gamma^2}\) (one atom for each ordinate \(\gamma>0\) of a zero \(\frac12+i\gamma\), with multiplicity), the weighted measure \(U_a^\sharp(dt)=2\sin^2(a\sqrt t)\,U(dt)\ge0\) and its Stieltjes transform \(\mathcal{S}_a(s)=\int U_a^\sharp(dt)/(t+s)\), one has \(g_a(\nu)=\nu\,\mathcal{S}_a(-\nu^2)\) and \(\langle\chi_n,\chi_m\rangle=(-1)^{n+m}\frac2a\int\frac{2\sin^2(a\sqrt t)\,(t+\nu_n\nu_m)}{(t-\nu_n^2)(t-\nu_m^2)}\,U(dt)\), and \(g_a\) is an odd Pick function. By the elementary half of Löwner’s theorem every finite Yoshida matrix is then positive semidefinite, for every \(a\). (iv) For each fixed \(a>0\), RH holds if and only if \(g_a\) is a Pick function.

Earlier editions: Theorem 17.7.1, page 692 (edition 1).

Source: ch/yo2.tex, label yo:thm:lowner.

The damped Yoshida form is a Fourier multiplier with symbol Re xi’/xi over pi

Volume II, Proposition 14.17.1, page 420 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), let \(\sigma_{\max}\) be the supremum of the real parts of the nontrivial zeros \(\rho\) of \(\zeta\), and for \(\alpha\ge\frac12\) put \(\varepsilon=\alpha-\frac12\) and \(v_\alpha(\tau)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\tau)\). Let \(\widehat F(t)=\int F(y)e^{ity}\,dy\), \(\gamma_\rho=-i(\rho-\frac12)\), \(\mathcal{Y}(F)=\sum_\rho\widehat F(\gamma_\rho)\) the functional of Weil’s explicit formula (for continuous, piecewise \(C^1\) functions \(F\) of compact support), \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), and \(\langle\varphi_1,\varphi_2\rangle_\alpha=\mathcal{Y}(e^{-\varepsilon|y|}\,\varphi_1*\widetilde{\varphi_2})\) the damped Yoshida form. For \(a>0\) let \(K(a)\) be the space of the functions \(\sum_{n\in\mathbb{Z}}c_n\chi_n\) with rapidly decreasing coefficients \(c_n\) and \(\chi_n(y)=(2a)^{-1/2}e^{\pi iny/a}1_{[-a,a]}(y)\), that is, the restrictions to \([-a,a]\) of smooth \(2a\) periodic functions, set to zero outside \([-a,a]\). Let \(\alpha\ge\sigma_{\max}\) and suppose that no zero has real part \(\alpha\). Then \(\rho_\varepsilon=\pi^{-1}v_\alpha\) is real analytic and positive, \((1+\tau^2)^{-1}\rho_\varepsilon\) is integrable over \(\mathbb{R}\), and for every \(a>0\) and every \(\varphi\in K(a)\), \(\langle\varphi,\varphi\rangle_\alpha=\int_{\mathbb{R}}|\widehat\varphi(\tau)|^2\rho_\varepsilon(\tau)\,d\tau=\frac1\pi\int_{\mathbb{R}}|\widehat\varphi(\tau)|^2v_\alpha(\tau)\,d\tau\). The identity holds as well for every bounded piecewise \(C^1\) function \(\varphi\) supported in \([-a,a]\).

Earlier editions: Proposition 17.8.1, page 694 (edition 1).

Source: ch/yo3.tex, label yo:prop:mult.

Positivity of the damped Yoshida forms: basepoints beyond the zeros, and RH

Volume II, Theorem 14.17.3, page 421 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho\) run over the nontrivial zeros of the Riemann zeta function with multiplicity, \(\sigma_{\max}=\sup_\rho\operatorname{Re}\rho\in[\frac12,1]\), \(\gamma_\rho=-i(\rho-\frac12)\), \(\widehat F(t)=\int F(y)e^{ity}\,dy\), \(\mathcal{Y}(F)=\sum_\rho\widehat F(\gamma_\rho)\) the functional of Weil’s explicit formula, \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), and for \(\alpha\ge\frac12\) and \(\varepsilon=\alpha-\frac12\) let \(\langle\varphi_1,\varphi_2\rangle_\alpha=\mathcal{Y}(e^{-\varepsilon|y|}\,\varphi_1*\widetilde{\varphi_2})\) be the damped Yoshida form (on the prime side the damping replaces \(\Lambda(n)n^{-1/2}\) by \(\Lambda(n)n^{-\alpha}\); at \(\alpha=\frac12\) it is the form of Weil and Yoshida). For \(a>0\) let \(C(a)\) be the space of the smooth functions supported in \([-a,a]\). (i) For \(\alpha\ge\sigma_{\max}\) the damped form is positive semidefinite on \(C_c^\infty(\mathbb{R})\), that is, \(\langle\varphi,\varphi\rangle_\alpha\ge0\) for all \(\varphi\in C_c^\infty(\mathbb{R})\); unconditionally this holds for every \(\alpha\ge1\). (ii) The Riemann Hypothesis holds if and only if the damped form is positive semidefinite for every \(\alpha>\frac12\). (iii) If the damped form at \(\alpha\) is positive semidefinite on \(C(a)\), so is the form at every \(\alpha'>\alpha\); thus \(a_0(\alpha)=\sup\{a:\langle\cdot,\cdot\rangle_\alpha\ge0\text{ on }C(a)\}\) is nondecreasing in \(\alpha\).

Earlier editions: Theorem 17.8.3, page 695 (edition 1).

Source: ch/yo3.tex, label yo:thm:damped.

Sharp coercivity of the damped Yoshida forms on every window

Volume II, Theorem 14.17.5, page 422 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(\sigma_{\max}\) the supremum of the real parts of the nontrivial zeros, \(v_\alpha(\tau)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\tau)\), \(b_\alpha=v_\alpha(0)=\xi'(\alpha)/\xi(\alpha)\) and \(m_\alpha=\inf_{\tau\in\mathbb{R}}v_\alpha(\tau)\). Let \(\langle\varphi,\varphi\rangle_\alpha\) be the damped Yoshida form \(\mathcal{Y}(e^{-(\alpha-1/2)|y|}\,\varphi*\widetilde\varphi)\), with \(\mathcal{Y}\) the functional of Weil’s explicit formula, \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), \(K(a)\) the space of the restrictions to \([-a,a]\) of smooth \(2a\) periodic complex valued functions (set to zero outside \([-a,a]\)), and \(\lambda_{\min}(\alpha,a)=\inf\{\langle\varphi,\varphi\rangle_\alpha/\|\varphi\|^2:\ 0\ne\varphi\in K(a)\}\) the least eigenvalue of the damped form on the window \([-a,a]\), where \(\|\cdot\|\) is the norm of \(L^2(\mathbb{R})\). Let \(V_\xi=(\log\xi)''(\frac12)\) and \(p_1=\frac12V_\xi=0.0231049931\ldots\), which is \(\sum_{\gamma>0}\gamma^{-2}\) under RH, and, under RH, \(p_2=\sum_{\gamma>0}\gamma^{-4}=3.71726\ldots\times10^{-5}\) (sums over the ordinates \(\gamma>0\) of the zeros \(\frac12+i\gamma\), with multiplicity). (i) If \(\alpha\ge\sigma_{\max}\) and no zero has real part \(\alpha\), then \(\lambda_{\min}(\alpha,a)\ge2m_\alpha\) for every \(a>0\), \(\lambda_{\min}(\alpha,a)\) is nonincreasing in \(a\), and \(\lambda_{\min}(\alpha,a)\to2m_\alpha\) as \(a\to\infty\). (ii) Under RH, \(m_\alpha=b_\alpha\) for every \(\alpha>\frac12\), and \(v_\alpha(\tau)>b_\alpha\) for \(\tau\ne0\): the minimum of \(\operatorname{Re}\xi'/\xi\) on each vertical line lies on the real axis, and only there. For \(\alpha\ge1\) this holds with no hypothesis; for \(\frac12<\alpha<1\) it holds with no hypothesis for \(0<|\tau|\le3\times10^{12}-1.21\) (the verification of RH to height \(3\times10^{12}\) by Platt and Trudgian is used); and RH holds if and only if \(v_\alpha(\tau)>b_\alpha\) for every \(\alpha\in(\frac12,1)\) and every \(\tau\ne0\). (iii) Hence RH holds if and only if \(\langle\varphi,\varphi\rangle_\alpha\ge2b_\alpha\|\varphi\|^2\) for all \(\alpha\in(\frac12,1]\), \(a>0\) and \(\varphi\in K(a)\), and under RH the constant is sharp: \(\lambda_{\min}(\alpha,a)\downarrow2b_\alpha\) as \(a\to\infty\). (iv) For every \(\alpha>\frac12\), with no hypothesis, \(2b_\alpha\le4p_1(\alpha-\frac12)\); under RH also \(4p_1(\alpha-\frac12)-4p_2(\alpha-\frac12)^3\le2b_\alpha\). The coercivity constant vanishes linearly at \(\alpha=\frac12\).

Earlier editions: Theorem 17.8.5, page 696 (edition 1).

Source: ch/yo3.tex, label yo:thm:coercive.

Unconditional sharp coercivity of the damped Yoshida form at alpha equal to 1

Volume II, Corollary 14.17.8, page 424 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\gamma_E\) be Euler’s constant, \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) and \(b_1=\xi'(1)/\xi(1)=1+\frac{\gamma_E}2-\frac12\log4\pi\). Let \(\langle\varphi,\varphi\rangle_1=\mathcal{Y}(e^{-|y|/2}\,\varphi*\widetilde\varphi)\) be the damped Yoshida form at \(\alpha=1\), with \(\mathcal{Y}\) the functional of Weil’s explicit formula and \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), let \(K(a)\) be the space of the restrictions to \([-a,a]\) of smooth \(2a\) periodic complex valued functions (set to zero outside \([-a,a]\)), \(\|\cdot\|\) the norm of \(L^2(\mathbb{R})\), and \(\lambda_{\min}(1,a)=\inf\{\langle\varphi,\varphi\rangle_1/\|\varphi\|^2:\ 0\ne\varphi\in K(a)\}\). For every \(a>0\) and every \(\varphi\in K(a)\), with no hypothesis on the zeros, \(\langle\varphi,\varphi\rangle_1\ge2b_1\|\varphi\|^2=(2+\gamma_E-\log4\pi)\|\varphi\|^2=0.0461914\ldots\|\varphi\|^2\), and no larger constant works for every \(a\): \(\lambda_{\min}(1,a)\downarrow2+\gamma_E-\log4\pi\) as \(a\to\infty\).

Earlier editions: Corollary 17.8.8, page 698 (edition 1).

Source: ch/yo3.tex, label yo:cor:half.

Uncertainty barrier for the tent coefficient above the verified height

Volume II, Proposition 14.18.1, page 427 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho\) run over the nontrivial zeros of the Riemann zeta function with multiplicity, \(\gamma_\rho=-i(\rho-\frac12)\) (complex for an off line zero, with \(\gamma_\rho=\gamma-i\delta\) for \(\rho=\frac12+\delta+i\gamma\)), and let \(D(x)=\sum_\rho4\sin^2(\gamma_\rho x/2)/\gamma_\rho^2\) be the tent coefficient, the value at the tent \((x-|y|)_+\) of the functional of Weil’s explicit formula. Suppose all zeros with \(0<\gamma\le H\) lie on the critical line, and let \(D_{\rm on}(x)\ge0\) be the part of \(D(x)\) from the zeros on the critical line. Then for every \(x>0\), \(|D(x)-D_{\rm on}(x)|\le\varepsilon_H(x):=8\cosh^2\frac x4\sum_{\gamma>H}\frac1{\gamma^2}\), the sum being over all zeros with ordinate \(\gamma>H\), counted with multiplicity. With \(H=3\times10^{12}\) (all zeros up to this height lie on the critical line, by the verification of Platt and Trudgian) the sum differs from \((\log(H/2\pi)+1)/(2\pi H)=1.4797\times10^{-12}\) by less than \(2\times10^{-24}\), so that \(\varepsilon_H(x)<1.184\times10^{-11}\cosh^2(x/4)\).

Earlier editions: Proposition 17.9.1, page 701 (edition 1).

Source: ch/yo4.tex, label yo:prop:barrier.

Computed positivity of the tent coefficient up to level 47.48 from the verification of RH

Volume II, Proposition 14.18.2, page 427 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\rho\) run over the nontrivial zeros of the Riemann zeta function with multiplicity, \(\gamma_\rho=-i(\rho-\frac12)\), and let \(D(x)=\sum_\rho4\sin^2(\gamma_\rho x/2)/\gamma_\rho^2\) be the tent coefficient, the value at the tent \((x-|y|)_+\) of the functional of Weil’s explicit formula. Let \(\gamma_1<\gamma_2<\dots\) be the ordinates of the first \(1700\) zeros in the upper half plane, all on the critical line, \(L(x)=\sum_{n\le1700}8\sin^2(\gamma_nx/2)/\gamma_n^2\), \(H=3\times10^{12}\) and \(\varepsilon_H(x)=8\cosh^2\frac x4\sum_{\gamma>H}\gamma^{-2}\), the sum over the zeros with ordinate \(\gamma>H\), counted with multiplicity. Since all zeros with \(0<\gamma\le H\) lie on the critical line (Platt and Trudgian), \(D(x)\ge L(x)-\varepsilon_H(x)\) for every \(x>0\), and a computation in double precision, which is not a ball arithmetic certificate, gives \(L(x)>\varepsilon_H(x)\) for \(0.05\le x\le47.48\): for \(x<46.688\) from the floor \(0.04070\) of \(L\) on \([0.05,47.1]\), found by branch and bound, and for \(46.688\le x\le47.48\) from a grid of step \(5\times10^{-4}\) and the Lipschitz constant \(10.85\) of \(L\). Hence \(D(x)>0\) for \(0.05\le x\le47.48\), from the verification of RH to \(H\) and the first \(1700\) zeros alone, with no prime sum.

Earlier editions: Proposition 17.9.2, page 701 (edition 1).

Source: ch/yo4.tex, label yo:prop:Dpos.

Open problem: the symbol inequality Re xi’/xi at least xi’/xi(alpha) below the line alpha equal to 1

Volume II, Problem 14.5, page 431 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(v_\alpha(\tau)=\operatorname{Re}\frac{\xi'}{\xi}(\alpha+i\tau)\) and \(b_\alpha=v_\alpha(0)=\xi'(\alpha)/\xi(\alpha)\). The inequality \(v_\alpha(\tau)\ge b_\alpha\) is a theorem of the book for every \(\alpha\ge1\) and every \(\tau\), and for \(\frac12<\alpha\le\alpha_{\max}\) (where \(\alpha_{\max}=7.35\) for \(\xi\)) up to height \(3\times10^{12}-1.21\) (using the verification of RH to height \(3\times10^{12}\) by Platt and Trudgian); for \(\frac12<\alpha<1\) and all \(\tau\) it is equivalent to RH. What is open is the region \(\alpha<1\) above the verified height. Determine the part of it in which \(v_\alpha(\tau)>b_\alpha\) can be proved with no hypothesis and with explicit constants; the Vinogradov Korobov region \(\alpha\ge1-c(\log|\tau|)^{-2/3}(\log\log|\tau|)^{-1/3}\), for a constant \(c>0\), is the natural target. A proof on every line of a strip \(\alpha_0<\alpha\le1\) and for all \(\tau\) would give \(\sigma_{\max}\le\alpha_0\), where \(\sigma_{\max}\) is the supremum of the real parts of the nontrivial zeros, which is why the target is a region that narrows with the height.

Earlier editions: Problem 17.1, page 703 (edition 1).

Source: ch/yo4.tex, label yo:op:symbol.

Open problem: a prime side proof of the symbol inequality at alpha equal to 1

Volume II, Problem 14.6, page 431 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(\Lambda\) the von Mangoldt function, \(\gamma_E\) Euler’s constant and \(\psi=\Gamma'/\Gamma\). The book proves \(\operatorname{Re}\frac{\xi'}{\xi}(1+i\tau)\ge\frac{\xi'}{\xi}(1)\) for all real \(\tau\) through the zeros, by the Cauchy kernels of the ordinates nearest to \(\tau\) and a bound on the gaps between them. Since \(\frac{\xi'}{\xi}(1)=1+\frac{\gamma_E}2-\frac12\log4\pi\) and \(\operatorname{Re}\frac1{i\tau}=0\), the statement can be written entirely in primes: for \(\tau\ne0\), \(\sum_{n\ge2}\frac{\Lambda(n)}{n}\cos(\tau\log n)\le\frac12\operatorname{Re}\psi\bigl(\frac{1+i\tau}2\bigr)-\frac12\log\pi+\frac1{1+\tau^2}-\frac{\xi'}{\xi}(1)\), the left side being the conditionally convergent series for \(-\operatorname{Re}\frac{\zeta'}{\zeta}(1+i\tau)\). Find a proof using only Mertens type information on the primes, in the spirit of a Fejér tapered form of the question; the right side grows like \(\frac12\log(\tau/2\pi)\) and the left side is \(O(\log\log\tau)\) under RH.

Earlier editions: Problem 17.2, page 704 (edition 1).

Source: ch/yo4.tex, label yo:op:primes.

Open problem: what does Löwner positivity on the lattice of one window say about the zeros

Volume II, Problem 14.7, page 431 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(a>0\) let \(\nu_n=\pi n/a\) and \(g_a(\nu)=\sum_\rho\sin^2(a\gamma_\rho)/(\gamma_\rho-\nu)\), where \(\rho\) runs over the nontrivial zeros of the Riemann zeta function in symmetric pairs and \(\gamma_\rho=-i(\rho-\frac12)\). Weil’s Hermitian form \(\langle\varphi_1,\varphi_2\rangle=\sum_\rho\widehat F(\gamma_\rho)\), with \(F=\varphi_1*\widetilde{\varphi_2}\), \(\widetilde\varphi(y)=\overline{\varphi(-y)}\) and \(\widehat F(t)=\int F(y)e^{ity}\,dy\), is positive semidefinite on the space \(K(a)\) of restrictions to \([-a,a]\) of smooth \(2a\) periodic functions if and only if the Löwner matrix of \(g_a\) on the lattice \(\{\nu_n\}=\frac\pi a\mathbb{Z}\), with the entries \((g_a(\nu_n)-g_a(\nu_m))/(\nu_n-\nu_m)\) off the diagonal and \(g_a'(\nu_n)\) on it, is positive semidefinite. Löwner’s theorem recovers the Pick property of \(g_a\), and with it RH, from the positivity of the Löwner matrices on all finite subsets of an interval. Open: what does positivity of the Löwner matrix on the lattice alone, for one fixed \(a\), imply about the location of the zeros? A positive answer that grew with \(a\) would quantify Yoshida’s threshold \(a_0\), the supremum of the half lengths \(a\) for which Weil’s form is positive semidefinite on the smooth functions supported in \([-a,a]\).

Earlier editions: Problem 17.3, page 704 (edition 1).

Source: ch/yo4.tex, label yo:op:lowner.

Open problem: the orders of the first failing rung and of Yoshida’s window threshold for one off line pair

Volume II, Problem 14.8, page 431 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\rho=\frac12+\delta+i\gamma\) with \(\gamma>0\) and \(0<\delta<\frac12\) be a hypothetical off line zero of the Riemann zeta function (with its symmetric companions) and let \(\sigma_{\max}\) be the supremum of the real parts of the zeros. Let \(W(x)=\sum_{\gamma>0}e^{(\rho-1/2)^2x}\) be the heat trace of the zeros, \(T_k=(-1)^kW^{(k)}\) its rungs, \(K_0\) the least order \(k\) for which \(T_k\) takes a negative value, and \(a_0\) Yoshida’s threshold, the supremum of the half lengths \(a\) for which Weil’s Hermitian form \(\langle\varphi_1,\varphi_2\rangle=\sum_\rho\widehat F(-i(\rho-\frac12))\) (all nontrivial zeros, \(F=\varphi_1*\widetilde{\varphi_2}\), \(\widetilde\varphi(y)=\overline{\varphi(-y)}\), \(\widehat F(t)=\int F(y)e^{ity}\,dy\)) is positive semidefinite on the smooth functions supported in \([-a,a]\). Determine the orders of \(K_0\) and of \(a_0\) for a single off line pair, in terms of its height and displacement (for \(K_0\) also of the spacing of the true ordinates near it), and compare them with the displacement \(\sigma_{\max}-\frac12\) and with the de Bruijn Newman constant \(\Lambda\), which satisfies \(\Lambda\le2(\sigma_{\max}-\frac12)^2\) (de Bruijn, in the normalisation of Rodgers and Tao). The book proves only that \(K_0\) is finite when RH fails, with no explicit bound, and gives for \(a_0\) only an upper bound from the single tent vector (a negative tent coefficient at level \(2a\) shows that the form is not positive semidefinite on the smooth functions supported in \([-a,a]\)); its numerical experiments on both are not proofs.

Earlier editions: Problem 17.4, page 704 (edition 1).

Source: ch/yo4.tex, label yo:op:orders.

Open problem: zero density estimates and the barrier, and a barrier for windows

Volume II, Problem 14.9, page 432 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(D(x)=\sum_\rho4\sin^2(\gamma_\rho x/2)/\gamma_\rho^2\) be the tent coefficient (the sum over the nontrivial zeros \(\rho\) of the Riemann zeta function with multiplicity, \(\gamma_\rho=-i(\rho-\frac12)\)), \(H=3\times10^{12}\) the height up to which RH is verified (Platt and Trudgian), and \(\varepsilon_H(x)=8\cosh^2\frac x4\sum_{\gamma>H}\gamma^{-2}\) the bound for the contribution of the zeros above \(H\) to \(D(x)\). This bound allows every zero above \(H\) to have displacement \(\delta\) from the critical line arbitrarily close to \(\frac12\), while zero density estimates make zeros with large \(\delta\) rare. Open: sharpen the barrier level using an explicit bound for the number \(N(\sigma,T)\) of zeros with real part at least \(\sigma\) and ordinate in \((0,T]\), and determine whether any prime side computation of \(D\) at feasible levels can, even in principle, exclude off line zeros beyond the verified height. For windows (smooth functions supported in \([-a,a]\)) no barrier is proved: decide whether a window whose level \(2a\) lies below the level up to which \(D\) is computed to be positive, in double precision, from the verification to \(H\) and finitely many certified zeros can see an off line zero above \(H\).

Earlier editions: Problem 17.5, page 704 (edition 1).

Source: ch/yo4.tex, label yo:op:density.

Volume II, Chapter 15. Zhu’s compact windows and the Landau Widom horizon

Dictionary: Zhu’s window form is Yoshida’s Weil form and, under RH, a sampling functional of the Thorin measure

Volume II, Proposition 15.2.1, page 435 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis, used only for the second set of identities (the sum over the zeros, the integral against the Thorin measure and the sampling reading of the least eigenvalue). The identity \(Q(f)=\langle f,f\rangle=\mathcal{Y}(f*f)\) holds with no hypothesis on the zeros.

Statement. Let \(\zeta\) be the Riemann zeta function, with nontrivial zeros written \(\rho=\frac12+i\gamma_\rho\) and counted with multiplicity (so that \(\gamma_\rho\) is real exactly for the zeros on the critical line), let \(\Lambda\) be the von Mangoldt function and \(\psi=\Gamma'/\Gamma\). For a continuous function \(g\) of compact support put \(\widehat g(t)=\int_{\mathbb{R}}g(y)e^{ity}\,dy\) and let \(\mathcal{Y}(g)=\sum_\rho\widehat g(\gamma_\rho)\) be Weil’s functional given by the explicit formula. Fix \(a>0\). Call a real even continuous function \(f\) supported in \([-a,a]\) admissible if Weil’s explicit formula applies to \(f*f\) (this holds for every real even \(f\in C_c^\infty(\mathbb{R})\) supported in \([-a,a]\)), and write \(F=\widehat f\), a real even function. Yoshida’s form is \(\langle f,f\rangle=\mathcal{Y}(f*f)\), and Zhu’s form (Xuefeng Zhu, 2026, arXiv:2608.24827) is \(Q(f)=2F(\tfrac i2)^2+\frac1{2\pi}\int_{\mathbb{R}}|F(t)|^2\,\Psi_a(t)\,dt\) with \(\Psi_a(t)=\operatorname{Re}\psi\bigl(\tfrac14+\tfrac{it}2\bigr)-\log\pi-\mathcal{P}_a(t)\), where \(\mathcal{P}_a(t)=\sum_{n\ge2,\ \log n<2a}2\Lambda(n)n^{-1/2}\cos(t\log n)\) is the prime comb. Then, for every admissible \(f\) and with no hypothesis on the zeros, \(Q(f)=\langle f,f\rangle=\mathcal{Y}(f*f)\). If the Riemann Hypothesis holds, so that every \(\gamma_\rho\) is real, then \(Q(f)=\sum_\rho|F(\gamma_\rho)|^2=2\int_{(0,\infty)}|F(\sqrt t)|^2\,U(dt)\), where \(U=\sum_{\gamma>0}\delta_{\gamma^2}\) (the sum over the positive ordinates \(\gamma\) with multiplicity) is the Thorin measure of the clock \(\sum_{\gamma>0}e_\gamma/\gamma^2\), with independent standard exponential variables \(e_\gamma\), a generalized gamma convolution with Laplace transform \(\prod_{\gamma>0}(1+s/\gamma^2)^{-1}\); and, with \(\|f\|^2=\int|f|^2=\frac1{2\pi}\int_{\mathbb{R}}|F|^2\) by Plancherel, \(\lambda^{\mathrm{even}}_{\min}(\tfrac12,a)=\inf_f Q(f)/\|f\|^2=\inf_f\dfrac{2\int_{(0,\infty)}|F(\sqrt t)|^2\,U(dt)}{\frac1{2\pi}\int_{\mathbb{R}}|F|^2}\), the infimum over the nonzero admissible \(f\). Thus \(\lambda^{\mathrm{even}}_{\min}(\tfrac12,a)\) is the lower sampling constant of the Thorin measure \(U\) for the transforms \(F\) of the admissible \(f\), which lie in the space of the real even entire functions of exponential type at most \(a\) that are square integrable on \(\mathbb{R}\) (a Paley Wiener space).

Earlier editions: Proposition 18.2.1, page 709 (edition 1).

Source: ch/zhu1.tex, label zh:prop:dict.

Two parameter bound: the damped least eigenvalue is at least the undamped one, and under RH at least \(2\xi'(\alpha)/\xi(\alpha)\)

Volume II, Theorem 15.5.2, page 446 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann Hypothesis, used for part (ii) when \(\alpha<1\), for the lower bound \(2b_\alpha\ge4p_1(\alpha-\frac12)-4p_2(\alpha-\frac12)^3\) at every \(\alpha>\frac12\) and for the closing maximum. Part (i), the bound \(\lambda_{\min}(\alpha,a)\ge2b_\alpha\) for \(\alpha\ge1\) and the upper bound \(2b_\alpha\le4p_1(\alpha-\frac12)\) hold with no hypothesis on the zeros.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, let \(\mathcal{Y}(g)=\sum_\rho\widehat g(\gamma_\rho)\) be Weil’s functional (the sum over the nontrivial zeros \(\rho=\frac12+i\gamma_\rho\) of the Riemann zeta function with multiplicity, where \(\widehat g(t)=\int_{\mathbb{R}}g(y)e^{ity}\,dy\)), and for \(\alpha\ge\frac12\) let \(\langle f,f\rangle_\alpha=\mathcal{Y}\bigl(e^{-(\alpha-1/2)|y|}\,f*\widetilde f\bigr)\) be the damped Weil form of a complex valued \(f\in C_c^\infty(\mathbb{R})\), with \(\widetilde f(y)=\overline{f(-y)}\). For \(a>0\) let \(C(a)\) be the set of these \(f\) supported in \([-a,a]\), let \(\|f\|^2=\int_{\mathbb{R}}|f|^2\), and let \(\lambda_{\min}(\alpha,a)=\inf\{\langle f,f\rangle_\alpha/\|f\|^2:\ 0\ne f\in C(a)\}\) be the least eigenvalue of the damped form on the window \([-a,a]\). Put \(b_\alpha=\xi'(\alpha)/\xi(\alpha)\) and \(p_1=\frac12(\log\xi)''(\frac12)=0.0231049931\ldots\), which equals \(\sum_{\gamma>0}\gamma^{-2}\) under the Riemann Hypothesis (RH), and, under RH, \(p_2=\sum_{\gamma>0}\gamma^{-4}=3.7172599\ldots\times10^{-5}\), the sums running over the positive ordinates \(\gamma\) of the zeros with multiplicity. Let \(a>0\) and \(\alpha>\frac12\). (i) With no hypothesis, \(\lambda_{\min}(\alpha,a)\ge\lambda_{\min}(\frac12,a)\). (ii) If RH holds, or if \(\alpha\ge1\), then \(\lambda_{\min}(\alpha,a)\ge2b_\alpha\). Here \(2b_\alpha\le4p_1(\alpha-\frac12)\) with no hypothesis, and under RH \(2b_\alpha\ge4p_1(\alpha-\frac12)-4p_2(\alpha-\frac12)^3\). Hence, under RH, \(\lambda_{\min}(\alpha,a)\ge\max\bigl(\lambda_{\min}(\frac12,a),\,2b_\alpha\bigr)\) for every \(\alpha>\frac12\) and every \(a>0\).

Earlier editions: Theorem 18.5.2, page 720 (edition 1).

Source: ch/zhu3.tex, label zh:thm:twoparam.

Open problem: the Landau Widom horizon as a theorem, the largest window on which Weil’s form is positive when the zeros are controlled up to a height

Volume II, Problem 15.1, page 448 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\mathcal{Y}(g)=\sum_\rho\widehat g(\gamma_\rho)\) be Weil’s functional (the sum over the nontrivial zeros \(\rho=\frac12+i\gamma_\rho\) of the Riemann zeta function with multiplicity, where \(\widehat g(t)=\int_{\mathbb{R}}g(y)e^{ity}\,dy\)), let \(C(a)\) be the set of complex valued \(f\in C_c^\infty(\mathbb{R})\) supported in \([-a,a]\), and say that Weil’s form is positive on the window \([-a,a]\) if \(\mathcal{Y}(f*\widetilde f)\ge0\) for every \(f\in C(a)\), where \(\widetilde f(y)=\overline{f(-y)}\). Determine the largest half length \(a(H,\delta_0)\) such that, if all zeros \(\beta+i\gamma\) with \(0<\gamma\le H\) lie on the line \(\beta=\frac12\) and every zero with \(\gamma>H\) has \(|\beta-\frac12|\le\delta_0\), Weil’s form is positive on the window \([-a,a]\) with \(a=a(H,\delta_0)\). Is \(a(H,\delta_0)\ge\frac12\log\frac H{2\pi}-C(\delta_0)\), with a constant \(C(\delta_0)\) that depends only on \(\delta_0\)? The book proves no such theorem. The passage from a window of half length \(a\) to the height \(T_*(a)=2\pi e^{2a}\) of the zeros that it resolves (the height at which the density \(a/\pi\) of the zeros of a function of exponential type \(a\) meets the density \(\frac1{2\pi}\log\frac t{2\pi}\) of the ordinates; the resolution height of Xuefeng Zhu, 2026, which the book calls the Landau Widom horizon) is a heuristic, and it suggests the bound for small \(\delta_0\).

Earlier editions: Problem 18.1, page 722 (edition 1).

Source: ch/zhu4.tex, label zh:op:horizon.

Open problem: derive Zhu’s Landau Widom decay law for the least eigenvalue of Weil’s form on a window from the Thorin side

Volume II, Problem 15.2, page 448 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\mathcal{Y}(g)=\sum_\rho\widehat g(\gamma_\rho)\) be Weil’s functional (the sum over the nontrivial zeros \(\rho=\frac12+i\gamma_\rho\) of the Riemann zeta function with multiplicity, where \(\widehat g(t)=\int_{\mathbb{R}}g(y)e^{ity}\,dy\)), let \(\lambda^{\mathrm{even}}_{\min}(\frac12,a)=\inf\mathcal{Y}(f*f)/\|f\|^2\) over the nonzero real even continuous functions \(f\) supported in \([-a,a]\) (for which the explicit formula applies to \(f*f\)), with \(\|f\|^2=\int|f|^2\), let \(N(T)\) be the number of ordinates \(0<\gamma\le T\) of the zeros with multiplicity, and let \(T_*=2\pi e^{2a}\). Derive the decay law of Xuefeng Zhu (2026, arXiv:2608.24827), \(-\log\lambda^{\mathrm{even}}_{\min}(\frac12,a)=C\,N(T_*)/\log N(T_*)\,(1+o(1))\) as \(a\to\infty\), which Zhu fits with \(C=20.13\ldots\) to certified upper bounds for \(a\le1.2\) and tests on \(a\) from \(1.4\) to \(2\) and states as a conjecture with \(C=2\pi^2=19.74\ldots\), from his test function with Fourier transform \(\widehat f(r)=h(r)\,G_T(-r^2)\), where \(h\) is a window kernel of exponential type \(a\) and \(G_T(u)=\prod_{0<\gamma_j\le T}(1+u/\gamma_j^2)\), optimising \(T\) and \(h\) jointly, and explain the constant through the Landau Widom plunge of the eigenvalues of the concentration operator of time and band limiting. What is known: the law is Zhu’s conjecture, fitted and not derived. Under RH Zhu proves only \(\lambda^{\mathrm{even}}_{\min}(\frac12,a)\le\exp(-ae^a)\) for \(a\) large, which uses the ordinates up to \(2\pi e^a\), half the exponent of \(T_*\), and is weaker than the law.

Earlier editions: Problem 18.2, page 723 (edition 1).

Source: ch/zhu4.tex, label zh:op:decay.

Open problem: certified damped windows for basepoints between one half and one

Volume II, Problem 15.3, page 449 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, let \(\mathcal{Y}(g)=\sum_\rho\widehat g(\gamma_\rho)\) be Weil’s functional (the sum over the nontrivial zeros \(\rho=\frac12+i\gamma_\rho\) of the Riemann zeta function with multiplicity, where \(\widehat g(t)=\int_{\mathbb{R}}g(y)e^{ity}\,dy\)), and for a basepoint \(\alpha\ge\frac12\) and \(f\in C_c^\infty(\mathbb{R})\) let \(\langle f,f\rangle_\alpha=\mathcal{Y}\bigl(e^{-(\alpha-1/2)|y|}\,f*\widetilde f\bigr)\) be the damped Weil form, with \(\widetilde f(y)=\overline{f(-y)}\). Xuefeng Zhu (2026, arXiv:2608.24827) proves by a computer assisted certificate that at \(\alpha=\frac12\) one has \(\langle f,f\rangle_\alpha\ge8.9\times10^{-18}\|f\|^2\) for every real even \(f\) supported in \([-0.8,0.8]\), where \(\|f\|^2=\int|f|^2\). Extend this certification to the damped forms at \(\alpha\in(\frac12,1)\), where under the Riemann Hypothesis the least eigenvalue on a window is at least \(2\xi'(\alpha)/\xi(\alpha)\), of order \(\alpha-\frac12\), in place of the least eigenvalue at \(\alpha=\frac12\), and measure how far the half length \(a\) of the window can be pushed. What is known: the book proves no damped analogue of Zhu’s reduction of positivity on a window to a finite matrix, and for \(\alpha<1\) the pole part of the damped form is not a positive rank one term; for \(\alpha\ge1\) no certificate is needed, because the damped form is positive on every window with margin \(2\xi'(\alpha)/\xi(\alpha)\) with no hypothesis.

Earlier editions: Problem 18.3, page 723 (edition 1).

Source: ch/zhu4.tex, label zh:op:damped.

Open problem: comb cancellation on average over Paley Wiener functions

Volume II, Problem 15.4, page 449 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Lambda\) be the von Mangoldt function and, for \(a>0\), let \(\mathcal{P}_a(t)=\sum_{n\ge2,\ \log n<2a}2\Lambda(n)n^{-1/2}\cos(t\log n)\) be the prime comb, whose supremum over \(t\in\mathbb{R}\) is \(A_a=\sum_{\log n<2a}2\Lambda(n)n^{-1/2}=(4+o(1))e^a\) as \(a\to\infty\) (it is attained at \(t=0\), and values arbitrarily close to it occur at arbitrarily large \(t\), as Xuefeng Zhu (2026, arXiv:2608.24827) shows). Let \(PW_a\) be the Paley Wiener space of the entire functions of exponential type at most \(a\) that are square integrable on \(\mathbb{R}\). Weil’s form on the window \([-a,a]\) sees the comb only through \(\int|F|^2\mathcal{P}_a\) with \(F\in PW_a\), an average over a frequency band of width about \(1/a\). Determine the supremum of \(\int_{|t|\ge T}|F|^2\mathcal{P}_a\Big/\int_{|t|\ge T}|F|^2\) over \(F\in PW_a\), as a function of \(T\) and \(a\). What is known: the pointwise supremum \(A_a\) bounds the ratio from above, and how much smaller the ratio is, uniformly in \(T\), is not known. A bound of smaller order than \(A_a\) would be a statement about short sums of \(\Lambda(n)n^{-1/2-it}\).

Earlier editions: Problem 18.4, page 723 (edition 1).

Source: ch/zhu4.tex, label zh:op:comb.

Volume II, Chapter 16. Truncations: the Biane Pitman Yor lattice, the beta lattice and Hermite coefficients

First order expansion of the zeros of the truncated lattice transforms

Volume II, Proposition 16.1.2, page 455 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(e_k,e_k'\) (\(k\ge1\)) be independent standard exponential random variables, \(\mathcal S_n=\sum_{k\le n}k^{-2}(e_k+e_k')\) and \(F_n(w)=\mathbb{E}\,\mathcal S_n^{\,w}\) for \(\operatorname{Re}w>-2\). As \(n\to\infty\), \(F_n\to F\) locally uniformly on \(\operatorname{Re}w>-2\), where \(F(w)=\mathbb{E}\bigl[(\sum_{k\ge1}k^{-2}(e_k+e_k'))^w\bigr]=2\pi^w\xi(2w)\) and \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) is the completed zeta function, so the zeros of \(F\) are the points \(\rho/2\) for the nontrivial zeros \(\rho\) of \(\zeta\). Let \(\rho\) be a simple zero of \(\xi\) and put [ ()== . ] There is a constant \(C\) (depending on \(\rho\)) such that for all large \(n\), \(F_n\) has exactly one zero \(w_n(\rho)\) in the disc \(\bigl|w-\frac\rho2-\frac{\kappa(\rho)}n\bigr|<C/n^2\); thus \(w_n(\rho)=\frac\rho2+\frac{\kappa(\rho)}n+O(n^{-2})\).

Earlier editions: Proposition 7.2.2, page 190 (edition 1).

Source: ch/rz4.tex, label rz:prop:firstorder.

Displacement constant of the truncated zeros in terms of Hardy’s function

Volume II, Theorem 16.1.3, page 456 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\rho=\frac12+i\gamma\) be a simple zero of \(\zeta\) on the critical line, \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), and \(\kappa(\rho)=\rho\,\xi(\rho-2)/(2\pi\,\xi'(\rho))\) the constant of the first order displacement of the zeros of the truncated lattice transforms \(F_n(w)=\mathbb{E}\bigl[(\sum_{k\le n}k^{-2}(e_k+e_k'))^w\bigr]\), with \(e_k,e_k'\) independent standard exponentials. Let \(\theta(t)=\arg\Gamma(\frac14+\frac{it}2)-\frac t2\log\pi\) be the Riemann Siegel theta function, \(Z(t)=e^{i\theta(t)}\zeta(\frac12+it)\) Hardy’s function, and \(a(\gamma)=e^{i\theta(\gamma)}\zeta(\frac52+i\gamma)\). Then \[\begin{align*} \operatorname{Re}\kappa(\rho)&=\frac{(\gamma^2-\frac{15}4)\operatorname{Im}a(\gamma)-4\gamma\operatorname{Re}a(\gamma)}{4\pi^2Z'(\gamma)},\\ \operatorname{Im}\kappa(\rho)&=\frac{4\gamma\operatorname{Im}a(\gamma)+(\gamma^2-\frac{15}4)\operatorname{Re}a(\gamma)}{4\pi^2Z'(\gamma)} . \end{align*}\]

Earlier editions: Theorem 7.2.3, page 191 (edition 1).

Source: ch/rz4.tex, label rz:thm:kappaexact.

Conjecture T is false for large n

Volume II, Corollary 16.1.4, page 457 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: counterexample, proved.

Statement. Let \(e_k,e_k'\) be independent standard exponential variables and \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) for \(\operatorname{Re}w>-2\). Conjecture T says that for every \(n\ge1\), \(F_n\) has no zeros with \(\operatorname{Re}w>\frac14\) (which, if it held for infinitely many \(n\), would imply the Riemann Hypothesis by Hurwitz’s theorem, since \(F_n\to2\pi^w\xi(2w)\)). Conjecture T is false: for all sufficiently large \(n\), \(F_n\) has a zero with \(\operatorname{Re}w>\frac14\) near \(\rho_{213}/2\), where \(\rho_{213}=\frac12+i\gamma_{213}\) and \(\gamma_{213}=415.4552\ldots\) is the ordinate of the 213th zero of the zeta function in the upper half plane. The proof uses that the first \(1.5\times10^9\) zeros lie on the critical line and are simple (van de Lune, te Riele and Winter), the first order displacement \(\kappa(\rho)=\rho\,\xi(\rho-2)/(2\pi\,\xi'(\rho))\) of the zero of \(F_n\) near \(\rho/2\), and a ball arithmetic enclosure \(233.95<\operatorname{Re}\kappa(\rho_{213})<233.96\). The statement is for all sufficiently large \(n\) and gives no explicit \(n\); the book notes that the first order regime requires \(n\) large compared with \(|\kappa(\rho_{213})|\approx2311\).

Earlier editions: Corollary 7.2.4, page 192 (edition 1).

Source: ch/rz4.tex, label rz:cor:Tfalse.

Symmetrised truncations: symmetry, limit and persistence of simple zeros on the line

Volume II, Proposition 16.1.6, page 458 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(e_k,e_k'\) be independent standard exponential variables, \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) for \(\operatorname{Re}w>-2\), and define the symmetrised truncation and its restriction to the line by [ K_n(w)=^{-w}F_n(w)+{w-}F_n(-w),Z_n(y)=K_n(+iy). ] Then \(K_n(\frac12-w)=K_n(w)\) and \(K_n(\bar w)=\overline{K_n(w)}\), so \(Z_n\) is real for real \(y\) and the zeros of \(K_n\) off the line \(\operatorname{Re}w=\frac14\) come in quadruples. As \(n\to\infty\), \(K_n\to4\,\xi(2w)\) locally uniformly on \(-\frac74<\operatorname{Re}w<\frac94\), where \(\xi\) is the completed zeta function. Every simple zero \(\frac14+i\frac\gamma2\) of \(\xi(2w)\) on the line \(\operatorname{Re}w=\frac14\) is, for all large \(n\), the limit of a zero of \(K_n\) lying exactly on that line.

Earlier editions: Proposition 7.2.6, page 193 (edition 1).

Source: ch/rz4.tex, label rz:prop:symtrunc.

Real rooted symmetrised truncations for infinitely many n imply the Riemann Hypothesis

Volume II, Theorem 16.1.7, page 458 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(e_k,e_k'\) be independent standard exponential variables, \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) for \(\operatorname{Re}w>-2\), and \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-\frac12}F_n(\tfrac12-w)\). If for infinitely many \(n\) all zeros of \(K_n\) in the strip \(0<\operatorname{Re}w<\frac12\) lie on the line \(\operatorname{Re}w=\frac14\), then the Riemann Hypothesis holds.

Earlier editions: Theorem 7.2.7, page 193 (edition 1).

Source: ch/rz4.tex, label rz:thm:symhurwitz.

Zeros of the approximate functional equation truncations of zeta, N equal to 2 and 3

Volume II, Proposition 16.1.8, page 459 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi\) be the factor in \(\zeta(s)=\chi(s)\zeta(1-s)\), \(\zeta_N(s)=\sum_{m\le N}m^{-s}+\chi(s)\sum_{m\le N}m^{s-1}\) and \(\vartheta\) the Riemann Siegel theta function. Write \(s=\sigma+i\tau\). (i) All but finitely many zeros of \(\zeta_2\) in \(0<\sigma<1\) lie on the critical line \(\sigma=\frac12\), and the number of zeros with \(0<\tau\le T\) is \(\vartheta(T)/\pi+O(1)\). (ii) Let \(\sigma^*\) be the root of \(2^{-\sigma}+3^{-\sigma}=1\). For every \(\sigma_0\in(\frac12,\sigma^*)\) there are zeros \(\sigma_k+i\tau_k\) of \(\zeta_3\) with \(\tau_k\to\infty\) and \(\sigma_k\to\sigma_0\). In particular \(\zeta_3\) has zeros off the critical line at arbitrarily large heights.

Earlier editions: Proposition 7.2.8, page 195 (edition 1).

Source: ch/rz4.tex, label rz:prop:zetaN.

Single term bound: no off line zeros of the symmetrised truncation above explicit heights

Volume II, Theorem 16.1.9, page 460 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. For \(n\ge1\) let \(\alpha_k,\beta_k\) (\(k\le n\)) be defined by the partial fraction expansion \(\prod_{j\le n}(1+\lambda/j^2)^{-2}=\sum_{k\le n}k^{-2}\bigl\{\alpha_k(1+\lambda/k^2)^{-1}+\beta_k(1+\lambda/k^2)^{-2}\bigr\}\) (so \(\beta_k=k^2\prod_{j\le n,\,j\ne k}(1-k^2/j^2)^{-2}\) and \(\alpha_k=-2\beta_k\sum_{j\le n,\,j\ne k}k^2/(j^2-k^2)\)), and put \(Q_n(u)=\sum_{k\le n}k^{-2u}(\alpha_k+\beta_ku)\). Then \(F_n(u-1)=\Gamma(u)Q_n(u)\) for \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) with \(e_k,e_k'\) independent standard exponentials, and the symmetrised truncation is \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-\frac12}F_n(\frac12-w)\). For \(y>|\alpha_1|/|\beta_1|\) let [ n(y)={k=2}^nk{-2},n’(y)={k=2}^nk{-2}+_n(y),, ] where \(\delta_n'\) is a bound for a derivative used in the proof and is not the derivative of \(\delta_n\). If \(\delta_n(y)<1\) and [ y-1y-->0, ] then \(K_n\) has no zeros off the line \(\operatorname{Re}w=\frac14\) in \(0<\operatorname{Re}w<\frac12\) at any height (imaginary part) at least \(y\). The condition holds for all \(y\ge Y_n\), with \(Y_1=4.81\), \(Y_2=5.85\), \(Y_3=7.11\), \(Y_4=9.08\), \(Y_5=12.8\), \(Y_6=20.8\), \(Y_8=104.1\) and \(Y_{10}=21232\) (values obtained by evaluating the bounds with the exact coefficients).

Earlier editions: Theorem 7.2.9, page 196 (edition 1).

Source: ch/rz4.tex, label rz:thm:Yn.

Conjecture S for n up to 6 and n = 8 above explicit heights

Volume II, Corollary 16.1.10, page 461 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-\frac12}F_n(\frac12-w)\) be the symmetrised truncation, with \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) and \(e_k,e_k'\) independent standard exponential variables. Conjecture S says that all zeros of \(K_n\) in the strip \(0<\operatorname{Re}w<\frac12\) lie on the line \(\operatorname{Re}w=\frac14\). Proved: for \(n=1,2,3,4,5,6,8\) the zeros of \(K_n\) in the strip at height at least \(Y_n\) lie on the line, with \(Y_1=4.81\), \(Y_2=5.85\), \(Y_3=7.11\), \(Y_4=9.08\), \(Y_5=12.8\), \(Y_6=20.8\), \(Y_8=104.1\) (the heights of the single term bound). Below these heights the book verifies the same statement for these \(n\) only numerically, by floating point argument principle counts and not by an enclosure (up to heights \(6,7,8\) for \(n=1,2,3\), up to \(70\) for \(4\le n\le6\) and up to \(140\) for \(n=8\)); that part is not a certified result.

Earlier editions: Corollary 7.2.10, page 196 (edition 1).

Source: ch/rz4.tex, label rz:cor:Ssmall.

Torus bound for the symmetrised truncations

Volume II, Theorem 16.1.11, page 461 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\alpha_k,\beta_k\) (\(k\le n\)) be defined by \(\prod_{j\le n}(1+\lambda/j^2)^{-2}=\sum_{k\le n}k^{-2}\bigl\{\alpha_k(1+\lambda/k^2)^{-1}+\beta_k(1+\lambda/k^2)^{-2}\bigr\}\), so that \(F_n(u-1)=\Gamma(u)Q_n(u)\) with \(Q_n(u)=\sum_{k\le n}k^{-2u}(\alpha_k+\beta_ku)=uP_n(u)+R_n(u)\), \(P_n(u)=\sum_{k\le n}\beta_kk^{-2u}\), \(R_n(u)=\sum_{k\le n}\alpha_kk^{-2u}\), for \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) with \(e_k,e_k'\) independent standard exponentials, and let \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-\frac12}F_n(\frac12-w)\). For real \(x\) and \(\omega=(\omega_p)_{p\le n}\) in the torus \(\mathbb{T}^{\pi(n)}\) of phases indexed by the primes \(p\le n\), put \(P_n(x,\omega)=\sum_{k\le n}\beta_kk^{-2x}\exp\bigl(-i\sum_{p\le n}v_p(k)\,\omega_p\bigr)\), where \(v_p(k)\) is the exponent of \(p\) in \(k\), let \(P_n'(x,\omega)=-\sum_{k\le n}2\log k\,\beta_kk^{-2x}\exp\bigl(-i\sum_{p\le n}v_p(k)\,\omega_p\bigr)\), \(r(x)=\sum_{k\le n}|\alpha_k|k^{-2x}\) and \(r'(x)=\sum_{k\le n}2\log k\,|\alpha_k|k^{-2x}\). Let \(m_n=\min|P_n(x,\omega)|\), \(M_n=\max\{-\operatorname{Re}(P_n'/P_n)(x,\omega)\}\) and \(\ell_n=\max|P_n'/P_n|(x,\omega)\) over \(\omega\in\mathbb{T}^{\pi(n)}\) and the band \(1.2\le x\le1.3\), and [ B_n(y)=_{0.3, } . ] If \(m_n>r(1.2)/y\), [ y-2y-M_n->0y-1y>B_n(y), ] then \(K_n\) has no zeros off the line \(\operatorname{Re}w=\frac14\) in \(0<\operatorname{Re}w<\frac12\) at height \(y\), and the same holds at every larger height.

Earlier editions: Theorem 7.2.11, page 197 (edition 1).

Source: ch/rz4.tex, label rz:thm:torus.

Conjecture S for n up to 24, 32 and 36 above explicit thresholds

Volume II, Corollary 16.1.12, page 462 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-\frac12}F_n(\frac12-w)\) be the symmetrised truncation, with \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) and \(e_k,e_k'\) independent standard exponential variables. Conjecture S says that all zeros of \(K_n\) in the strip \(0<\operatorname{Re}w<\frac12\) lie on the line \(\operatorname{Re}w=\frac14\). Proved, by torus bounds whose final boxes are enclosed in ball arithmetic: for \(n=16,17,\dots,24,32,36\) the zeros of \(K_n\) in the strip at height at least the threshold lie on the line, with thresholds \(30,34,37,41,46,50,56,62,69\) for \(n=16,\dots,24\) and \(213\), \(558\) for \(n=32\), \(36\); and the same holds for \(n=7\) and \(9\le n\le15\) above the thresholds \(14,24,31,41,57,84,131,229\) for \(n=7,9,10,11,12,13,14,15\). For all these \(n\) the statement is also verified numerically below the thresholds, by floating point argument principle counts, which are not certified. Together with the case \(n=1,\dots,6,8\) above explicit heights, this covers every \(n\le24\) and \(n=32,36\). The torus bounds behind the thresholds are proved in the book; in Polson’s 2026 book On Hilbert’s 8th Problem the corresponding corollary supposes that the extremes of the torus function, estimated by global optimisation and not enclosed, are bounds.

Supersedes: Corollary 30.11 of Polson (2026) is conditional: it supposes that the torus extremes, estimated by global optimisation and not enclosed, are bounds.

Earlier editions: Corollary 7.2.12, page 198 (edition 1).

Source: ch/rz4.tex, label rz:cor:Storus.

Thorin positivity of R is Weil’s criterion

Volume II, Proposition 16.1.14, page 465 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(R(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\). Let \(D(x)=\sum_\rho e^{(\rho-\frac12)x}\) be Weil’s distribution, summed over the nontrivial zeros \(\rho\) of \(\zeta\), whose action on a test function \(h\) is \(\sum_\rho\hat h(\gamma_\rho)\) with \(\gamma_\rho=-i(\rho-\frac12)\) and \(\hat h\) the Fourier transform of \(h\). Then \(R\) is the Laplace transform of a generalised gamma convolution (whose Thorin measure is then \(\sum_{\gamma>0}\delta_{\gamma^2}\) over the ordinates of the zeros) if and only if \(D\) is positive definite, that is, \(\sum_\rho\hat h(\gamma_\rho)\ge0\) for every \(h=g*\tilde g\) with \(g\) smooth and compactly supported and \(\tilde g(x)=\overline{g(-x)}\). Both conditions are equivalent to the Riemann Hypothesis: the first by Proposition 3.4 of Polson’s 2026 book On Hilbert’s 8th Problem, the second by Weil’s criterion (Weil 1952; Bombieri 2000). The proposition combines the two; the correspondence between Thorin measures and Weil’s distribution behind it is formal.

Supersedes: Theorem 30.13 of Polson (2026) states the correspondence between Thorin positivity of \(R\) and positive definiteness of Weil’s distribution as a theorem.

Earlier editions: Proposition 7.2.14, page 200 (edition 1).

Source: ch/rz4.tex, label rz:thm:weil.

Weil positivity on the Bernstein cone

Volume II, Theorem 16.1.15, page 467 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. For \(y>0\) let \(W(y)=\frac12\sum_\rho e^{y(\rho-\frac12)^2}\), the sum over all nontrivial zeros \(\rho\) of the Riemann zeta function (the heat trace of the zeros), and put \(\gamma_\rho=-i(\rho-\frac12)\). Let \(\hat h(r)=\int_0^\infty e^{-yr^2}g(y)\,dy\) with \(g\ge0\), \(g\not\equiv0\) and \(\int_0^1g(y)\,y^{-1/2}\bigl(1+\log(1/y)\bigr)\,dy+\int_1^\infty g(y)\,dy<\infty\), so that \(\hat h\) is a completely monotone function of \(r^2\). Then [ h()=2_0^W(y),g(y),dy>0, ] the sum being over all nontrivial zeros. In particular Weil’s functional \(\sum_\rho\hat h(\gamma_\rho)\) is positive for every test function of Thorin type \(\hat h(r)=\prod_k(1+r^2/t_k)^{-1}\) (a nonempty product with \(t_k>0\) and \(\sum_k1/t_k<\infty\)), for \(\hat h(r)=\operatorname{sech}(cr)\) with \(c>0\), and for \(\hat h(r)=(1+r^2/a)^{-m}\) with \(a>0\) and \(m>\frac12\). No hypothesis on the zeros is made. The positivity of \(W(y)\) for every \(y>0\), which with the identity gives the result, is Theorem A of Polson’s 2026 book On Hilbert’s 8th Problem; the book gives a second proof of it, which uses the verification of the Riemann Hypothesis up to height \(50\) and interval enclosures, and corrects the integrability condition and the examples of Theorem 30.14 of that book.

Supersedes: Theorem 30.14 of Polson (2026) has the integrability condition with \(\log(1/y)\) in place of \(1+\log(1/y)\), Thorin products without nonemptiness or \(\sum_k1/t_k<\infty\), and \((1+r^2/a)^{-m}\) without \(m>\frac12\), and its proof uses the verification of the Riemann Hypothesis to height \(3\times10^{12}\).

Earlier editions: Theorem 7.2.15, page 203 (edition 1).

Source: ch/rz4.tex, label rz:thm:bernsteincone.

The de Bruijn Newman flow as a path of GGCs

Volume II, Theorem 16.1.16, page 469 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\bigr)\exp(-\pi n^2e^{4u})\), \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\) (so that \(H_0(z)=\frac18\xi(\frac12+\frac{iz}2)\) for the completed zeta function \(\xi\)) and \(R_t(s)=H_t(0)/H_t(2i\sqrt s)\), so that \(R_0(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)=R(s)\). Let \(\Lambda\) be the de Bruijn Newman constant, so that \(H_t\) has only real zeros exactly when \(t\ge\Lambda\). Then for every \(t\ge\Lambda\), \(R_t\) is the Laplace transform of a generalised gamma convolution with Thorin measure \(\sum_k\delta_{(z_k(t)/2)^2}\), where \(z_k(t)>0\) are the positive zeros of \(H_t\) listed with multiplicity. In particular \(R_t\) is a GGC for every \(t\ge0.2\), unconditionally (using the bound \(\Lambda\le0.2\) of Platt and Trudgian, which improves the bound \(0.22\) of the Polymath project). The Riemann Hypothesis holds if and only if \(R_t\) is a GGC for arbitrarily small \(t>0\) (and, since \(\Lambda\ge0\) by Rodgers and Tao, if and only if \(\Lambda=0\)).

Earlier editions: Theorem 7.2.16, page 205 (edition 1).

Source: ch/rz4.tex, label rz:thm:dbn.

No GGC multiplier can create the positivity of R

Volume II, Proposition 16.1.17, page 470 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(R(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\), \(G(s)=\Gamma(\frac14+\frac{\sqrt s}2)\,\pi^{-\sqrt s/2}/\Gamma(\frac14)\) and \(\varphi(s)=R(s)G(s)=\zeta(\frac12)/\bigl((1-4s)\zeta(\frac12+\sqrt s)\bigr)\), and let \(c=\frac12(\log\pi-\psi(\frac14))\approx2.68609\) with \(\psi\) the digamma function. Let \(V\) be the Laplace transform of any generalised gamma convolution (GGC). Then \(V\cdot R\) is the Laplace transform of a GGC if and only if the Riemann Hypothesis holds. For \(\varphi\) only one direction holds for every \(V\): if \(V\cdot\varphi\) is the Laplace transform of a GGC, then the Riemann Hypothesis holds. The converse depends on \(V\): it fails for \(V\equiv1\) (since \(\varphi\) is not a GGC, unconditionally), and holds for \(V(s)=e^{-c\sqrt s}/G(s)\), which is the Laplace transform of a GGC and for which \(V\varphi=e^{-c\sqrt s}R\) is the floor of the reciprocal Xi function. In Polson’s 2026 book On Hilbert’s 8th Problem the same is claimed with \(\varphi\) in place of \(R\), that is, both directions for every multiplier, which fails for \(V\equiv1\); the book keeps one direction.

Supersedes: Proposition 30.16 of Polson (2026) states that the same is true with \(\varphi\) in place of \(R\), that is, both directions for every generalised gamma convolution multiplier, which fails for the multiplier \(1\) by Theorem 28.9 of that book.

Earlier editions: Proposition 7.2.17, page 205 (edition 1).

Source: ch/rz4.tex, label rz:prop:nomult.

Conjecture S for infinitely many n

Volume II, Problem 16.1, page 471 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(e_k,e_k'\) be independent standard exponential variables, \(F_n(w)=\mathbb{E}\bigl[\bigl(\sum_{k\le n}k^{-2}(e_k+e_k')\bigr)^w\bigr]\) for \(\operatorname{Re}w>-2\), and \(K_n(w)=\pi^{-w}F_n(w)+\pi^{w-\frac12}F_n(\frac12-w)\) the symmetrised truncation. Conjecture S: for every \(n\ge1\) all zeros of \(K_n\) in the strip \(0<\operatorname{Re}w<\frac12\) lie on the line \(\operatorname{Re}w=\frac14\). Problem: prove that this holds for infinitely many \(n\); by a Hurwitz argument this would imply the Riemann Hypothesis. The book proves it above explicit heights for \(n\le24\) and \(n=32,36\) and checks it numerically below those heights. Since the one sided Conjecture T (no zero of \(F_n\) with \(\operatorname{Re}w>\frac14\)) and the Hermite Biehler inequality for \(\pi^{-w}F_n(w)\) fail for large \(n\), a proof must establish real rootedness without a one sided inequality, for instance by an interlacing argument in \(n\), or by following the zeros of \(K_n(\frac14+iy)\) continuously in a parameter interpolating between \(n\) and \(n+1\) and excluding collisions of zeros on the line.

Earlier editions: Problem 7.1, page 207 (edition 1).

Source: ch/rz4.tex, label rz:prob:S.

Torus method beyond n = 36

Volume II, Problem 16.2, page 471 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(\alpha_k,\beta_k\) (\(k\le n\)) be the partial fraction coefficients of the Laplace transform \(\prod_{j\le n}(1+\lambda/j^2)^{-2}\) of \(\sum_{k\le n}k^{-2}(e_k+e_k')\), with \(\beta_k=k^2\prod_{j\le n,\,j\ne k}(1-k^2/j^2)^{-2}\), and let \(P_n(x,\omega)=\sum_{k\le n}\beta_kk^{-2x}\exp\bigl(-i\sum_{p\le n}v_p(k)\,\omega_p\bigr)\) be the torus function of the Dirichlet polynomial \(P_n(u)=\sum_{k\le n}\beta_kk^{-2u}\) (phases \(\omega_p\) indexed by the primes \(p\le n\), \(v_p(k)\) the exponent of \(p\) in \(k\)). The torus method proves Conjecture S, that all zeros of the symmetrised truncation \(K_n\) in \(0<\operatorname{Re}w<\frac12\) lie on \(\operatorname{Re}w=\frac14\), above explicit heights for \(n=7\), \(9\le n\le24\) and \(n=32,36\) (the cases \(n\le6\) and \(n=8\) come from a single term bound); at \(n=40\) the threshold, estimated by optimisation and not certified, is already near \(4900\), and with the band \(1.2\le x\le1.3\) the method ends between \(n=40\) and \(n=44\), where the torus minimum of \(|P_n|\) in the band reaches zero. Problem: extend the torus bound with narrower bands around \(x=\frac54\), and find the first \(n\) at which \(P_n\) has torus zeros with \(x\ge\frac54\).

Earlier editions: Problem 7.2, page 207 (edition 1).

Source: ch/rz4.tex, label rz:prob:torus.

From finite order to all orders for the floor

Volume II, Problem 16.3, page 472 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(W(y)=\frac12\sum_\rho e^{y(\rho-\frac12)^2}\), the sum over the nontrivial zeros \(\rho\) of the Riemann zeta function, let \(c=\frac12(\log\pi-\psi(\frac14))\approx2.68609\), and let \(k(y)=W(y)+c/(2\sqrt{\pi y})\) be the canonical function of the floor \(e^{-c\sqrt s}\,\xi(\frac12)/\xi(\frac12+\sqrt s)\) of the reciprocal Xi function. The book proves, by a two range argument, that \((-1)^nk^{(n)}(y)>0\) for every \(y>0\) and every \(n\le10^{10}\), the bound on \(n\) in that argument coming only from the height \(3\times10^{12}\) to which the Riemann Hypothesis is verified, and the small \(y\) branch of the proof holds for every \(n\) on \((0,0.05/n]\). This is a separate proof of a smaller range: the same signs hold for every \(n\le10^{17}\) by Theorem 7.10 of Polson’s arXiv paper 1708.02653 (version 24, 2026), which proves them for \(W\), and the book proves them for every \(n\le1.16\times10^{25}\) by sharper estimates of the heat trace; the signs for every \(n\) would make \(k\) completely monotone, which is equivalent to the Riemann Hypothesis. Problem: can the two regimes be made to overlap for all \(n\)? This asks for a prime side estimate for \((-1)^nk^{(n)}(y)\) on a range \(y\le y_n\) with \(y_n\) growing at least linearly in \(n\). In the proof the obstruction is the factor \(2^n\) in the Cauchy estimate on the circle of radius \(y/2\); a circle of radius close to \(y\) loses only a factor of order \(n\), but that argument also stops at a finite order.

Earlier editions: Problem 7.3, page 207 (edition 1).

Source: ch/rz4.tex, label rz:prob:allorders.

Beyond the Bernstein cone

Volume II, Problem 16.4, page 472 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Weil’s functional \(\sum_\rho\hat h(\gamma_\rho)\), with \(\gamma_\rho=-i(\rho-\frac12)\) for the nontrivial zeros \(\rho\) of the Riemann zeta function, is unconditionally positive for every test function \(\hat h\) completely monotone in \(r^2\) under an integrability condition (the Bernstein cone). Problem: find the largest cone of test functions, closed under the operations that preserve the Thorin structure, on which positivity of Weil’s functional can be proved unconditionally, and determine how far it can localise in height.

Earlier editions: Problem 7.4, page 208 (edition 1).

Source: ch/rz4.tex, label rz:prob:cone.

Deforming the binary product

Volume II, Problem 16.5, page 472 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. With the necklace weights \(b_n(q)\) of the Vere-Jones factorisation, each Euler factor \((1-p^{-\sigma-u})^{-1}\) of \(\zeta(\sigma+u)\) equals \(\prod_{n\ge1}(1+p^{-nu})^{b_n(p^{-\sigma})}\), with integer multiplicities \(0\) and \(1\) at \(\sigma=0\) (Euler’s binary product) and necklace multiplicities \(b_n(p^{-1/2})\) at \(\sigma=\frac12\). The pseudo spectral trace formula writes the heat trace of the zeros of \(\zeta\) (under the Riemann Hypothesis \(\sum_{\gamma>0}e^{-x\gamma^2}\), and in general \(\frac12\sum_\rho e^{x(\rho-\frac12)^2}\)), net of its smooth part, as \(e^{x/4}\) plus the sum over primes \(p\) and \(n\ge1\) of \(b_n(p^{-1/2})\) times the heat trace \(\sum_{k\ge0}e^{-x((2k+1)\pi/(n\log p))^2}\) of the arithmetic progression \((2k+1)\pi/(n\log p)\), net of its mean \(n\log p/(4\sqrt{\pi x})\). Problem: follow this formula along the path \(q=p^{-\sigma}\), \(0\le\sigma\le\frac12\), and determine where and how the positivity of the interference pattern of the progressions can fail.

Earlier editions: Problem 7.5, page 208 (edition 1).

Source: ch/rz4.tex, label rz:prob:binary.

Regularising the autocorrelation of the Pólya kernel

Volume II, Problem 16.6, page 472 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(p(\tau)=|\Gamma(\frac14+\frac{i\tau}2)|^2/(2^{3/2}\pi^{3/2})\) be the Meixner density, \(\Phi\) the Pólya kernel (an even function with Fourier transform \(\Xi(\tau)=\xi(\frac12+i\tau)\)) and \(\mathcal C(v)=\int p(\tau)\,|\zeta(\frac12+i\tau)|^2e^{iv\tau}\,d\tau\). Then \((\Phi*\Phi)(v)=2^{-3/2}\bigl(\frac{d^2}{dv^2}-\frac14\bigr)^2\mathcal C(v)\). Formally \(\mathcal C=\operatorname{sech}(\cdot)^{1/2}*\mathcal A\), where \(\operatorname{sech}(v)^{1/2}\) is the characteristic function of the Meixner law and \(\mathcal A=\sum_{m,n\ge1}(mn)^{-1/2}\delta_{\log(m/n)}\) is a positive, positive definite measure. The lattice series diverges: truncated at \(m,n\le X\) it grows linearly in \(X\), through the square of the pole term of the partial sums of \(\zeta\), and the factor \((\frac{d^2}{dv^2}-\frac14)^2\) does not remove this growth. Problem: find the regularised form of \(\operatorname{sech}(\cdot)^{1/2}*\mathcal A\) and decide whether it has a Thorin or Lee and Yang structure in \(v\). In Polson’s 2026 book On Hilbert’s 8th Problem the problem is posed up to a logarithmic divergence removed by the pole; the book finds linear growth that the factor does not remove.

Supersedes: Open Problem 30.22 of Polson (2026) poses the problem up to a logarithmic divergence removed by the pole.

Earlier editions: Problem 7.6, page 208 (edition 1).

Source: ch/rz4.tex, label rz:prob:meixner.

Unimodality of w and of the density f

Volume II, Problem 16.7, page 472 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. Corrects a statement of Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(w(x)=e^{x/4}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\Lambda(n)\,n^{-1/2}e^{-(\log n)^2/4x}\) be the canonical function of the normalised reciprocal zeta function \(\varphi(s)=\zeta(\frac12)/\bigl((1-4s)\zeta(\frac12+\sqrt s)\bigr)\) and \(f\) the density of the law with Laplace transform \(\varphi\), which satisfies Steutel’s equation \(xf(x)=\int_0^xf(x-y)\,w(y)\,dy\). The book proves that \(w'>0\) on \((0,0.004]\) and \(w'<0\) on \([4.52,\infty)\); a grid evaluation (computed, not proved) finds exactly one sign change of \(w'\) on \([0.004,4.6]\), at \(x_m=0.015851\ldots\), and numerically \(f\) rises from \(0.365089\) to its maximum \(0.366883\) at \(x=0.024046\) and then decreases. Problem: prove that \(w\) is unimodal, and prove that \(f\) is unimodal with mode near \(0.0240\), for instance from Steutel’s equation and the unimodality of \(w\). Yamazato’s theorem, that every self decomposable law is unimodal, does not apply to the law of \(\varphi\), which is infinitely divisible and not self decomposable. In Polson’s 2026 book On Hilbert’s 8th Problem the unimodality of \(w\) is stated as a theorem, whose proof uses a bound on \(w''\) that is not derived, and the problem asks only for the unimodality of \(f\); the book proves only the two end ranges of \(w'\) and asks for both.

Supersedes: Theorem 28.12 of Polson (2026) states that \(w\) is unimodal, with a proof that uses a bound on \(w''\) not derived there, and Open Problem 30.23 asks only for the unimodality of \(f\).

Earlier editions: Problem 7.7, page 208 (edition 1).

Source: ch/rz4.tex, label rz:prob:unimodal.

Infinite divisibility across the strip

Volume II, Problem 16.8, page 473 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(\sigma\in(0,1)\) let \(\varphi_\sigma(s)=\zeta(\sigma)/\bigl((1-s/(1-\sigma)^2)\zeta(\sigma+\sqrt s)\bigr)\) and \(w_\sigma(x)=e^{(1-\sigma)^2x}-\frac{1}{\sqrt{4\pi x}}\sum_{n\ge2}\Lambda(n)\,n^{-\sigma}e^{-(\log n)^2/4x}\), its canonical function, with \(\Lambda\) the von Mangoldt function. Problem: is \(w_\sigma(x)>0\) for every \(\sigma\in(0,1)\) and every \(x>0\)? Positivity gives infinite divisibility of \(\varphi_\sigma\) with Lévy density \(w_\sigma(x)/x\). The book proves \(w_{1/2}>0\) and checks \(w_\sigma>0\) numerically only for \(x\le10^3\) and at nine values of \(\sigma\). For \(\frac12<\sigma<1\) the method of the case \(\sigma=\frac12\) may apply, since the terms of the zeros below the verified height are Poisson kernels of positive width, so the question reduces to the sign of the explicit pole and archimedean terms at large \(x\), whose limit is governed by \(\zeta'(\sigma)/\zeta(\sigma)>0\). For the completed functions \(R_\sigma(s)=\xi(\sigma)/\xi(\sigma+\sqrt s)\) the answer is known: \(R_\sigma\) is infinitely divisible for every \(\sigma>\frac12\) with no hypothesis; since \(\varphi_\sigma\) differs from \(R_\sigma\) by the gamma factor and elementary factors in \(\sqrt s\), this does not settle the question for \(w_\sigma\).

Earlier editions: Problem 7.8, page 208 (edition 1).

Source: ch/rz4.tex, label rz:prob:strip.

Meixner Pollaczek truncations with real zeros

Volume II, Problem 16.9, page 473 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(Z\) be Hardy’s function and \(p(\tau)=|\Gamma(\frac14+\frac{i\tau}2)|^2/(2^{3/2}\pi^{3/2})\) the Meixner density. The truncations of the expansion of the real even function \((\tau^2+\frac14)Z(\tau)\) in the even Meixner Pollaczek polynomials \(P^{(1/4)}_{2m}(\tau/2;\frac\pi2)\), which are orthogonal for \(p\), are real even polynomials. The symmetric expansion captures the first ordinate at degree \(40\) but converges slowly, and the real zeros of its truncations do not track the ordinates of the zeros of the zeta function. Problem: is there a weighting of its coefficients, or a Jensen type construction from them, whose polynomials are real rooted and whose zeros converge to the ordinates?

Earlier editions: Problem 7.9, page 209 (edition 1).

Source: ch/rz4.tex, label rz:prob:mptrunc.

Nyman Beurling criterion and the sech expansion

Volume II, Problem 16.10, page 473 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. For \(\operatorname{Re}\sqrt s>\frac12\) the book writes \(1/\zeta(\frac12+\sqrt s)=1+\sum_{N\ge2}A_N\operatorname{sech}(\sqrt s\log N)\) with explicit Möbius type weights \(A_N\) supported on the powers \(N=n^m\) with \(n\) squarefree and \(m\) odd, and \(A_N=\mu(N)/(2\sqrt N)\) for squarefree \(N\) (\(\mu\) the Möbius function). Problem: relate the completeness question of Lai and Vere-Jones for the dilates \(\{\operatorname{sech}(\lambda x)\}\) to the Nyman Beurling criterion for the Riemann Hypothesis, and decide whether the sech expansion admits an \(L^2\) version on the boundary \(\operatorname{Re}\sqrt s=\frac12\).

Earlier editions: Problem 7.10, page 209 (edition 1).

Source: ch/rz4.tex, label rz:prob:nymanbeurling.

Product representations of the law of R

Volume II, Problem 16.11, page 473 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. First stated in Polson (2026), On Hilbert’s 8th Problem: the reciprocal Xi clock, Thorin measures, van Dantzig pairs, zero free approximations, hitting times and the reciprocal zeta function, first edition, Independently published.

Status: open.

Statement. Let \(X_R\) be a random variable with Laplace transform \(R(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\) and \(X_\varphi\) one with Laplace transform \(\varphi(s)=\zeta(\frac12)/\bigl((1-4s)\zeta(\frac12+\sqrt s)\bigr)\). The class of generalised gamma convolutions (GGC) is closed under independent sums, independent products and weak limits, so an unconditional representation of \(X_R\) built from independent GGC variables by sums, products and weak limits would make \(R\) a GGC and prove the Riemann Hypothesis; for \(X_\varphi\) the book proves that no such representation exists, and the harmonic mean is not a closure operation of the class. The factorisation of \(\varphi\) into \(R\) and the gamma factor \(\Gamma(\frac14+\frac{\sqrt s}2)\pi^{-\sqrt s/2}/\Gamma(\frac14)\) gives, formally, \(X_R\) as \(X_\varphi\) minus a positive stable law of index \(\frac12\) and a compensated series of independent variables \(Y_{c_n}\), where \(\mathbb{E}e^{-sY_\kappa}=\kappa/(\kappa+\sqrt s)\) and \(c_n=2n+\frac12\) for \(n\ge0\). Question: can the compensation be absorbed into a product?

Earlier editions: Problem 7.11, page 209 (edition 1).

Source: ch/rz4.tex, label rz:prob:xrproduct.

Truncations converge locally uniformly, and GRH for chi 4 is equivalent to zero free compact subsets of the left half strip for large n

Volume II, Theorem 16.2.1, page 476 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Suppose \(R_n\to0\) in probability, as for the hard cutoff \(R_n=0\) and for every nonnegative remainder with \(\mathbb ER_n=\mathbb ET_n\). Then \(M_n\to M\) locally uniformly on \(0<\Re z<\frac12\), and the Generalised Riemann Hypothesis (GRH) for \(\chi_4\) holds if and only if every compact subset of \(0<\Re z<\frac14\) is free of zeros of \(M_n\) for all sufficiently large \(n\).

Earlier editions: Theorem 28.1.1, page 1000 (edition 1).

Source: ch/tr1.tex, label tr:thm:local.

The tail cumulants of the beta lattice have an explicit expansion in the truncation index

Volume II, Lemma 16.3.1, page 477 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Then for fixed \(j\ge1\), \(q_j(n)=\lambda_*^{-j}\{n^{1-2j}/(2j-1)-\frac j{12}n^{-2j-1}+O(n^{-2j-3})\}\). In particular \(c_n=\mathbb ET_n=\frac1{\lambda_*n}-\frac1{12\lambda_*n^3}+O(n^{-5})\), \(v_n=\operatorname{Var}T_n=q_2(n)=\frac1{3\lambda_*^2n^3}-\frac1{6\lambda_*^2n^5}+O(n^{-7})\) and \(\kappa_3(T_n)=\frac2{5\lambda_*^3n^5}+O(n^{-7})\).

Earlier editions: Lemma 28.2.1, page 1001 (edition 1).

Source: ch/tr1.tex, label tr:lem:tail.

The truncated Mellin transform differs from M by a multiple of M(z+r), and each simple zero is displaced by a multiple of the first unmatched cumulant difference

Volume II, Theorem 16.3.2, page 477 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Fix \(m\ge0\), \(r=m+1\), \(p=2r-1\), and let \(R_n\ge0\) be independent of \(\omega_n\) with \(\kappa_j(R_n)=\kappa_j(T_n)\) for \(1\le j<r\), such that for a constant \(C\) independent of \(n\) and \(j\), \(\log\mathbb Ee^{-tR_n}=\sum_{j\ge1}(-1)^j\kappa_j(R_n)t^j/j!\) for \(0\le t<n^2/C\) and \(|\kappa_j(R_n)|/j!\le C^jn^{1-2j}\). Put \(\Delta_{r,n}=\kappa_r(T_n)-\kappa_r(R_n)\) and \((z)_r=z(z+1)\cdots(z+r-1)\). Then uniformly for \(z\) in a compact subset \(K\) of \(\Re z>0\), \(M_n(z)-M(z)=-\frac{(-1)^r\Delta_{r,n}}{r!}(z)_rM(z+r)+O_K(n^{-p-2}+n^{-2p})\). Let \(\rho=\frac12+i\gamma\) be a simple zero of \(\xi_4\) and \(z_0=\rho/2\). For all large \(n\) a fixed small disc about \(z_0\) contains exactly one zero \(z_n\) of \(M_n\), it is simple, and \(z_n-z_0=\frac{(-1)^r\Delta_{r,n}}{r!}(z_0)_r\frac{M(z_0+r)}{M'(z_0)}+O(n^{-p-2}+n^{-2p})\) with \(\frac{M(z_0+r)}{M'(z_0)}=\frac{(2\pi)^r\xi_4(\rho+2r)}{2\xi_4'(\rho)}\). If \(\Delta_{r,n}=d\,n^{-p}+o(n^{-p})\) with \(d\ne0\), then \(z_n-z_0=D_rn^{-p}+o(n^{-p})\) with \(D_r=\frac{(-1)^rd\,(2\pi)^r(z_0)_r\,\xi_4(\rho+2r)}{2\,r!\,\xi_4'(\rho)}\ne0\).

Earlier editions: Theorem 28.2.2, page 1001 (edition 1).

Source: ch/tr1.tex, label tr:thm:drift.

The hard cutoff zeros drift at the rate 1 over n and the mean corrected zeros at 1 over n cubed, with the coefficient delta m

Volume II, Corollary 16.3.3, page 478 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Fix \(m\ge0\), \(r=m+1\), \(p=2r-1\), and let \(R_n\ge0\) be independent of \(\omega_n\) with \(\kappa_j(R_n)=\kappa_j(T_n)\) for \(1\le j<r\), such that for a constant \(C\) independent of \(n\) and \(j\), \(\log\mathbb Ee^{-tR_n}=\sum_{j\ge1}(-1)^j\kappa_j(R_n)t^j/j!\) for \(0\le t<n^2/C\) and \(|\kappa_j(R_n)|/j!\le C^jn^{1-2j}\). Put \(\Delta_{r,n}=\kappa_r(T_n)-\kappa_r(R_n)\) and \((z)_r=z(z+1)\cdots(z+r-1)\). Let \(\rho=\frac12+i\gamma\) be a simple zero of \(\xi_4\), \(z_0=\rho/2\), and \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\,\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\,\pi^{m+1}\,\xi_4'(\rho)}\). Let \(z_n\) be the zero of \(M_n\) in a fixed small disc about \(z_0\) (unique and simple for large \(n\)). For the hard cutoff \(R_n=0\), \(z_n=z_0+\delta_0/n+O(n^{-2})\). For the mean correction \(R_n=\mathbb ET_n\) (a constant), \(z_n=z_0+\delta_1/n^3+O(n^{-5})\). For a remainder that satisfies the hypotheses above at order \(m\) and has \(\kappa_{m+1}(R_n)=o(n^{-(2m+1)})\) the displacement is \(\delta_mn^{-(2m+1)}(1+o(1))\).

Earlier editions: Corollary 28.2.3, page 1002 (edition 1).

Source: ch/tr1.tex, label tr:cor:named.

Remainders matching mean and variance but not the third cumulant give different drifts, so the formula of delta 2 needs the matching hypothesis

Volume II, Proposition 16.3.4, page 479 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Let \(c_n=\mathbb ET_n\), \(v_n=\operatorname{Var}T_n\), \(\rho=\frac12+i\gamma\) a simple zero of \(\xi_4\), \(z_0=\rho/2\), \(z_n\) the zero of \(M_n\) in a fixed small disc about \(z_0\) (unique and simple for large \(n\)), and \(\delta_2=-\frac{(z_0)_3\,\xi_4(\rho+6)}{30\,\pi^3\,\xi_4'(\rho)}\) with \((z_0)_3=z_0(z_0+1)(z_0+2)\). Let \(R_n\) have the gamma law with shape \(c_n^2/v_n\) and scale \(v_n/c_n\). Then \(R_n\) satisfies, for a constant \(C\) independent of \(n\) and \(j\), \(\log\mathbb Ee^{-tR_n}=\sum_{j\ge1}(-1)^j\kappa_j(R_n)t^j/j!\) for \(0\le t<n^2/C\) and \(|\kappa_j(R_n)|/j!\le C^jn^{1-2j}\), and \(z_n=z_0+\frac49\delta_2\,n^{-5}+O(n^{-7})\). A remainder \(R_n=G_{1,n}+G_{2,n}\), a sum of two independent gamma variables with the same mean and variance and the same natural scale, gives \(z_n=z_0-\frac19\delta_2\,n^{-5}+O(n^{-7})\). A bounded remainder \(R_n=b_n+hB_n\) with \(B_n\) a Bernoulli variable, \(h>0\) small and fixed, matched to the mean and variance, has \(z_n-z_0=c\,n^{-3}+O(n^{-5})\) with \(c\ne0\).

Earlier editions: Proposition 28.2.4, page 1004 (edition 1).

Source: ch/tr1.tex, label tr:prop:gamma.

The sign of the real part of the drift is the sign of minus sin Theta m times the derivative of the Hardy function of chi 4

Volume II, Theorem 16.4.1, page 480 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Put \(G(s)=(4/\pi)^{s/2}\Gamma(\frac{s+1}2)\), so that \(\xi_4=\frac2{\sqrt\pi}GL\), let \(\vartheta_\beta(x)=\arg G(\frac12+ix)\) and \(Z_\beta(x)=e^{i\vartheta_\beta(x)}L(\frac12+ix,\chi_4)\) (real), the Riemann Siegel function and the Hardy function of \(\chi_4\). Let \(\rho=\frac12+i\gamma\), \(\gamma>0\), be a simple zero of \(\xi_4\), \(z_0=\rho/2\), \((z)_r=z(z+1)\cdots(z+r-1)\), and for \(m\ge0\) let \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\,\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\,\pi^{m+1}\,\xi_4'(\rho)}\). Then \(\delta_m=\frac{i\,C_m(\gamma)e^{i\Theta_m(\gamma)}}{Z_\beta'(\gamma)}\) with \(C_m(\gamma)>0\) and \(\Theta_m(\gamma)=\vartheta_\beta(\gamma)+\arg L(\rho+2m+2,\chi_4)-\sum_{j=0}^m[\arctan\frac{1/2+2j}\gamma+\arctan\frac{3/2+2j}\gamma]\), so that \(\operatorname{sgn}\Re\delta_m=\operatorname{sgn}[-\sin\Theta_m(\gamma)Z_\beta'(\gamma)]\), zero included. For a remainder \(R_n\ge0\) independent of \(\omega_n\) with \(\kappa_j(R_n)=\kappa_j(T_n)\) for \(1\le j\le m\) such that, for a constant \(C\) independent of \(n\) and \(j\), \(\log\mathbb Ee^{-tR_n}=\sum_{j\ge1}(-1)^j\kappa_j(R_n)t^j/j!\) for \(0\le t<n^2/C\) and \(|\kappa_j(R_n)|/j!\le C^jn^{1-2j}\), and with \(\eta_m=\lim_n[\kappa_{m+1}(T_n)-\kappa_{m+1}(R_n)]/\kappa_{m+1}(T_n)\) the displacement \(z_n-z_0\) of the zero \(z_n\) of \(M_n\) near \(z_0\) is \(\eta_m\delta_mn^{-(2m+1)}(1+o(1))\) when the limit exists and \(\eta_m\ne0\), and the sign of the real part is \(\operatorname{sgn}[-\eta_m\sin\Theta_m(\gamma)Z_\beta'(\gamma)]\) when \(\eta_m\sin\Theta_m\ne0\).

Earlier editions: Theorem 28.3.1, page 1005 (edition 1).

Source: ch/tr1.tex, label tr:thm:phase.

L(s,chi 4) has 379 zeros below height 500, all simple and on the line (certified), with five anomalous Gram parities

Volume II, Result 16.4.1, page 481 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\vartheta_\beta\) and \(Z_\beta\) the Riemann Siegel and Hardy functions of \(\chi_4\) with \(\vartheta_\beta(x)=\arg[(4/\pi)^{s/2}\Gamma(\frac{s+1}2)]_{s=1/2+ix}\). Put \(Z_\beta(x)=e^{i\vartheta_\beta(x)}L(\frac12+ix,\chi_4)\) (real), \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\), and for a zero \(\rho\) with \(z_0=\rho/2\) and \((z)_r=z(z+1)\cdots(z+r-1)\) let \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\,\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\,\pi^{m+1}\,\xi_4'(\rho)}\), the drift coefficient of the zeros of the truncated Mellin transforms of the beta lattice (\(m=0\) for the hard cutoff, \(m=1\) for the mean correction). Locating the sign changes of \(Z_\beta\) on \(0<x<500\) on two independent grids (steps \(0.02\) and \(0.0137\)) and refining at two precisions gives \(379\) zeros, all simple and on the critical line (the minimal gap is \(0.1336\)), in agreement with \(\vartheta_\beta(500)/\pi=379.15\). The count and the signs of \(\Re\delta_m\) below are certified by the argument principle in ball arithmetic, without a Turing bound. Measuring the Gram parity by \(\lfloor\vartheta_\beta(\gamma)/\pi\rfloor\) minus the zero number, \(374\) zeros have offset \(-1\) and five do not, at \(\gamma=138.7502\), \(290.2521\), \(364.4197\), \(400.8264\) and \(422.5104\) (zero numbers \(78\), \(195\), \(258\), \(291\), \(309\)). The mean correction has \(\Re\delta_1<0\) exactly at these five zeros and at no other, and the hard cutoff has \(\Re\delta_0<0\) at the last four.

Earlier editions: Result 28.3.1, page 1005 (edition 1).

Source: ch/tr1.tex, label tr:res:gram.

Certified zeros of L(s, chi_4) below height 3000 and certified signs of the truncation drifts

Volume II, Proposition 16.4.2, page 482 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal character modulo \(4\) and \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\). For a simple zero \(\rho\) of \(\xi_4\) and \(z_0=\rho/2\) put \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\pi^{m+1}\xi_4'(\rho)}\) with \((z)_k=z(z+1)\cdots(z+k-1)\). Every zero \(\rho\) of \(L(s,\chi_4)\) with \(|\operatorname{Im}\rho|<500\) and \(|\operatorname{Re}\rho-\frac12|<\frac32\) is simple and lies on the critical line, and there are \(379\) of them with positive ordinate. The same holds below height \(3000\), with \(3130\) zeros. For each of these zeros and each \(m\le5\) the sign of \(\operatorname{Re}\delta_m\) is determined, and the numbers of zeros with \(\operatorname{Re}\delta_m<0\) for \(m=0,\dots,5\) are \(4,5,8,11,19,29\) below height \(500\) and \(72,93,95,97,103,115\) below height \(3000\). In particular \(\operatorname{Re}\delta_1<0\) at \(\gamma=138.7502\). The result is certified in ball arithmetic by the argument principle on the rectangle \([-1,2]\times[-T,T]\). It is not new in substance as to the generalised Riemann Hypothesis for \(\chi_4\), which is verified to much greater height in the literature; the new content is the self contained certificate and the first certified list of the signs of \(\operatorname{Re}\delta_m\).

Source: ch/tr1.tex, label tr:prop:certify.

One sided criterion fails at order m if some critical zero is approached from the left

Volume II, Corollary 16.4.3, page 483 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Fix \(m\ge0\) and a remainder \(R_n\ge0\) independent of \(\omega_n\) with \(\kappa_j(R_n)=\kappa_j(T_n)\) for \(1\le j\le m\) such that, for a constant \(C\) independent of \(n\) and \(j\), \(\log\mathbb Ee^{-tR_n}=\sum_{j\ge1}(-1)^j\kappa_j(R_n)t^j/j!\) for \(0\le t<n^2/C\) and \(|\kappa_j(R_n)|/j!\le C^jn^{1-2j}\), for which \(\eta_m=\lim_n[\kappa_{m+1}(T_n)-\kappa_{m+1}(R_n)]/\kappa_{m+1}(T_n)\) exists, and for a simple zero \(\rho\) of \(\xi_4\) let \(z_0=\rho/2\), \((z)_r=z(z+1)\cdots(z+r-1)\) and \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\,\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\,\pi^{m+1}\,\xi_4'(\rho)}\). If some simple zero \(\rho=\frac12+i\gamma\) of \(\xi_4\) on the critical line has \(\Re(\eta_m\delta_m)<0\), then for all large \(n\) the function \(M_n\) has a zero in \(0<\Re z<\frac14\), so no criterion of the form that \(M_n\) has no zero in \(0<\Re z<\frac14\) for infinitely many \(n\) can hold at order \(m\) for this remainder.

Earlier editions: Corollary 28.3.2, page 1006 (edition 1).

Source: ch/tr1.tex, label tr:cor:nogo.

Infinitely many Gram anomalies with a margin force infinitely many zeros with negative real drift

Volume II, Proposition 16.4.4, page 483 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Put \(G(s)=(4/\pi)^{s/2}\Gamma(\frac{s+1}2)\), so that \(\xi_4=\frac2{\sqrt\pi}GL\), let \(\vartheta_\beta(x)=\arg G(\frac12+ix)\) and \(Z_\beta(x)=e^{i\vartheta_\beta(x)}L(\frac12+ix,\chi_4)\) (real), the Riemann Siegel function and the Hardy function of \(\chi_4\). For a simple zero \(\rho=\frac12+i\gamma\), \(\gamma>0\), of \(\xi_4\) let \(z_0=\rho/2\), \((z)_r=z(z+1)\cdots(z+r-1)\), \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\,\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\,\pi^{m+1}\,\xi_4'(\rho)}\) and \(\Theta_m(\gamma)=\vartheta_\beta(\gamma)+\arg L(\rho+2m+2,\chi_4)-\sum_{j=0}^m[\arctan\frac{1/2+2j}\gamma+\arctan\frac{3/2+2j}\gamma]\). Let \(\sigma=2m+\frac52\), \(\epsilon_\sigma=\sum_{k\ge1}(2k+1)^{-\sigma}<1\) and \(B_m(\gamma)=\arcsin\epsilon_\sigma+\sum_{j=0}^m[\arctan\frac{1/2+2j}\gamma+\arctan\frac{3/2+2j}\gamma]\). Then \(|\Theta_m(\gamma)-\vartheta_\beta(\gamma)|\le B_m(\gamma)\). If infinitely many simple critical zeros satisfy \(-\sin\vartheta_\beta(\gamma)Z_\beta'(\gamma)<0\) and \(\operatorname{dist}(\vartheta_\beta(\gamma),\pi\mathbb Z)>B_m(\gamma)\), then infinitely many zeros have \(\Re\delta_m<0\).

Earlier editions: Proposition 28.3.3, page 1007 (edition 1).

Source: ch/tr1.tex, label tr:prop:gram.

If the symmetrised truncations have only critical zeros for infinitely many n then GRH holds for chi 4

Volume II, Theorem 16.6.1, page 485 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Put \(g_n(z)=(2\pi)^{-z}M_n(z)\) and \(F_n(z)=g_n(z)+g_n(\frac12-z)\), for remainders with \(R_n\to0\) in probability. If for infinitely many \(n\) all zeros of \(F_n\) in \(0<\Re z<\frac12\) lie on \(\Re z=\frac14\), then the Generalised Riemann Hypothesis holds for \(\chi_4\).

Earlier editions: Theorem 28.5.1, page 1009 (edition 1).

Source: ch/tr1.tex, label tr:thm:symhurwitz.

Each simple critical zero is the limit of exactly one zero of the symmetrised truncation, and that zero is critical

Volume II, Proposition 16.6.2, page 485 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_4\) be the nonprincipal Dirichlet character modulo \(4\), \(L(s,\chi_4)\) the Dirichlet beta function, \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) its completion (entire and symmetric about \(s=\frac12\)), \(\lambda_*=2\pi^2\), \(\lambda_k=\lambda_*(k-\frac12)^2\), \(\omega=\sum_{k\ge1}e_k/\lambda_k\) with independent standard exponential variables \(e_k\) (so \(\mathbb Ee^{-\lambda\omega}=\operatorname{sech}\sqrt{\lambda/2}\)), \(\omega_n=\sum_{k\le n}e_k/\lambda_k\) and \(T_n=\omega-\omega_n\) the tail with cumulants \(\kappa_j(T_n)=(j-1)!\,q_j(n)\), \(q_j(n)=\sum_{k>n}\lambda_k^{-j}\). The Mellin transform is \(M(z)=\mathbb E\,\omega^{-z}=(2\pi)^z\xi_4(2z)\), and the nontrivial zeros of \(L(s,\chi_4)\) are the points \(z=\rho/2\) of the strip \(0<\Re z<\frac12\) where \(M\) vanishes. A truncation is a random variable \(R_n\ge0\) independent of \(\omega_n\), with \(M_n(z)=\mathbb E(\omega_n+R_n)^{-z}\). Put \(g_n(z)=(2\pi)^{-z}M_n(z)\) and \(F_n(z)=g_n(z)+g_n(\frac12-z)\), for remainders with \(R_n\to0\) in probability. Every simple zero of \(\xi_4(2z)\) on \(\Re z=\frac14\) is, for all large \(n\), the limit of exactly one zero of \(F_n\) in a small disc about it, and that zero lies on \(\Re z=\frac14\).

Earlier editions: Proposition 28.5.2, page 1009 (edition 1).

Source: ch/tr1.tex, label tr:prop:symlocal.

Does the window of the shifted Williams family tend to infinity (open problem)

Volume II, Problem 16.12, page 489 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(S=\sum_{n\ge1}\frac{2}{\pi^2n^2}G_n\) with independent \(G_n\sim\mathrm{Gamma}(2,1)\), so that \(2\xi(z)=\mathbb{E}[(\pi S/2)^{z/2}]\), where \(\xi(z)=\frac12z(z-1)\pi^{-z/2}\Gamma(z/2)\zeta(z)\), let \(S_N\) be the sum of the terms with \(n\le N\) and \(d_N=\mathbb{E}(S-S_N)=\frac4{\pi^2}\sum_{n>N}n^{-2}\), and let \(\xi_N(z)=\frac12\mathbb{E}[(\pi(S_N+d_N)/2)^{z/2}]\) be the shifted Williams approximant. Put \(F_N(y)=(\xi_N(\frac12+iy)+\xi_N(\frac12-iy))/(2\xi_N(\frac12))\) and \(T_N=\inf\{|y|:F_N(y)=0,\ \operatorname{Im}y\ne0\}\). Decide whether \(T_N\to\infty\), for example by showing that \(\xi_N\) has no zero to the right of \(\frac12\) below a height growing with \(N\). Granted that \(F_N\) converges locally uniformly to \(\Xi/\Xi(0)\), where \(\Xi(y)=\xi(\frac12+iy)\), this is equivalent to the Riemann Hypothesis when all zeros of \(\Xi\) are simple (Hurwitz’s theorem). An argument that uses only the Gamma convolution structure, kernel positivity or the modularity of the theta function would equally prove the Riemann Hypothesis for the Davenport and Heilbronn function, so the Euler product must enter. The computed values at \(N=5,10\) are \(T_N=30.8\) and \(59.8\).

Source: ch/tr1.tex, label ht:prob:williams.

Hybrid Gamma and theta approximants of Xi with a faster growing window (open problem)

Volume II, Problem 16.13, page 489 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(T_N\) be the height of the first nonreal zero of the even real entire approximants \(F_N\) of \(\Xi/\Xi(0)\) (where \(\Xi\) is Riemann’s Xi function on the line \(\operatorname{Re}s=\frac12\)). Find approximants whose window \(T_N\) grows faster than that of the shifted Williams family (the approximants \(F_N\) built from the Gamma sum truncated at \(N\) terms with its tail replaced by its mean), built as Gamma sums for large values of the variable and from the theta expansion for small values, and determine whether any such hybrid keeps a positivity property that controls \(T_N\). Any such hybrid must carry the Euler product, because positivity of the kernel, the functional equation and the theta transformation hold for the Davenport and Heilbronn function, which has zeros off the critical line.

Source: ch/tr1.tex, label ht:prob:hybrid.

The Hermite coefficients of Xi are generated by the Gaussian smoothing G a, with an explicit moment formula

Volume II, Theorem 16.8.1, page 491 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the P'olya kernel, positive and even, and \(\Xi(t)=\int_{\mathbb R}\Phi(u)\cos(tu)\,du\) the Riemann Xi function. Let \(h_n(x)=\pi^{-1/4}(2^nn!)^{-1/2}H_n(x)e^{-x^2/2}\) be the orthonormal Hermite functions, with \(H_n\) the physicists’ Hermite polynomials. For a scale \(a>0\) put \(c_n(a)=\sqrt a\int_{\mathbb R}\Phi(u)h_n(au)\,du\), \(t_m=a\sqrt{4m+1}\) and \(b_m(a)=\sqrt{2\pi}\,(-1)^mc_{2m}(a)=\int_{\mathbb R}\Xi(t)h_{2m}(t/a)a^{-1/2}\,dt\). Let \(G_a(\tau)=\int_{\mathbb R}\Phi(u)e^{-a^2u^2/2}\cos(\tau u)\,du\) (the Gaussian smoothing of \(\Xi\) of variance \(a^2\)) and \(\nu_k(a)=\int\Phi(u)e^{-a^2u^2/2}u^{2k}du>0\). Then for every \(a>0\), \(\sum_{m\ge0}b_m(a)\frac{z^{2m}}{\sqrt{(2m)!}}=\sqrt{2\pi}\,\pi^{-1/4}\sqrt a\;e^{z^2/2}\,G_a(\sqrt2\,az)\), and \(b_m(a)=\sqrt{2\pi}\,\pi^{-1/4}\sqrt a\,\sqrt{(2m)!}\sum_{k=0}^m\frac{(-1)^k2^ka^{2k}\nu_k(a)}{(2k)!\,2^{m-k}(m-k)!}\).

Earlier editions: Theorem 28.7.1, page 1013 (edition 1).

Source: ch/tr2.tex, label tr:thm:heat.

The normalised Hermite coefficient equals Xi at the turning point up to an explicit error of order a to the fourth

Volume II, Theorem 16.9.1, page 492 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the P'olya kernel, positive and even, and \(\Xi(t)=\int_{\mathbb R}\Phi(u)\cos(tu)\,du\) the Riemann Xi function. Let \(h_n(x)=\pi^{-1/4}(2^nn!)^{-1/2}H_n(x)e^{-x^2/2}\) be the orthonormal Hermite functions, with \(H_n\) the physicists’ Hermite polynomials. For a scale \(a>0\) put \(c_n(a)=\sqrt a\int_{\mathbb R}\Phi(u)h_n(au)\,du\), \(t_m=a\sqrt{4m+1}\) and \(b_m(a)=\sqrt{2\pi}\,(-1)^mc_{2m}(a)=\int_{\mathbb R}\Xi(t)h_{2m}(t/a)a^{-1/2}\,dt\). Let \(A_m(a)=\sqrt2\,\pi^{1/4}\sqrt a\sqrt{(2m)!}/(2^mm!)\) and \(M_k(a_0)=\int_{\mathbb R}\Phi(u)|u|^ke^{a_0^2u^2/2}du\). Then for every \(a_0>0\), \(0<a\le a_0\) and integer \(m\ge0\), \(|b_m(a)/A_m(a)-\Xi(t_m)|\le\frac{M_4(a_0)}{12}a^4\) and \(b_m(a)/A_m(a)=\Xi(t_m)+a^4Q(t_m)+R_m(a)\) with \(|R_m(a)|\le\frac{M_8(a_0)}{672}a^8\), where \(Q(t)=\frac{\Xi'''(t)}{6t}-\frac{\Xi''(t)}{4t^2}+\frac{\Xi'(t)}{4t^3}\) for \(t\ne0\) and \(Q(0)=\frac{\Xi^{(4)}(0)}{12}=\frac1{12}\int_{\mathbb R}\Phi(u)u^4du\) is an even entire function.

Earlier editions: Theorem 28.8.1, page 1014 (edition 1).

Source: ch/tr2.tex, label tr:thm:herm.

At a real zero of Xi the Hermite coefficient at the matching scale does not vanish for large m

Volume II, Proposition 16.9.2, page 494 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the P'olya kernel, positive and even, and \(\Xi(t)=\int_{\mathbb R}\Phi(u)\cos(tu)\,du\) the Riemann Xi function. Let \(h_n(x)=\pi^{-1/4}(2^nn!)^{-1/2}H_n(x)e^{-x^2/2}\) be the orthonormal Hermite functions, with \(H_n\) the physicists’ Hermite polynomials. For a scale \(a>0\) put \(c_n(a)=\sqrt a\int_{\mathbb R}\Phi(u)h_n(au)\,du\), \(t_m=a\sqrt{4m+1}\) and \(b_m(a)=\sqrt{2\pi}\,(-1)^mc_{2m}(a)=\int_{\mathbb R}\Xi(t)h_{2m}(t/a)a^{-1/2}\,dt\). Let \(A_m(a)=\sqrt2\,\pi^{1/4}\sqrt a\sqrt{(2m)!}/(2^mm!)\) and \(M_k(a_0)=\int_{\mathbb R}\Phi(u)|u|^ke^{a_0^2u^2/2}du\). Let \(Q(t)=\frac{\Xi'''(t)}{6t}-\frac{\Xi''(t)}{4t^2}+\frac{\Xi'(t)}{4t^3}\) for \(t\ne0\). At every positive real zero \(\gamma\) of \(\Xi\), of any multiplicity, \(b_m(a_m)\ne0\) for all large \(m\), where \(a_m=\gamma/\sqrt{4m+1}\). If \(Q(\gamma)\ne0\) then \(b_m(a_m)=\frac{2Q(\gamma)}{\sqrt\gamma}a_m^5+O(a_m^9)\). So \(b_m(a)=\sqrt{2a}\,m^{-1/4}\,\Xi(a\sqrt{4m+1})(1+O(a^2))\) fails at the zeros of \(\Xi\).

Earlier editions: Proposition 28.8.2, page 1016 (edition 1).

Source: ch/tr2.tex, label tr:prop:zeros.

The index of the first negative Hermite coefficient satisfies 4 a squared m star tends to gamma 1 squared

Volume II, Corollary 16.10.1, page 494 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the P'olya kernel, positive and even, and \(\Xi(t)=\int_{\mathbb R}\Phi(u)\cos(tu)\,du\) the Riemann Xi function. Let \(h_n(x)=\pi^{-1/4}(2^nn!)^{-1/2}H_n(x)e^{-x^2/2}\) be the orthonormal Hermite functions, with \(H_n\) the physicists’ Hermite polynomials. For a scale \(a>0\) put \(c_n(a)=\sqrt a\int_{\mathbb R}\Phi(u)h_n(au)\,du\), \(t_m=a\sqrt{4m+1}\) and \(b_m(a)=\sqrt{2\pi}\,(-1)^mc_{2m}(a)=\int_{\mathbb R}\Xi(t)h_{2m}(t/a)a^{-1/2}\,dt\). Let \(Q(t)=\frac{\Xi'''(t)}{6t}-\frac{\Xi''(t)}{4t^2}+\frac{\Xi'(t)}{4t^3}\) (\(t\ne0\)), \(\gamma_1=14.134725\ldots\) the first positive zero of \(\Xi\) (simple), \(m^*(a)=\min\{m\ge0:b_m(a)<0\}\), and \(y_{z,t}\) the even solution of \(y''+(t^2-zu^2)y=0\) with \(y(0)=1\), \(y'(0)=0\). Then for every \(0<\delta<\gamma_1\) and \(a\) small, \(b_m(a)>0\) for all \(m\) with \(t_m\le\gamma_1-\delta\), the index \(m^*(a)\) exists, and \(\lim_{a\to0}4a^2m^*(a)=\gamma_1^2\). If \(\gamma_a\) is the unique zero near \(\gamma_1\) of \(t\mapsto\int\Phi(u)y_{a^4,t}(u)du\), then \(\gamma_a=\gamma_1-a^4Q(\gamma_1)/\Xi'(\gamma_1)+O(a^8)\) and \(m^*(a)=\lfloor(\gamma_a^2/a^2-1)/4\rfloor+1\) for small \(a\).

Earlier editions: Corollary 28.9.1, page 1016 (edition 1).

Source: ch/tr2.tex, label tr:cor:first.

A verified sign change count of the Hermite coefficients together with an upper bound on the zero count proves real and simple zeros

Volume II, Proposition 16.11.2, page 497 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the P'olya kernel, positive and even, and \(\Xi(t)=\int_{\mathbb R}\Phi(u)\cos(tu)\,du\) the Riemann Xi function. Let \(h_n(x)=\pi^{-1/4}(2^nn!)^{-1/2}H_n(x)e^{-x^2/2}\) be the orthonormal Hermite functions, with \(H_n\) the physicists’ Hermite polynomials. For a scale \(a>0\) put \(c_n(a)=\sqrt a\int_{\mathbb R}\Phi(u)h_n(au)\,du\), \(t_m=a\sqrt{4m+1}\) and \(b_m(a)=\sqrt{2\pi}\,(-1)^mc_{2m}(a)=\int_{\mathbb R}\Xi(t)h_{2m}(t/a)a^{-1/2}\,dt\). Let \(A_m(a)=\sqrt2\,\pi^{1/4}\sqrt a\sqrt{(2m)!}/(2^mm!)\) and \(M_k(a_0)=\int_{\mathbb R}\Phi(u)|u|^ke^{a_0^2u^2/2}du\). Let \(\widetilde b_m/A_m\) be computed values of \(b_m/A_m\) with rigorous errors \(e_m\), and \(E_m\) a rigorous bound for \(|b_m/A_m-\Xi(t_m)|\) (for example \(M_4(a_0)a^4/12\)). If \(|\widetilde b_m/A_m|>e_m+E_m\) then \(\operatorname{sgn}\widetilde b_m=\operatorname{sgn}\Xi(t_m)\). Suppose there are \(S\) sign changes at pairs of adjacent sampling points \(t_m\) below \(T\), all with this property. Then \(\Xi\) has \(S\) zeros in disjoint intervals, and if an independent rigorous method gives that the number \(N(T)\) of zeros \(t\) of \(\Xi\) with \(0<\Re t<T\) (the zeros \(\frac12+it\) of \(\zeta\) with ordinate in \((0,T)\)), counted with multiplicity, is at most \(S\) (Turing’s method), then all zeros \(t\) of \(\Xi\) with \(0<\Re t<T\) are real and simple.

Earlier editions: Proposition 28.10.2, page 1019 (edition 1).

Source: ch/tr2.tex, label tr:prop:verified.

If each simple zero gives one sign change the mean run of the Hermite coefficient signs is asymptotic to pi T over a squared log of T over 2 pi

Volume II, Proposition 16.12.1, page 498 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the zeros \(t\) of \(\Xi\) with \(T\le\Re t\le T+H\) are real and simple and the sign changes of \((b_m(a))\) with \(t_m\in[T,T+H]\) correspond one to one to them.

Statement. Let \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) be the P'olya kernel, positive and even, and \(\Xi(t)=\int_{\mathbb R}\Phi(u)\cos(tu)\,du\) the Riemann Xi function. Let \(h_n(x)=\pi^{-1/4}(2^nn!)^{-1/2}H_n(x)e^{-x^2/2}\) be the orthonormal Hermite functions, with \(H_n\) the physicists’ Hermite polynomials. For a scale \(a>0\) put \(c_n(a)=\sqrt a\int_{\mathbb R}\Phi(u)h_n(au)\,du\), \(t_m=a\sqrt{4m+1}\) and \(b_m(a)=\sqrt{2\pi}\,(-1)^mc_{2m}(a)=\int_{\mathbb R}\Xi(t)h_{2m}(t/a)a^{-1/2}\,dt\). Let \(N(T)\) be the number of zeros \(t\) of \(\Xi\) with \(0<\Re t<T\) (the zeros \(\frac12+it\) of \(\zeta\) with ordinate in \((0,T)\)), counted with multiplicity. Assume the zeros \(t\) of \(\Xi\) with \(T\le\Re t\le T+H\) are real and simple and the sign changes of \((b_m(a))\) with \(t_m\in[T,T+H]\) correspond one to one to them. Then the mean number of coefficient steps per sign change is the number \(TH/(2a^2)+O(H^2/a^2+1)\) of grid points in the window divided by \(N(T+H)-N(T)=\frac H{2\pi}\log\frac T{2\pi}+O(H^2/T+\log T)\), and for fixed \(a\), \(H\to\infty\) and \(H=o(T)\) the ratio is asymptotic to \(\pi T/(a^2\log(T/2\pi))\).

Earlier editions: Proposition 28.11.1, page 1020 (edition 1).

Source: ch/tr2.tex, label tr:prop:run.

Gram anomalies with margin occur infinitely often for L(s, chi_4) (open problem)

Volume II, Problem 16.14, page 500 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\vartheta_\beta(x)=\arg G(\frac12+ix)\) with \(G(s)=(4/\pi)^{s/2}\Gamma(\frac{s+1}2)\) and let \(Z_\beta(x)=e^{i\vartheta_\beta(x)}L(\frac12+ix,\chi_4)\) be the Hardy function of \(\chi_4\). For \(\sigma=2m+\frac52\), \(\epsilon_\sigma=\sum_{k\ge1}(2k+1)^{-\sigma}\) and \(B_m(\gamma)=\arcsin\epsilon_\sigma+\sum_{j=0}^m[\arctan\frac{1/2+2j}\gamma+\arctan\frac{3/2+2j}\gamma]\), if infinitely many simple critical zeros \(\gamma\) satisfy \(-\sin\vartheta_\beta(\gamma)Z_\beta'(\gamma)<0\) and \(\operatorname{dist}(\vartheta_\beta(\gamma),\pi\mathbb{Z})>B_m(\gamma)\), then infinitely many zeros have \(\operatorname{Re}\delta_m<0\), where, for a simple zero \(\rho\) and \(z_0=\rho/2\), \(\delta_m=\frac{(-1)^{m+1}(z_0)_{m+1}\xi_4(\rho+2m+2)}{2(m+1)(2m+1)\pi^{m+1}\xi_4'(\rho)}\) with \(\xi_4(s)=(4/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi_4)\) is the drift of the \(m\)th order truncation of the beta lattice. Problem: prove that such Gram anomalies with margin occur infinitely often for \(L(s,\chi_4)\), adapting the arguments known for zeta. Together with the implication this would make the no one sided criterion (that the \(m\)th order truncation \(M_n\) has no zero in \(0<\operatorname{Re}z<\frac14\) for infinitely many \(n\)) unconditional at each order \(m\).

Source: ch/tr2.tex, label tr:prob:gram.

The symmetrised hard cutoff F_8 for chi_4 has only critical zeros (open problem)

Volume II, Problem 16.15, page 500 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(F_8(z)=g_8(z)+g_8(\frac12-z)\) with \(g_8(z)=(2\pi)^{-z}\mathbb{E}\,\omega_8^{-z}\), \(\omega_8=\sum_{k\le8}e_k/\lambda_k\) and \(e_k\) independent standard exponentials. Prove that \(F_8\), the symmetrised hard cutoff with eight atoms of the beta lattice \(\lambda_k=2\pi^2(k-\frac12)^2\) associated with \(L(s,\chi_4)\), has only critical zeros, by combining the phase of \(g_8\) at low height with a modulus inequality and a Newton polygon bound at large height.

Source: ch/tr2.tex, label tr:prob:F8.

Largest n with only critical zeros for the symmetrised mean corrected truncation (open problem)

Volume II, Problem 16.16, page 500 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Determine the largest \(n\) for which the symmetrised mean corrected truncation (with \(n\) atoms of the beta lattice and the constant remainder \(c_n=\sum_{k>n}\lambda_k^{-1}\)) has only critical zeros in the strip \(0<\operatorname{Re}z<\frac12\) (in the variable \(z=\rho/2\)), and decide whether the first failure is related to the first Gram anomaly of \(L(s,\chi_4)\) at \(\gamma=138.75\).

Source: ch/tr2.tex, label tr:prob:largest.

A truncation remainder whose drift sign is independent of Gram parity (open problem)

Volume II, Problem 16.17, page 500 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For the truncations of the beta lattice of \(L(s,\chi_4)\), the side from which a truncated zero approaches the critical line is set by the Gram parity through the phase \(\vartheta_\beta(\gamma)\) of the gamma factor \(G(s)=(4/\pi)^{s/2}\Gamma(\frac{s+1}2)\). Problem: find a remainder \(R_n\), not a cumulant match, whose drift \(\eta\delta_m\) has a sign independent of the Gram parity. This needs a correction whose phase cancels \(\vartheta_\beta(\gamma)\), which suggests building \(R_n\) from the functional equation and not from the tail.

Source: ch/tr2.tex, label tr:prob:remainder.

Orders m at least one for the zeta lattice truncations and Gram’s law (open problem)

Volume II, Problem 16.18, page 500 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Carry out for zeta, with the lattice of atoms \(k^2\) used in the book for the truncations of \(\xi\), the analysis of the drift at orders \(m\ge1\) (the case \(m=0\) being known exactly), and compare the Gram parity of the corrected phase with the classical failures of Gram’s law for zeta, the first near the Gram point \(g_{126}\approx282.455\), where \(Z(g_{126})=-0.0276\).

Source: ch/tr2.tex, label tr:prob:zeta.

Volume II, Chapter 17. Friable approximants of Xi after Miyatake and the cosh ladder

The symmetrised finite Euler product of Xi has zeros near the lines of its poles with real part tending to one

Volume II, Proposition 17.1.5, page 504 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P\ge2\), let \(\zeta_P(s)=\prod_{p\le P}(1-p^{-s})^{-1}\) be the Euler product over the primes \(p\le P\), \(\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)\), \(E_P(s)=\frac12s(s-1)\Gamma_{\mathbb{R}}(s)\zeta_P(s)\) and \(F_P(s)=E_P(s)+E_P(1-s)\), a function with poles on \(\operatorname{Re}s=0\) and \(\operatorname{Re}s=1\). Let \(p\le P\) be prime. For infinitely many integers \(k\), \(F_P\) has a zero \(s_k\) with \(\big|s_k-1-\frac{2\pi ik}{\log p}\big|=O(|k|^{-1/2})\), and for \(P=2\) every \(k\) with \(|k|\) large enough qualifies. So \(\operatorname{Re}s_k\to1\), and the zeros \(1-\bar s_k\) have real parts tending to \(0\).

Earlier editions: Proposition 22.1.5, page 801 (edition 1).

Source: ch/mi1.tex, label mi:prop:polezeros.

The friable approximant of Xi is exponentially close to Xi in a strip, with the accuracy set by the first omitted prime

Volume II, Theorem 17.2.1, page 505 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. Let \(\delta_P=\sum_{m\notin\mathcal F(P)}e^{-\pi m^2}/(\pi m^2)\). For \(|\operatorname{Im}t|\le\frac32\), \(|\Xi_P(t)-\Xi(t)|\le|t^2+\frac14|\,\delta_P\), and \(\delta_P\le e^{-\pi q^2}/(\pi q^2(1-e^{-2\pi q}))\).

Earlier editions: Theorem 22.2.1, page 801 (edition 1).

Source: ch/mi1.tex, label mi:thm:approx.

Certified tracking: a Rouche criterion for a real zero of the friable approximant near a simple zero of Xi

Volume II, Corollary 17.2.2, page 505 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. Let \(\delta_P=\sum_{m\notin\mathcal F(P)}e^{-\pi m^2}/(\pi m^2)\). If \(\gamma\) is a simple zero of \(\Xi\) and \(|\Xi(t)|>|t^2+\frac14|\delta_P\) on the circle \(|t-\gamma|=r\le\frac32\), then \(\Xi_P\) has exactly one zero \(\gamma_P\) in the disc \(|t-\gamma|<r\), it is real, and \(\gamma_P-\gamma=-(\Xi_P-\Xi)(\gamma)/\Xi'(\gamma)\,(1+O(\eta))\), where \(\eta\) is the ratio of \(\max|\Xi_P-\Xi|\) on the disc to \(\min|\Xi|\) on the circle.

Earlier editions: Corollary 22.2.2, page 802 (edition 1).

Source: ch/mi1.tex, label mi:cor:rouche.

The folded kernel of the friable approximant has a kink of exact size c_P, so Xi_P tends to c_P on the real line and has finitely many real zeros

Volume II, Theorem 17.2.3, page 506 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. Fold \(\Xi_P\) as \(\Xi_P(t)=(t^2+\frac14)\int_0^\infty k_P(u)\cos(tu)\,du\) with kernel \(k_P(u)=e^{-u/2}-2e^{u/2}\psi_P(e^{2u})\) and put \(c_P=-k_P'(0)\). Then \(c_P=\sum_{m\notin\mathcal F(P)}(4\pi m^2-1)e^{-\pi m^2}=4\pi q^2e^{-\pi q^2}(1+O(q^{-2}))>0\), and \(\Xi_P(t)=c_P+O_P(t^{-2})\) as \(t\to\infty\) along \(\mathbb{R}\). Hence \(\Xi_P\) has finitely many real zeros.

Earlier editions: Theorem 22.2.3, page 802 (edition 1).

Source: ch/mi1.tex, label mi:thm:kink.

No reweighting of Riemann’s theta series by summable weights has infinitely many real zeros

Volume II, Theorem 17.2.4, page 507 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. For real weights \(w=(w_n)\ne0\) with \(\sum_n|w_n|n^{-\sigma}<\infty\) for every \(\sigma>0\) (this includes the weights \(1_{\mathcal F(P)}\), for which the sum is \(\prod_{p\le P}(1-p^{-\sigma})^{-1}\), and so \(\Xi_P\)), put \(\psi_w(x)=\sum_nw_ne^{-\pi n^2x}\) and \(\Xi_w(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_w(e^{2u})e^{u/2}\cos(tu)\,du\). Then \(\Xi_w\) is even, entire of order at most one, and has finitely many real zeros and infinitely many nonreal zeros.

Earlier editions: Theorem 22.2.4, page 802 (edition 1).

Source: ch/mi1.tex, label mi:thm:noproxy.

Theta series truncated to N terms of Xi: constants, real zeros and nonreal zeros for N equal to 1 and 2 (computed)

Volume II, Result 17.2, page 507 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\psi_N(x)=\sum_{n\le N}e^{-\pi n^2x}\) and \(\Xi_w(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_N(e^{2u})e^{u/2}\cos(tu)\,du\), the weights being \(w_n=1\) for \(n\le N\) and \(0\) otherwise; this is even and entire of order at most one, with finitely many real zeros and infinitely many nonreal zeros, and it tends to \(c_E=\sum_{m>N}(4\pi m^2-1)e^{-\pi m^2}>0\) on \(\mathbb{R}\). Computed: \(c_E=1.71806\cdot10^{-4}\) for \(N=1\) and \(c_E=5.89126\cdot10^{-11}\) for \(N=2\). In \(0<t\le60\) the function \(\Xi_w\) has \(2\) real zeros for \(N=1\), at \(14.1659\) and \(19.5639\), and \(6\) for \(N=2\), at \(14.1347\), \(21.0220\), \(25.0109\), \(30.4243\), \(32.9388\) and \(37.4954\), against the \(13\) zeros of the Riemann Xi function \(\Xi(t)=\xi(\frac12+it)\) there. Its nonreal zeros with \(0<\operatorname{Re}t\le60\) and \(0<\operatorname{Im}t\le12\) are \(22.915+4.666i\) and \(27.930+9.028i\) for \(N=1\), and \(41.190+1.462i\), \(45.418+4.421i\), \(49.549+7.364i\) and \(53.720+10.074i\) for \(N=2\). These values come from the argument principle and are a computation at these two values of \(N\) and this range only. The computation follows a note of Nicholas Polson merged into the book.

Earlier editions: Result 22.2, page 803 (edition 1).

Source: ch/mi1.tex, label mi:res:trunc.

The deficit Xi_P minus Xi is positive on the real line and decreases as the prime bound grows

Volume II, Theorem 17.3.1, page 508 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. For all real \(t\), \(D_P(t)>0\), and \(D_P>D_{P'}\) when \(P<P'\) are primes. Hence \(\Xi_P>\Xi_{P'}>\Xi\) on \(\mathbb{R}\).

Earlier editions: Theorem 22.3.1, page 804 (edition 1).

Source: ch/mi1.tex, label mi:thm:positive.

The transform of no theta term vanishes on the real line, and the Laplace law is a mixture of the Polya law and a pole law

Volume II, Result 17.3.2, page 508 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), \(\Xi(t)=\xi(\frac12+it)\), \(\psi(x)=\sum_{n\ge1}e^{-\pi n^2x}\) for \(x>0\), and for \(n\ge1\) and complex \(t\) let \(C_{\pi n^2}(t)=2\int_0^\infty e^{u/2}e^{-\pi n^2e^{2u}}\cos(tu)\,du\) and \(W(t)=4\int_0^\infty\psi(e^{2u})e^{u/2}\cos(tu)\,du=2\sum_{n\ge1}C_{\pi n^2}(t)\). Then \(2\Xi(t)=1-(t^2+\frac14)W(t)\) for every complex \(t\), and \(C_{\pi n^2}(t)>0\) for every \(n\ge1\) and every real \(t\), so \(W>0\) on \(\mathbb{R}\). Hence the zeros of \(\Xi\) in \(\mathbb{C}\) are exactly the solutions of \((t^2+\frac14)W(t)=1\), and the Riemann Hypothesis says that every solution is real: the transform of no term of the theta series vanishes on the real line, and the zeros of \(\Xi\) come from the cancellation between the pole term and the whole series. Further, \(C_{\pi n^2}(t)/C_{\pi n^2}(0)\) is the characteristic function of the symmetric density proportional to \(e^{|u|/2}\exp(-\pi n^2e^{2|u|})\), the law of \(\pm\frac12\log X\) with an independent fair sign, where \(X\) has density proportional to \(x^{-3/4}e^{-\pi n^2x}\) on \([1,\infty)\). So \(W(t)/W(0)\) is the characteristic function of the mixture of these laws with weights \(2C_{\pi n^2}(0)/W(0)\), with density \(p_V(u)=2e^{|u|/2}\psi(e^{2|u|})/W(0)\), where \(W(0)=2\sum_{n\ge1}(\pi n^2)^{-1/4}\Gamma(\frac14,\pi n^2)=0.0230338\) (computed). Let \(\Phi(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\), an even function with \(\int_{\mathbb{R}}\Phi=\xi(\frac12)\), so that \(\Phi/\xi(\frac12)\) is the Polya density with characteristic function \(\Xi(t)/\Xi(0)\), and \(\varkappa(u)=\frac14e^{-|u|/2}\) the Laplace density. Then \(\varkappa=(1-\upsilon)\,\varkappa\ast\frac{\Phi}{\xi(1/2)}+\upsilon\,p_V\) with \(\upsilon=W(0)/4=1-2\xi(\frac12)=0.00575844\) (computed): the Laplace law is the mixture, with weights \(1-\upsilon\) and \(\upsilon\), of the Polya law plus an independent Laplace variable and of the law \(p_V\). The result is from notes of Nicholas Polson merged into the book.

Earlier editions: Result 22.3.2, page 804 (edition 1).

Source: ch/mi1.tex, label mi:res:laws.

Uniform Lorentzian bound: the deficit of the friable approximant is a sum of Lorentzians with an explicit uniform relative error

Volume II, Theorem 17.3.4, page 509 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(a\ge9\pi\) put \(b=a-\frac14\), \(f_a(w)=e^{-ae^w}e^{w/4}\), \(C_a(t)=\int_0^\infty f_a(w)\cos(tw/2)\,dw\) and \(\mathcal L_a(t)=e^{-a}b/(b^2+t^2/4)\). For all real \(t\), \(|C_a(t)-\mathcal L_a(t)|\le\delta(a)\mathcal L_a(t)\) with \(\delta(a)=\frac1b\big[\frac{2ab^2}{(b-1)^3}+\frac{2ab}{(b-1)^2}+\frac{2a^2}{(b-2)^3}+\frac a{b-1}\big]\le5.61/a\). The deficit of the friable approximant \(\Xi_P\) (the difference of \(\Xi_P\) and Riemann’s \(\Xi\), for \(\mathcal F(P)\) the integers with all prime factors at most \(P\)) is \(D_P(t)=(t^2+\frac14)\sum_{m\notin\mathcal F(P)}C_{\pi m^2}(t)\), so it is a sum of Lorentzians of half widths about \(2\pi m^2\) over the omitted integers \(m\), with relative error \(\delta(\pi q^2)\) for \(q\) the least prime above \(P\).

Earlier editions: Theorem 22.3.4, page 805 (edition 1).

Source: ch/mi1.tex, label mi:thm:lorentz.

Curvature: the logarithm of the deficit of the friable approximant is uniformly almost convex for t at least 14

Volume II, Theorem 17.3.5, page 510 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. For \(a\ge9\pi\) and \(b=a-\frac14\) let \(\delta=\delta(a)\) be the constant of the uniform Lorentzian bound, \(r_1(a)=\frac1{2(1-\delta)b}\big(1+\frac{a^2}{(b-2)^2}+\frac a{(b-1)^2}+\frac{2a}{b-1}\big)\) and \(r_2(a)=\frac1{4(1-\delta)b}\big(\frac4b+\frac{2a^2}{(b-2)^3}+\frac{2a}{(b-1)^3}+\frac{4a}{(b-1)^2}\big)\); these bound \(|C_a'|/C_a\) and \(|C_a''|/C_a\) uniformly in \(t\), where \(C_a\) is the cosine transform of \(e^{-ae^w}e^{w/4}\) at \(t/2\). For every prime \(P\) and \(t\ge14\), with \(a_q=\pi q^2\), \((\log D_P)''(t)\ge-2/t^2-r_2(a_q)-r_1(a_q)^2\ge-0.0239\).

Earlier editions: Theorem 22.3.5, page 806 (edition 1).

Source: ch/mi1.tex, label mi:thm:curv.

Real zeros of the friable approximant live in the negative humps of Xi, in pairs, and surviving humps are nested in P

Volume II, Theorem 17.4.1, page 511 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. A negative hump is a maximal interval \(I_j\subset(0,\infty)\) on which \(\Xi<0\), of length \(\ell_j\). (1) Every real zero of \(\Xi_P\) lies in a negative hump, and \(\Xi_P(\gamma)=D_P(\gamma)>0\) at every zero \(\gamma\) of \(\Xi\). (2) Each hump holds an even number of zeros of \(\Xi_P\) counted with multiplicity, and the number \(R_P\) of positive real zeros is even. (3) Suppose every zero of \(\Xi\) whose real part lies within \(\frac12\) of \(I_j\) is real (by the verification of the Riemann Hypothesis of Platt and Trudgian (2021) this holds for every hump below \(3\cdot10^{12}\)) and \(\ell_j<18.29\). Then \(I_j\) holds no zero, one double zero or two simple zeros of \(\Xi_P\), and two simple zeros exactly when \(\max_{I_j}(|\Xi|-D_P)>0\). (4) If \(I_j\) carries zeros of \(\Xi_P\) then it carries zeros of \(\Xi_{P'}\) for every prime \(P'>P\).

Earlier editions: Theorem 22.4.1, page 807 (edition 1).

Source: ch/mi1.tex, label mi:thm:humps.

Count: the number of positive real zeros of the friable approximant is twice the number of negative humps of Xi deeper than the deficit

Volume II, Theorem 17.4.2, page 512 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. Let \(N(T)\) count the zeros of \(\Xi\) in \((0,T]\) with multiplicity, \(A(t)=\frac12(t^2+\frac14)\pi^{-1/4}|\Gamma(\frac14+\frac{it}2)|\), so that \(\Xi=-AZ\) with \(Z\) Hardy’s function, and \(I_j\) the negative humps of \(\Xi\). Assume that for every hump meeting \((0,\tau_P^+)\) the local hypotheses hold (all zeros of \(\Xi\) within \(\frac12\) of the hump are real, by the verification of the Riemann Hypothesis of Platt and Trudgian (2021) below height \(3\cdot10^{12}\), and its length is below \(18.29\)). Then \(R_P=2\#\{j:\max_{I_j}(|\Xi|-D_P)>0\}\). Let \(\tau_P^+\) solve \(D_P(\tau)=0.63A(\tau)\tau^{1/6}\log\tau\); the solution is unique for \(\tau\ge14\) and \(\Xi_P>0\) on \((\tau_P^+,\infty)\) (this uses Hiary’s bound \(|\zeta(\frac12+it)|\le0.63t^{1/6}\log t\)). Then \(R_P\le2\lceil N(\tau_P^+)/2\rceil\).

Earlier editions: Theorem 22.4.2, page 808 (edition 1).

Source: ch/mi1.tex, label mi:thm:count.

Conjecture: the number of real zeros of the friable approximant is the zero count at the breakdown height plus a correction from the peaks of Hardy’s function

Volume II, Conjecture 17.5.1, page 514 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. Let \(N(T)\) count the zeros of \(\Xi\) in \((0,T]\), \(R_P\) the number of positive real zeros of \(\Xi_P\), \(\tau_P^*\) the breakdown height, the solution of \(A(\tau)=D_P(\tau)\) (about \(4q^2\), a heuristic formula checked by computation) and \(m_Z(T)\) the mean of \(\log\max|Z|\) over the negative humps near \(T\), for \(Z\) Hardy’s function. Conjecture: \(R_P=N(\tau_P^*)+\frac2{\pi^2}\log\frac{\tau_P^*}{2\pi}\,m_Z(\tau_P^*)+O\big((\log q)^{1/2}(\log\log q)^{1/2}\big)\).

Earlier editions: Conjecture 22.5.1, page 810 (edition 1).

Source: ch/mi2.tex, label mi:conj:survey.

Truncated two term family is not Lee Yang for N at least 3 (certified)

Volume II, Proposition 17.7.2, page 517 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\rho_{*,N}(u)\propto\sum_{n\le N}(2\pi^2n^4\cosh9u-3\pi n^2\cosh5u)e^{-2\pi n^2\cosh4u}\), with characteristic function continued to complex \(t\) as \(\psi_N(2t)=\frac14\sum_{n\le N}[T_n(t)+T_n(-t)]\), \(T_n(t)=2\pi^2n^4K_{\frac94+\frac{it}2}(2\pi n^2)-3\pi n^2K_{\frac54+\frac{it}2}(2\pi n^2)\), \(K_\nu\) the modified Bessel function of the second kind, and \(\psi_\infty\) the limit as \(N\to\infty\). For \(N=3\), \(\psi_3(2t)\) has exactly one zero in the box of half width \(0.025\) around \(188.433681-0.274981i\). For every \(N\) with \(4\le N\le\infty\), \(\psi_N(2t)\) has exactly one zero in the box of half width \(0.02\) around \(188.434477-0.192499i\). So no \(\rho_{*,N}\) with \(N\ge3\), and not the limit, is a Lee Yang spin. Certified in ball arithmetic: a winding number around the box for \(N=3\); for \(N=4\) a boundary walk with point values in ball arithmetic and a Lipschitz bound, and Rouché’s theorem for the terms with \(n\ge5\).

Source: ch/mi3.tex, label mi:prop:starnonreal.

Kink budget: the limit of the friable approximant with a smooth even kernel replacing the pole term

Volume II, Proposition 17.8.1, page 518 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. Let \(c_P=\sum_{m\notin\mathcal F(P)}(4\pi m^2-1)e^{-\pi m^2}\) be the kink constant. Replace the pole kernel \(e^{-u/2}\) in the folded kernel \(k_P(u)=e^{-u/2}-2e^{u/2}\psi_P(e^{2u})\) by an even kernel \(\kappa\), smooth on \([0,\infty)\) with all derivatives \(O(e^{-u/2})\), and let \(\Xi_P^\kappa(t)=(t^2+\frac14)\int_0^\infty(k_P+\kappa-e^{-u/2})(u)\cos(tu)\,du\). Then \(\lim_{t\to\infty}\Xi_P^\kappa(t)=c_P-\frac12-\kappa'(0+)\) along \(\mathbb{R}\).

Earlier editions: Proposition 22.8.1, page 814 (edition 1).

Source: ch/mi3.tex, label mi:prop:kinkbudget.

Reweighting: among weights in [0,1] vanishing on a given set, the indicator of the complement has the smallest deficit

Volume II, Proposition 17.8.2, page 519 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. For weights \(w=(w_n)\) let \(\Xi_w(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_w(e^{2u})e^{u/2}\cos(tu)\,du\) with \(\psi_w(x)=\sum_nw_ne^{-\pi n^2x}\), and \(D_w=\Xi_w-\Xi=(t^2+\frac14)\sum_n(1-w_n)C_{\pi n^2}\), where \(C_a\) is the cosine transform of \(e^{-ae^w}e^{w/4}\) at \(t/2\). If \(0\le w_n\le w_n'\le1\) for every \(n\) then \(D_w\ge D_{w'}\ge0\) on \(\mathbb{R}\), with \(D_{w'}>0\) unless every \(w_n'=1\). So among weights in \([0,1]\) that vanish on a given set of integers, the indicator of its complement has the smallest deficit.

Earlier editions: Proposition 22.8.2, page 814 (edition 1).

Source: ch/mi3.tex, label mi:prop:reweight.

The friable approximants for P = 2 and 3 are uniformly close to Xi on the imaginary axis and have no zero there

Volume II, Proposition 17.9.1, page 520 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\mathcal F(P)\) be the set of positive integers all of whose prime factors are at most \(P\) (the \(P\) friable integers), \(\psi_P(x)=\sum_{n\in\mathcal F(P)}e^{-\pi n^2x}\), and let the friable approximant be \(\Xi_P(t)=\frac12-2(t^2+\frac14)\int_0^\infty\psi_P(e^{2u})e^{u/2}\cos(tu)\,du\) (the same formula with all integers \(n\) gives \(\Xi\)). Let \(q\) be the least prime above \(P\) and \(D_P=\Xi_P-\Xi\) the deficit. For \(P=2\) and \(P=3\), \(\sup_{y\in\mathbb{R}}|\Xi_P(iy)/\Xi(iy)-1|\) is at most \(6.8\times10^{-13}\) for \(P=2\) and at most \(3.4\times10^{-35}\) for \(P=3\). In particular \(\Xi_P\) has no zero on \(i\mathbb{R}\). The bound rests on the proved estimate \(|\Xi_P(iy)-\Xi(iy)|\le|y^2-\frac14|\sum_{m\notin\mathcal F(P)}(\pi m^2)^{-s}\Gamma(s,\pi m^2)\) with \(s=\frac12(|y|+\frac12)\), whose supremum after division by \(\Xi(iy)=\xi(\frac12+y)>0\) was computed at \(30\) digits on a grid of step \(0.1\) on \([0,40]\) and shown to decrease beyond it; it is a computation, not a certified enclosure.

Earlier editions: Proposition 22.9.1, page 815 (edition 1).

Source: ch/mi3.tex, label mi:prop:axis.

Shift identity: Xi is the average of two shifts of the transform of Phi divided by cosh, which contains Miyatake’s identity

Volume II, Proposition 17.11.1, page 522 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. For \(b>0\) put \(g_b(z)=\int_{\mathbb{R}}\Phi(u)\operatorname{sech}(bu)\,e^{izu}\,du\) (the transform of \(\Phi/\cosh(bu)\)), an entire even function of order at most one, real on \(\mathbb{R}\). For every \(b>0\) and \(z\in\mathbb{C}\), \(\Xi(z)=\frac12\{g_b(z+ib)+g_b(z-ib)\}\), that is \(\Xi=\cos(b\partial)g_b\) with \(\partial=d/dz\). Miyatake’s identity is the case \(b=\frac92\) (with his transform \(g(x)=g_{9/2}(2x)/(4\pi^2)\)).

Earlier editions: Proposition 22.11.1, page 818 (edition 1).

Source: ch/mi4.tex, label mi:prop:shiftb.

Levy area, Polya Gamma clock and Thorin atoms: the transform of Phi over cosh is Xi smoothed by Levy’s stochastic area

Volume II, Theorem 17.11.2, page 523 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. For \(b>0\) put \(g_b(z)=\int_{\mathbb{R}}\Phi(u)\operatorname{sech}(bu)\,e^{izu}\,du\) (the transform of \(\Phi/\cosh(bu)\)), an entire even function of order at most one, real on \(\mathbb{R}\). Let \(\mathcal A=\frac12\int_0^1(W^1dW^2-W^2dW^1)\) be Lévy’s stochastic area of a planar Brownian motion \((W^1,W^2)\), with \(\mathbb{E}e^{i\lambda\mathcal A}=\operatorname{sech}(\lambda/2)\), let \(\varpi\) be a Pólya Gamma variable \(\mathrm{PG}(1,0)\) with \(\mathbb{E}e^{-s\varpi}=\operatorname{sech}\sqrt{s/2}\), and \(\mathcal H_\lambda(z)=\int_{\mathbb{R}}\Phi(u)e^{\lambda u^2}e^{izu}\,du\) the de Bruijn Newman family. For real \(x\), \(g_b(x)=\mathbb{E}\,\Xi(x+2b\mathcal A)=\int_{\mathbb{R}}\Xi(x-y)\frac1{2b}\operatorname{sech}\frac{\pi y}{2b}\,dy=\mathbb{E}_\varpi[\mathcal H_{-2b^2\varpi}(x)]\), and \(2b\mathcal A\overset d=2b\sqrt\varpi\,\mathcal N\overset d=\sum_{k\ge1}\frac{2b}{(2k-1)\pi}L_k\) with \(\mathcal N\) standard normal and \(L_k\) independent standard Laplace variables. Hence \(\Xi\mapsto g_b\) is convolution with an extended generalised gamma convolution whose Thorin measure has atoms at \(\pm(2k-1)\pi/(2b)\).

Earlier editions: Theorem 22.11.2, page 819 (edition 1).

Source: ch/mi4.tex, label mi:thm:levyarea.

Logistic partition: 2 xi is the sum of two logistic weighted expectations of a power of a random variable

Volume II, Proposition 17.11.3, page 523 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Upsilon=V^{1/2}\) with \(V\) the random variable of the Biane Pitman Yor identity for \(\xi\) (as defined in the book), and \(\varpi\sim\mathrm{PG}(1,0)\) a Pólya Gamma variable. For \(c>0\), \(2\xi(s)=\mathbb{E}[\Upsilon^s/(1+\Upsilon^c)]+\mathbb{E}[\Upsilon^{s+c}/(1+\Upsilon^c)]\); with \(c=2b\) and \(s=\frac12+iz\) the two terms are \(g_b(z+ib)\) and \(g_b(z-ib)\), for \(g_b\) the transform of \(\Phi/\cosh(bu)\). The logistic weight is the Pólya Gamma mixture \(\Upsilon^c/(1+\Upsilon^c)=\frac12\Upsilon^{c/2}\mathbb{E}_\varpi\exp(-\frac12\varpi c^2(\log\Upsilon)^2)\) (Polson, Scott and Windle 2013).

Earlier editions: Proposition 22.11.3, page 819 (edition 1).

Source: ch/mi4.tex, label mi:prop:logistic.

Exponential tail: the transform of Phi over cosh has a residue term C_b e^{-pi z/(2b)} exactly when b exceeds 2

Volume II, Theorem 17.12.1, page 524 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. For \(b>0\) put \(g_b(z)=\int_{\mathbb{R}}\Phi(u)\operatorname{sech}(bu)\,e^{izu}\,du\) (the transform of \(\Phi/\cosh(bu)\)), an entire even function of order at most one, real on \(\mathbb{R}\). Let \(b>2\), \(\pi/(2b)<\tau<\min(\pi/4,3\pi/(2b))\) and \(Y>0\). Then, uniformly for \(\operatorname{Re}z\ge0\) and \(|\operatorname{Im}z|\le Y\), \(g_b(z)=C_be^{-\pi z/(2b)}+O(e^{-\tau\operatorname{Re}z})\) with \(C_b=\frac{2\pi}b\,\Phi(\frac{i\pi}{2b})\). For \(b<2\) there is no residue and \(g_b(x)=O(e^{-\tau x})\) for every \(\tau<\pi/4\).

Earlier editions: Theorem 22.12.1, page 820 (edition 1).

Source: ch/mi4.tex, label mi:thm:tail.

Cusp asymptotics: the residue constant C_b of the transform of Phi over cosh as b decreases to 2

Volume II, Theorem 17.12.2, page 524 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. For \(b>2\) put \(C_b=\frac{2\pi}b\Phi(\frac{i\pi}{2b})\) (the residue constant of the transform of \(\Phi/\cosh(bu)\)) and \(\delta=\pi(b-2)/(4b)\). As \(b\downarrow2\), \(C_b=-\frac{\pi^3}{32b}(2\delta)^{-1/2}\delta^{-4}e^{-\frac\pi8\cot\delta}(1+O(b-2))\).

Earlier editions: Theorem 22.12.2, page 820 (edition 1).

Source: ch/mi4.tex, label mi:thm:cusp.

For b above 2 with C_b nonzero the transform of Phi over cosh has finitely many real zeros and is not in the Laguerre Polya class

Volume II, Corollary 17.12.3, page 525 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(tu)\,du\) and \(\Phi\) is positive and even. For \(b>0\) put \(g_b(z)=\int_{\mathbb{R}}\Phi(u)\operatorname{sech}(bu)\,e^{izu}\,du\) (the transform of \(\Phi/\cosh(bu)\)), an entire even function of order at most one, real on \(\mathbb{R}\). For \(b>2\) with \(C_b=\frac{2\pi}b\Phi(\frac{i\pi}{2b})\ne0\), \(g_b\) has finitely many real zeros, infinitely many nonreal zeros, and is not in the Laguerre Pólya class. In particular Miyatake’s transform, for which \(b=\frac92\) and \(C_{9/2}>0\), is not.

Earlier editions: Corollary 22.12.3, page 821 (edition 1).

Source: ch/mi4.tex, label mi:cor:notLP.

Polar part of the Miyatake kernel in the variable cosh t: a simple pole at cos(2 pi/9) with positive residue r_0

Volume II, Proposition 17.13.1, page 527 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi\) be Riemann’s kernel, \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\), analytic for \(|\operatorname{Im}u|<\frac\pi4\), and let \(\psi_{\mathrm M}(t)=\Phi(t/2)/(8\pi^2\cosh(9t/4))\), an even function of \(t\) on the strip \(|\operatorname{Im}t|<\frac\pi2\). The map \(t\mapsto c=\cosh t\) sends this strip, modulo \(t\mapsto-t\), one to one onto the half plane \(\operatorname{Re}c>0\), so \(\psi_{\mathrm M}\) is a function of \(c\) there. As a function of \(c\), \(\psi_{\mathrm M}\) is analytic on \(\operatorname{Re}c>0\) except for a simple pole at \(c_0=\cos\frac{2\pi}9\) with residue \(r_0=\frac{\sin(2\pi/9)}{18\pi^2}\Phi(\frac{i\pi}9)=\frac{\sin(2\pi/9)}{8\pi^3}C_{9/2}>0\), where \(C_{9/2}=\frac{4\pi}9\Phi(\frac{i\pi}9)>0\) is the residue constant of the transform of \(\Phi\) divided by \(\cosh(9u/2)\). Consequently \(\psi_{\mathrm M}(t)=\sum_{n\ge1}n^4e^{-2\pi n^2\cosh t}+\psi_{\mathrm E}(t)+\psi_{\mathrm A}(t)\), where \(\psi_{\mathrm E}(t)=r_0e^{-2\pi(\cosh t-c_0)}/(\cosh t-c_0)=r_0\int_{2\pi}^\infty e^{c_0a}e^{-a\cosh t}\,da\) (the integral for \(\operatorname{Re}\cosh t>c_0\)) and \(\psi_{\mathrm A}\) is analytic in \(c\) on all of \(\operatorname{Re}c>0\). Moreover \(r_0=0.009206406367\), computed in interval arithmetic with all digits shown certified. The result is from a note of Nicholas Polson.

Earlier editions: Proposition 22.13.1, page 823 (edition 1).

Source: ch/mi4.tex, label mi:prop:polar.

The exponent is forced: an exponential mixture of Pólya Bessel functions has finitely many real zeros unless its exponent is cos(2 pi/9)

Volume II, Proposition 17.13.2, page 527 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0<\theta<\frac\pi2\) and \(a_0>0\), let \(K_\nu\) be the modified Bessel function of the second kind, and put \(\mathcal E_\theta(z)=\int_{\mathbb{R}}\frac{\cosh(9u/2)\,e^{-a_0(\cosh2u-\cos\theta)}}{\cosh2u-\cos\theta}e^{izu}\,du=\int_{a_0}^\infty e^{a\cos\theta}\,\frac12\{K_{\frac94+\frac{iz}2}(a)+K_{\frac94-\frac{iz}2}(a)\}\,da\), the exponential mixture with exponent \(\cos\theta\) of the Pólya functions \(\frac12\{K_{\frac94+\frac{iz}2}(a)+K_{\frac94-\frac{iz}2}(a)\}\) (the transforms of \(\cosh(9u/2)e^{-a\cosh2u}\)). For \(\theta/2<\tau<\frac\pi4\) and \(Y>0\), uniformly for \(\operatorname{Re}z\ge0\) and \(|\operatorname{Im}z|\le Y\), \(\mathcal E_\theta(z)=\frac{\pi\cos(9\theta/4)}{\sin\theta}e^{-\theta z/2}+O(e^{-\tau\operatorname{Re}z})\). If \(\theta\ne\frac{2\pi}9\), then \(\mathcal E_\theta\) has finitely many real zeros and infinitely many nonreal zeros, and it is not in the Laguerre Pólya class. The conclusion leaves only \(\theta=\frac{2\pi}9\), where \(\cos(9\theta/4)=0\); whether \(\mathcal E_{2\pi/9}\) is in the Laguerre Pólya class is open. The result is from a note of Nicholas Polson.

Earlier editions: Proposition 22.13.2, page 823 (edition 1).

Source: ch/mi4.tex, label mi:prop:rigid.

A Meixner convolution: the polar transform is a Pólya frequency density convolved with a Laguerre Pólya function

Volume II, Proposition 17.13.3, page 528 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(c_1=\cos\frac\pi9\) and \(c_0=\cos\frac{2\pi}9=2c_1^2-1\), and let \(\mathcal U(u)=\frac{\cosh(9u/2)}{2(\cosh u-c_1)}=\cosh\frac{7u}2+2c_1\cosh\frac{5u}2+(1+2c_0)\cosh\frac{3u}2+(1+2c_1)\cosh\frac u2\), so that \(\cosh(9u/2)/(\cosh2u-c_0)=\mathcal U(u)/(\cosh u+c_1)\). Let \(P(z)=\int_{\mathbb{R}}e^{-2\pi\cosh2u}e^{izu}\,du\) and \(\Xi^{\mathcal U}(z)=\int_{\mathbb{R}}\mathcal U(u)e^{-2\pi\cosh2u}e^{izu}\,du=\sum_jm_j\cos(\frac{b_j}2\partial)P\) with \((b_j,m_j)=(7,1),(5,2c_1),(3,1+2c_0),(1,1+2c_1)\), where \(\cos(\beta\partial)f(z)=\frac12\{f(z+i\beta)+f(z-i\beta)\}\). (i) \(\mathcal U\) is a universal factor in the sense of Pólya, and \(\Xi^{\mathcal U}\) is in the Laguerre Pólya class. (ii) For \(0\le\varphi<\pi\) let \(q_\varphi(x)=\frac{(1+\cos\varphi)\sinh(\varphi x)}{\sin\varphi\,\sinh(\pi x)}\), with \(q_0(x)=2x/\sinh(\pi x)\). Then \(q_\varphi\) is a probability density with \(\int_{\mathbb{R}}q_\varphi(x)e^{iux}\,dx=\frac{1+\cos\varphi}{\cosh u+\cos\varphi}\) and variance \(1/(1+\cos\varphi)\), and it is a Pólya frequency function (by Schoenberg’s characterisation). Moreover \(\frac{1+\cos\varphi}{\cosh u+\cos\varphi}=\mathbb{E}\,e^{-\varsigma_\varphi u^2}\) with \(\varsigma_\varphi=\sum_{k\ge0}\big(\frac{e_k}{((2k+1)\pi-\varphi)^2}+\frac{e_k'}{((2k+1)\pi+\varphi)^2}\big)\), \(\mathbb{E}\varsigma_\varphi=\frac1{2(1+\cos\varphi)}\), for independent standard exponential variables \(e_k,e_k'\). Hence for every kernel \(k\) for which the integrals converge, the transform of \(k(u)(1+\cos\varphi)/(\cosh u+\cos\varphi)\) is \(q_\varphi*\hat k\) (\(\hat k\) the transform of \(k\)); for \(k=\Phi\), Riemann’s kernel \(\Phi(u)=2\sum_{n\ge1}(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})e^{-\pi n^2e^{2u}}\) (so that \(\Xi(z)=\int_{\mathbb{R}}\Phi(u)e^{izu}\,du\)), it is \(\mathbb{E}\,\mathcal H_{-\varsigma_\varphi}\) with \(\mathcal H_\lambda(z)=\int_{\mathbb{R}}\Phi(u)e^{\lambda u^2}e^{izu}\,du\) the de Bruijn Newman family, a flow at a random negative time. (iii) Let \(r_0=\frac{\sin(2\pi/9)}{18\pi^2}\Phi(\frac{i\pi}9)=0.009206406367\), the residue at \(\cosh t=c_0\) of the Miyatake kernel \(\Phi(t/2)/(8\pi^2\cosh(9t/4))\) as a function of \(\cosh t\), and let \(\Xi^{\mathrm E}(z)=8\pi^2r_0\int_{\mathbb{R}}\frac{\cosh(9u/2)\,e^{-2\pi(\cosh2u-c_0)}}{\cosh2u-c_0}e^{izu}\,du\), the transform of the polar part. Then \(\Xi^{\mathrm E}=\frac{8\pi^2r_0e^{2\pi c_0}}{1+c_1}\,q_{\pi/9}*\Xi^{\mathcal U}\). The result is from a note of Nicholas Polson, using Pólya’s theorems on universal factors and on the zeros of \(K_{iz}\) and Schoenberg’s theorem.

Earlier editions: Proposition 22.13.3, page 824 (edition 1).

Source: ch/mi4.tex, label mi:prop:meixner.

The bottom of the cosh ladder is at most the largest real part of a zero of zeta minus one half

Volume II, Proposition 17.14.1, page 530 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Xi(t)=\xi(\frac12+it)\), an even entire function, real on \(\mathbb{R}\), whose zeros are the ordinates of the nontrivial zeros of the zeta function. For real \(b\) let \(F_b(z)=\int_{\mathbb{R}}\Phi(u)\cosh(bu)e^{izu}\,du=\frac12\{\Xi(z+ib)+\Xi(z-ib)\}\), so that \(F_b(x)=\operatorname{Re}\xi(\frac12+b+ix)\) on \(\mathbb{R}\). Let \(B_{\cosh}=\inf\{b\ge0: F_b\text{ has only real zeros}\}\) (the bottom of the cosh ladder; \(B_{\cosh}\le\frac12\) by de Bruijn’s strip theorem) and \(\sigma_{\max}=\sup\operatorname{Re}\rho\) over the nontrivial zeros \(\rho\) of zeta. Then \(B_{\cosh}\le\sigma_{\max}-\frac12\).

Earlier editions: Proposition 22.14.1, page 826 (edition 1).

Source: ch/mi5.tex, label mi:prop:BTheta.

Conjecture: the bottom of the cosh ladder lies between the asymptotic and the global quasi Riemann abscissa

Volume II, Conjecture 17.14.2, page 531 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. With \(F_b(z)=\int\Phi(u)\cosh(bu)e^{izu}\,du\), so that \(F_b(x)=\operatorname{Re}\xi(\frac12+b+ix)\) on \(\mathbb R\), the cosh ladder of Riemann’s xi function, \(B_{\cosh}=\inf\{b\ge0: F_b\text{ has only real zeros}\}\) its bottom rung, \(\sigma_{\max}=\sup\operatorname{Re}\rho\) and \(\sigma_{\max}^\infty=\limsup_{\gamma\to\infty}\beta\) over the nontrivial zeros \(\rho=\beta+i\gamma\) of zeta: conjecturally \(\sigma_{\max}^\infty-\frac12\le B_{\cosh}\le\sigma_{\max}-\frac12\). The upper bound is proved; the lower bound is conjectured. Under the conjecture, lowering the unconditional bound \(B_{\cosh}\le\frac12\) is equivalent to a quasi Riemann hypothesis whenever \(\sigma_{\max}\) is approached at large height.

Earlier editions: Conjecture 22.14.2, page 827 (edition 1).

Source: ch/mi5.tex, label mi:conj:quasi.

Paired form: zeta has no zero to the right of alpha exactly when the real and imaginary parts of xi on the line alpha interlace

Volume II, Theorem 17.14.3, page 532 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\alpha>\frac12\), \(E_\alpha(z)=\xi(\alpha-iz)\) and \(E^\#(z)=\overline{E(\bar z)}\). The following are equivalent: (i) \(\zeta(s)\ne0\) for \(\operatorname{Re}s\ge\alpha\); (ii) \(E_\alpha\) is Hermite Biehler, that is \(|E_\alpha(z)|>|E_\alpha^\#(z)|\) for \(\operatorname{Im}z>0\); (iii) \(\operatorname{Re}\xi(\alpha+it)\) and \(\operatorname{Im}\xi(\alpha+it)\), as entire functions of \(t\), have only real simple zeros, and these strictly interlace. Hence the paired bottom rung, the infimum of the \(\alpha-\frac12\) for which these hold, is exactly \(\sigma_{\max}-\frac12\) (with \(\sigma_{\max}\) the supremum of the real parts of the nontrivial zeros), and the Riemann Hypothesis holds if and only if (iii) holds for every \(\alpha>\frac12\).

Earlier editions: Theorem 22.14.3, page 828 (edition 1).

Source: ch/mi5.tex, label mi:thm:paired.

Real zeros of the real and imaginary parts of xi on a vertical line right of one half have asymptotically full density

Volume II, Theorem 17.14.4, page 534 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(N^\pm(T)\) the number of nontrivial zeros \(\beta+i\gamma\) of zeta with \(|\gamma|\le T\), \(N^\pm_{>\alpha}(T)\) the number of them with \(\beta>\alpha\), and \(\theta_\alpha(t)=\arg\xi(\alpha+it)\). For \(\alpha>\frac12\) with \(\beta\ne\alpha\) for every zero, \(\theta_\alpha(T)-\theta_\alpha(-T)=\pi(N^\pm(T)-2N^\pm_{>\alpha}(T))+O(\log T)\). Hence \(\operatorname{Re}\xi(\alpha+it)\) and \(\operatorname{Im}\xi(\alpha+it)\) each have at least \(N^\pm(T)-2N^\pm_{>\alpha}(T)-O(\log T)\) real zeros in \([-T,T]\). If moreover \(F_b\), for \(b=\alpha-\frac12\) and \(F_b(z)=\operatorname{Re}\xi(\alpha+iz)\), satisfies the Riemann von Mangoldt formula with error \(O(\log T)\), its nonreal zeros with \(|\operatorname{Re}z|\le T\) number at most \(2N^\pm_{>\alpha}(T)+O(\log T)\).

Earlier editions: Theorem 22.14.4, page 830 (edition 1).

Source: ch/mi5.tex, label mi:thm:density.

Interlacing across rungs: criteria for the zeros of the real and imaginary parts of xi on two lines to alternate or to fail to alternate

Volume II, Proposition 17.14.5, page 535 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function. For \(\alpha>1\) let \(A_\alpha(t)=\operatorname{Re}\xi(\alpha+it)\) and \(B_\alpha(t)=\operatorname{Im}\xi(\alpha+it)\) for real \(t\), and write \(\xi(\alpha+it)=|\xi(\alpha+it)|e^{i\theta_\alpha(t)}\) with \(\theta_\alpha\) continuous and \(\theta_\alpha(0)=0\). Let \(1<\alpha<\alpha'\), \(h=\alpha'-\alpha\), \(D=\theta_{\alpha'}-\theta_\alpha\), \(\mathcal I=\int_\alpha^{\alpha'}M(\sigma)\,d\sigma=\log\frac{\zeta(\alpha)}{\zeta(\alpha')}\) where \(M(\sigma)=-\zeta'(\sigma)/\zeta(\sigma)=\sum_n\Lambda(n)n^{-\sigma}\) (\(\Lambda\) the von Mangoldt function), \(\arg\zeta(\sigma+it)=\operatorname{Im}\log\zeta(\sigma+it)\) with \(\log\zeta(s)=\sum_n\Lambda(n)n^{-s}/\log n\), and \(J=\sum_p(\arctan p^{-\alpha}-\arctan p^{-\alpha'})\le\mathcal I\) with the sum over primes \(p\). Then \(D(t)=\frac{h\pi}4+\arg\zeta(\alpha'+it)-\arg\zeta(\alpha+it)+O(1/t)\), so \(|D(t)-\frac{h\pi}4|\le\mathcal I+O(1/t)\). Two sequences of zeros alternate if between any two consecutive zeros of either there is exactly one zero of the other. (i) If \([\frac{h\pi}4-\mathcal I,\frac{h\pi}4+\mathcal I]\subset(k\pi,(k+1)\pi)\) for an integer \(k\), the zeros of \(A_\alpha\) and \(A_{\alpha'}\) alternate for large \(t\), and so do those of \(B_\alpha\) and \(B_{\alpha'}\). (ii) If \([\frac{h\pi}4-\mathcal I,\frac{h\pi}4+\mathcal I]\subset(k\pi-\frac\pi2,k\pi+\frac\pi2)\) for an integer \(k\), the zeros of \(A_\alpha\) and \(B_{\alpha'}\) alternate for large \(t\), and so do those of \(B_\alpha\) and \(A_{\alpha'}\). (iii) If a multiple of \(\pi\) lies in \((\frac{h\pi}4-J,\frac{h\pi}4+J)\), the zeros of \(A_\alpha\) and \(A_{\alpha'}\) fail to alternate at arbitrarily large \(t\) (two consecutive zeros of one function with no zero of the other between them); if an odd multiple of \(\frac\pi2\) does, so do those of \(A_\alpha\) and \(B_{\alpha'}\). The result is from a note of Nicholas Polson.

Earlier editions: Proposition 22.14.5, page 831 (edition 1).

Source: ch/mi5.tex, label mi:prop:pairs.

Non interlacing: the zeros of A_alpha and of the Bessel functions K do not alternate with the real zeros of Xi

Volume II, Proposition 17.14.6, page 536 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(\Xi(t)=\xi(\frac12+it)\), \(\vartheta\) the Riemann Siegel theta function, \(N(T)=\frac{\vartheta(T)}\pi+1+S(T)\) the number of zeros \(\gamma\) of \(\Xi\) with \(0<\gamma\le T\) and \(S(T)=\frac1\pi\arg\zeta(\frac12+iT)\). (i) For \(\alpha>1\), \(A_\alpha(t)=\operatorname{Re}\xi(\alpha+it)\) has \(N(T)-S(T)+O_\alpha(1)\) zeros in \((0,T]\); hence the zeros of \(A_\alpha\) do not alternate with the real zeros of \(\Xi\) for large \(t\). (ii) For \(a>0\) and \(K_\nu\) the modified Bessel function of the second kind, \(\operatorname{Re}K_{\frac94+\frac{it}2}(a)\) has \(\frac T{2\pi}(\log\frac Ta-1)+O_a(1)\) zeros in \((0,T]\); hence two of these functions with different \(a\) do not alternate for large \(t\), and none alternates with the real zeros of \(\Xi\) for large \(t\) if \(a\le2\pi\), or, under the Riemann Hypothesis, if \(a>2\pi\). Alternation means that between any two consecutive zeros of either sequence there is exactly one zero of the other. The result is from a note of Nicholas Polson.

Earlier editions: Proposition 22.14.6, page 832 (edition 1).

Source: ch/mi5.tex, label mi:prop:noninter.

Collision of one pair of zeros under a random vertical average: the critical spread depends on the variance and the fourth moment

Volume II, Proposition 17.14.8, page 538 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(G\) be a real entire function of order less than two, \(a\in\mathbb{R}\) with \(G(a)\ne0\), \(\tau_j=G^{(j)}(a)/G(a)\) and \(F_\eta(z)=((z-a)^2+\eta^2)G(z)\) for \(\eta>0\). Let \(\ell\) be a symmetric random variable, bounded or standard normal, with \(\mathbb{E}\ell^2=1\) and fourth moment \(m_4=\mathbb{E}\ell^4\), and put \(d_c=\tau_1^2-\frac{m_4-1}2\tau_2\). For every \(K>|d_c|\) there are \(r>0\) and \(\eta_0>0\) such that for \(0<\eta<\eta_0\) and \(s>0\) with \(|s^2-\eta^2|\le K\eta^4\) the function \(z\mapsto\mathbb{E}F_\eta(z+is\ell)\) has exactly two zeros in \(|z-a|<r\), and there is \(s_c\) with \(s_c^2=\eta^2-d_c\,\eta^4+O(\eta^6)\) such that the two zeros are a nonreal conjugate pair for \(s<s_c\), a double real zero at \(a+\tau_1\eta^2+O(\eta^4)\) for \(s=s_c\), and real and distinct for \(s>s_c\). The result is from a note of Nicholas Polson.

Earlier editions: Proposition 22.14.8, page 834 (edition 1).

Source: ch/mi5.tex, label mi:prop:collision.

Volume II, Chapter 18. Ferromagnetic and extreme value approximants with certified real zeros

Cosine and Gaussian tilts of the Pólya law with variance at least 9/64 if there is no zero beyond 7/8

Volume II, Proposition 18.1.2, page 542 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The Riemann zeta function has no zero with real part greater than 7/8 (announced by OpenAI in October 2026, not refereed, and not proved in the book).

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\). For \(\lambda_1,\dots,\lambda_N>0\) and \(\mu\ge0\) put \(m(t)=e^{\mu t^2}\prod_{k\le N}\cosh(\lambda_kt)\) and \(c(m)=2\mu+\sum_{k\le N}\lambda_k^2\), so that \(m(t)=\mathbb{E}e^{tY}\) for \(Y=\sqrt{2\mu}\,Z+\sum_k\lambda_k\varepsilon_k\) with \(Z\) standard normal and \(\varepsilon_k\) independent fair random signs, and \(c(m)=\operatorname{Var}Y\); the same applies with \(N=\infty\) when \(\sum_k\lambda_k^2<\infty\). A law is ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function vanishes only on the imaginary axis. Assume that \(\zeta\) has no zero with real part greater than \(\frac78\). Then the zeros of \(z\mapsto\int\Phi(t)m(t)e^{izt}\,dt\) lie in the strip \(|\operatorname{Im}z|\le(\max(\frac9{64}-c(m),0))^{1/2}\). In particular the law with density proportional to \(m\,p\) is ferromagnetic whenever \(\operatorname{Var}Y\ge\frac9{64}\); for example \(\cosh(at)\,p(t)\) is ferromagnetic for \(a\ge\frac38\) and \(\frac{\sinh(ct)}{ct}\,p(t)\) for \(c\ge\frac{3\sqrt3}8\). With no hypothesis on the zeros the same statements hold with \(\frac14\) in place of \(\frac9{64}\), by theorems of de Bruijn (1950), so that \(\cosh(at)\,p(t)\) is ferromagnetic for \(a\ge\frac12\).

Earlier editions: Proposition 13.5.2, page 549 (edition 1).

Source: ch/fm4.tex, label fm:prop:bruijn.

Ferromagnetic laws with density proportional to a multiplier times exp(-a cosh(l t)), Bessel transform and clock

Volume II, Theorem 18.1.4, page 544 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A symmetric real random variable \(X\) is called ferromagnetic if \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\) and its moment generating function \(L_X(z)=\mathbb{E}e^{zX}\), which is then entire, vanishes only on the imaginary axis; by Newman’s theorem \(L_X(s)=e^{bs^2}\prod_k(1+s^2/c_k^2)\) with \(b\ge0\) and \(c_k>0\). Let \(K_\nu(a)=\int_0^\infty e^{-a\cosh u}\cosh(\nu u)\,du\) be the Macdonald function, for \(a>0\) and complex order \(\nu\). Let \(a,\ell>0\), \(\mu\ge0\) and \(\lambda_1,\dots,\lambda_N\ge0\), let \(Z\) be standard normal and \(\varepsilon_1,\dots,\varepsilon_N\) independent signs \(\pm1\) with probability \(\frac12\) each, independent of \(Z\), let \(Y=\sqrt{2\mu}\,Z+\sum_{k\le N}\lambda_k\varepsilon_k\), so that \(m(t)=\mathbb{E}e^{tY}=e^{\mu t^2}\prod_{k\le N}\cosh(\lambda_kt)\), and let \(q\) be the probability density on \(\mathbb{R}\) proportional to \(m(t)e^{-a\cosh\ell t}\). Then: (i) \(q\) is ferromagnetic, it has Gaussian moments of all orders, and its Gaussian component is zero (\(b=0\)). (ii) \(\int_{\mathbb{R}}m(t)e^{-a\cosh\ell t}\,dt=\frac2\ell\,\mathbb{E}K_{Y/\ell}(a)\) and the moment generating function of \(q\) is \(L_q(s)=\mathbb{E}K_{(s+Y)/\ell}(a)/\mathbb{E}K_{Y/\ell}(a)\) for complex \(s\); for \(\mu=0\) the expectations are averages over the \(2^N\) values of \(Y\). (iii) The clock \(T_q\) of \(q\) is the generalised gamma convolution, with drift \(b=0\) and a unit Thorin atom at the square of each positive zero of the characteristic function of \(q\), whose Laplace transform is \(\mathbb{E}e^{-uT_q}=1/L_q(\sqrt u)\); here \(\mathbb{E}e^{-uT_q}=\mathbb{E}K_{Y/\ell}(a)/\mathbb{E}K_{(\sqrt u+Y)/\ell}(a)\) for \(u\ge0\). For \(m\equiv1\), \(T_q=\ell^{-2}\sum_je_j/\nu_j(a)^2\) with \(e_j\) independent standard exponential variables and \(0<\nu_1(a)\le\nu_2(a)\le\cdots\) the zeros of \(\nu\mapsto K_{i\nu}(a)\) on \((0,\infty)\) listed with multiplicity.

Earlier editions: Theorem 13.5.4, page 551 (edition 1).

Source: ch/fm4.tex, label fm:thm:bessel.

Laws on a Bessel kernel as logarithms of mixtures of generalised inverse Gaussian laws

Volume II, Proposition 18.1.5, page 545 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a,\ell>0\), \(\mu\ge0\), \(\lambda_1,\dots,\lambda_N\ge0\), \(Z\) standard normal, \(\varepsilon_k\) independent signs, \(Y=\sqrt{2\mu}\,Z+\sum_{k\le N}\lambda_k\varepsilon_k\), \(m(t)=\mathbb{E}e^{tY}\) and \(q\) the probability density proportional to \(m(t)e^{-a\cosh\ell t}\). Let \(K_v(a)=\int_0^\infty e^{-a\cosh u}\cosh(vu)\,du\), and for real \(v\) let \(G_v\) have the generalised inverse Gaussian density \(g^{v-1}e^{-\frac a2(g+1/g)}/(2K_v(a))\) on \((0,\infty)\). Let \(V\) be a random index with \(\mathbb{E}h(V)=\mathbb{E}[h(Y/\ell)K_{Y/\ell}(a)]/\mathbb{E}K_{Y/\ell}(a)\) and, given \(V\), let \(G_V\) have the density of index \(V\). Then \(q\) is the law of \(\ell^{-1}\log G_V\), and also the law of \(\varepsilon\,\ell^{-1}\log G_{|V|}\), with \(\varepsilon\) a sign independent of \(V\) and \(G_{|V|}\). For \(\mu=0\), \(V\) takes the values \(\ell^{-1}\sum_k\eta_k\lambda_k\), \(\eta\in\{\pm1\}^N\), with probabilities proportional to \(K_{\ell^{-1}\sum_k\eta_k\lambda_k}(a)\).

Earlier editions: Proposition 13.5.5, page 552 (edition 1).

Source: ch/fm4.tex, label fm:prop:besselgig.

No sequence of laws on a convergent Bessel kernel with Laguerre Pólya multipliers converges in law to the Pólya law

Volume II, Proposition 18.1.8, page 547 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(p=\Phi/\xi(\frac12)\) be the Pólya density, where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), a function with a natural boundary on the lines \(\operatorname{Im}t=\pm\pi/4\). Let \(a_n\to a\) and \(\ell_n\to\ell\) with \(a,\ell>0\), and let \(q_n\) be the probability density proportional to \(m_n(t)e^{-a_n\cosh\ell_nt}\), where \(m_n(t)=\psi_n(it)\) for an even entire function \(\psi_n\) of the Laguerre Pólya class with \(\psi_n(0)>0\), that is \(\psi_n(z)=\psi_n(0)e^{-\alpha_nz^2}\prod_j(1-z^2/z_{n,j}^2)\) with \(\alpha_n\ge0\), \(z_{n,j}>0\) and \(\sum_jz_{n,j}^{-2}<\infty\). Then \(q_n\) does not converge in law to \(p\). The multipliers \(e^{\mu t^2}\prod_{k\le N}\cosh(\lambda_kt)\) with \(\mu,\lambda_k\ge0\) are of this form, and so are \(e^{\mu t^2}\prod_{k\ge1}\cosh(\lambda_kt)\) with \(\sum_k\lambda_k^2<\infty\). Consequently a sequence of laws of Theorem fm:thm:bessel (multipliers \(e^{\mu t^2}\prod_{k\le N}\cosh(\lambda_kt)\)), which are ferromagnetic, converging in law to \(p\) would prove RH, by the closure of ferromagnetic laws under convergence in law, and its kernel parameters \((a_n,\ell_n)\) would have no limit point in \((0,\infty)^2\); whether such a sequence exists is not known.

Earlier editions: Proposition 13.5.8, page 554 (edition 1).

Source: ch/fm4.tex, label fm:prop:fixedkernel.

Coupled spins with Bessel kernel laws and a ferromagnetic interaction have ferromagnetic nonnegative combinations

Volume II, Corollary 18.1.9, page 548 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A symmetric real random variable \(X\) is called ferromagnetic if \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\) and its moment generating function \(L_X(z)=\mathbb{E}e^{zX}\), which is then entire, vanishes only on the imaginary axis. Let \(q_1,\dots,q_N\) be probability densities on \(\mathbb{R}\), each proportional to \(m_i(t)e^{-a_i\cosh\ell_it}\) with \(a_i,\ell_i>0\) and \(m_i(t)=e^{\mu_it^2}\prod_{k\le N_i}\cosh(\lambda_{ik}t)\) with \(\mu_i,\lambda_{ik}\ge0\) (the laws of the Bessel kernel theorem, which are ferromagnetic with Gaussian moments of all orders). Let \(J=(J_{ij})\) be a symmetric matrix with \(J_{ij}\ge0\) and \(J_{ii}=0\), and let \((X_1,\dots,X_N)\) have density on \(\mathbb{R}^N\) proportional to \(\exp\big(\sum_{i<j}J_{ij}t_it_j\big)\prod_iq_i(t_i)\). Then \(M=\sum_ic_iX_i\) is ferromagnetic for all \(c_1,\dots,c_N\ge0\), not all \(0\).

Earlier editions: Corollary 13.5.9, page 555 (edition 1).

Source: ch/fm4.tex, label fm:cor:coupled.

Zeros of the transform of two weakly coupled Bessel spins move by order J at a simple zero and by plus or minus the square root of J at a common zero

Volume II, Proposition 18.1.10, page 549 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(q_1,q_2\) be laws of the Bessel kernel theorem (densities proportional to \(m_i(t)e^{-a_i\cosh\ell_it}\) with \(a_i,\ell_i>0\) and \(m_i(t)=e^{\mu_it^2}\prod_{k\le N_i}\cosh(\lambda_{ik}t)\), \(\mu_i,\lambda_{ik}\ge0\)), with characteristic functions \(\varphi_1,\varphi_2\). For \(J>0\) let \((X_1,X_2)\) have density proportional to \(e^{Jt_1t_2}q_1(t_1)q_2(t_2)\), so that the characteristic function of \(X_1+X_2\) is proportional to \(Z_J(\theta)=\sum_{n\ge0}\frac{(-J)^n}{n!}\varphi_1^{(n)}(\theta)\varphi_2^{(n)}(\theta)\). As \(J\downarrow0\): (i) if \(\alpha\) is a simple zero of \(\varphi_1\) and \(\varphi_2(\alpha)\ne0\), then \(Z_J\) has a zero \(\alpha+J\varphi_2'(\alpha)/\varphi_2(\alpha)+O(J^2)\). (ii) If \(\alpha\) is a simple zero of both \(\varphi_1\) and \(\varphi_2\), then \(Z_J\) has the two real zeros \(\alpha\pm\sqrt J+\frac J4\big(\frac{\varphi_1''}{\varphi_1'}+\frac{\varphi_2''}{\varphi_2'}\big)(\alpha)+O(J^{3/2})\), at distance \(2\sqrt J+O(J^{3/2})\).

Earlier editions: Proposition 13.5.10, page 556 (edition 1).

Source: ch/fm4.tex, label fm:prop:levelrep.

Mellin transform of the clock of the Bessel K pair has no zero in a box where the uniform pair has two (computed)

Volume II, Result 18.1.3, page 549 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Let \(q\) be the law with characteristic function \(\varphi_q(\theta)=K_{i\theta/4}(\pi)/K_0(\pi)\) (\(K_\nu\) the modified Bessel function of the second kind), the ferromagnetic law with density proportional to \(e^{-\pi\cosh4t}\) on \(\mathbb{R}\); for the sum of two independent copies the number of zeros of the characteristic function in \((0,\theta]\) has the main terms of the Riemann von Mangoldt formula. Let \(\mathcal D\) be the Mellin transform of the Laplace transform of the clock (the sum of \(e_k/c_k^2\) over the zeros \(\pm c_k\) of the characteristic function, \(e_k\) independent standard exponential variables). The transform of the pair of two copies, and that of a single copy, have no zero in the box \(-0.95<\operatorname{Re}w<-0.05\), \(\frac12<\operatorname{Im}w<11\), in which the transform of the uniform pair \(\pi U+\pi U'\) has the two zeros \(\frac{\rho_1-1}2\) and \(\frac{\rho_2-1}2\), where \(\rho_k=\frac12+i\gamma_k\) are the first two zeros of \(\zeta\) on the critical line (computed: the argument principle on the boundaries of the boxes \(\frac12<\operatorname{Im}w<\frac{17}2\) and \(\frac{17}2<\operatorname{Im}w<11\), sampled, not a ball proof, with the uniform pair computed by the same quadrature as a control, which gives winding \(1\) in each box). The result follows a note of Nicholas Polson merged into the book.

Earlier editions: Result 13.5.3, page 556 (edition 1).

Source: ch/fm4.tex, label fm:res:besselmellin.

The components of the theta series in Riemann’s formula and their mixture are not ferromagnetic

Volume II, Proposition 18.1.11, page 550 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(n\ge1\) let \(r_n(t)=e^{|t|/2}\exp(-\pi n^2e^{2|t|})\), and consider the laws on \(\mathbb{R}\) with densities proportional to \(r_n\) and to \(\sum_{m\ge1}r_m\) (the components of the theta series in Riemann’s formula for the xi function and their mixture). Each of these laws is symmetric, has finite moments of all orders (indeed \(\mathbb{E}e^{b|X|}<\infty\) for every \(b\ge0\), so the characteristic function is entire of order at most one), and has a characteristic function positive on \(\mathbb{R}\), and none of them is ferromagnetic, that is, none has a characteristic function with only real zeros: each characteristic function has infinitely many zeros and none of them is real. For \(n=1\) the characteristic function has \(5\) zeros in the box \(0\le\operatorname{Re}\theta\le40\), \(0\le\operatorname{Im}\theta\le30\), the two of least modulus being \(13.0885+10.8707i\) and \(20.1987+15.0786i\) (computed by the argument principle). The result is from a note of Nicholas Polson merged into the book.

Earlier editions: Proposition 13.5.11, page 557 (edition 1).

Source: ch/fm4.tex, label fm:prop:nocomponent.

Zero completed laws of the Pólya variable, the Maxwell construction and a criterion for RH

Volume II, Theorem 18.1.12, page 551 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), with characteristic function \(\varphi_X\). Let \(\gamma_1\le\gamma_2\le\cdots\) be the positive ordinates of the zeros \(\frac12\pm i\gamma_k\) of \(\zeta\) on the critical line, with multiplicity, and put \(s_K=\sum_{k\le K}\gamma_k^{-2}\), \(\beta_K=\frac12\operatorname{Var}X - s_K\) and \(\varphi_K(\theta)=e^{-\beta_K\theta^2}\prod_{k\le K}(1-\theta^2/\gamma_k^2)\). Let \(M_1,M_2,\dots\) be independent symmetric Maxwell variables (density \(x^2e^{-x^2/2}/\sqrt{2\pi}\), characteristic function \((1-\theta^2)e^{-\theta^2/2}\), variance \(3\)), each ferromagnetic, and let \(Z\) be a standard normal variable independent of them. A law is ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function vanishes only on the imaginary axis. (i) With no hypothesis, \(\beta_K>0\) for every \(K\). If \(3s_K\le\operatorname{Var}X\), then \(\varphi_K\) is the characteristic function of \(X_K=(\operatorname{Var}X-3s_K)^{1/2}Z+\sum_{k\le K}M_k/\gamma_k\), whose law \(f_K\) is ferromagnetic, with \(\operatorname{Var}X_K=\operatorname{Var}X\) and \(\kappa_4(X_K)=-12\sum_{k\le K}\gamma_k^{-4}\). The condition \(3s_K\le\operatorname{Var}X\) holds for \(K\le16\) and fails for \(K=17\), a certified computation in ball arithmetic (\(3s_{16}=0.0456803\), \(\operatorname{Var}X=0.0462100\), \(3s_{17}=0.0463006\)). (ii) For every \(K\), if the inverse Fourier transform \(f_K\) of \(\varphi_K\) is nonnegative, then \(f_K\) is a ferromagnetic probability density. (iii) The Riemann Hypothesis holds if and only if \(\varphi_K\to\varphi_X\) pointwise as \(K\to\infty\). Under the Riemann Hypothesis, \(f_K\) converges in law to the Pólya law along every sequence of \(K\) for which \(f_K\ge0\).

Earlier editions: Theorem 13.5.12, page 558 (edition 1).

Source: ch/fm4.tex, label fm:thm:zc.

Hermite series for the Pólya density convolved with a Gaussian converges unconditionally

Volume II, Proposition 18.1.14, page 552 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable with characteristic function \(\varphi_X\) whose zeros are at \(\pm\gamma_k\), \(\gamma_k\) running over the ordinates of the zeros of \(\zeta\) with positive real part, with multiplicity (not assumed real). Let \(e_j(N)\) be the \(j\)th elementary symmetric function of \(\gamma_1^{-2},\dots,\gamma_N^{-2}\) and \(e_j=\lim_Ne_j(N)\). Then all \(e_j(N)\ge0\), they increase with \(N\) to \(e_j\), and \(\sum_je_j\sqrt{(2j)!}\,v^{-j}<\infty\) for every \(v>0\). With \(g_v\) the \(N(0,v)\) density, \(\mathrm{He}_n\) the Hermite polynomial \(\mathrm{He}_n(y)=\mathbb{E}(y+iZ)^n\) (\(Z\) standard normal) and \(f_{N,v}(x)=g_v(x)\sum_{j\le N}e_j(N)v^{-j}\mathrm{He}_{2j}(x/\sqrt v)\), the functions \(f_{N,v}\) converge, uniformly after multiplication by \(e^{x^2/4v}\), to the density of \(X+\sqrt vZ\) with \(Z\) an independent standard normal, for every \(v>0\), with no hypothesis on the zeros.

Source: ch/fm4.tex, label fm:prop:hermiteinf.

Positive Hermite syntheses at Gaussian variance at least half the variance, zeta and Davenport and Heilbronn

Volume II, Proposition 18.1.15, page 553 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable, \(f_{N,v}\) the density with Fourier transform \(e^{-v\theta^2/2}\prod_{k\le N}(1-\theta^2/\gamma_k^2)\) over the ordinates \(\gamma_k\) of the zeros of \(\zeta\) in the order of the book, and let \(v\ge\frac12\operatorname{Var}X=\sum_k\operatorname{Re}\gamma_k^{-2}=0.0231\). Then \(f_{N,v}\ge0\) for every \(N\) ending a conjugate pair, and \(f_{N,v}\) converges weakly to the law of \(X+\sqrt vZ\), \(Z\) an independent standard normal. This holds for the zeta function with no hypothesis on the zeros. It holds equally for the Davenport and Heilbronn function, which has zeros off the critical line, for \(v\ge\frac12\operatorname{Var}X_{\mathrm{DH}}=0.0981\); for that function the input that the nonreal ordinates satisfy \(|\operatorname{Im}\gamma|/\operatorname{Re}\gamma<0.02\) rests on computed zeros to height \(200\) and a bound \(|\operatorname{Im}\gamma|\le1.5\). For zeta the input is that a nonreal ordinate has \(\operatorname{Re}\gamma>3\cdot10^{12}\) and \(|\operatorname{Im}\gamma|<\frac12\) (Platt and Trudgian 2021 and the zero free region).

Source: ch/fm4.tex, label fm:prop:heightpos.

RH is equivalent to an identity in law between the smoothed Pólya variable and a series of Maxwell variables

Volume II, Theorem 18.1.16, page 555 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\), and let \(Z\) be a standard normal variable independent of \(X\). Let \(\gamma_1\le\gamma_2\le\cdots\) be the positive ordinates of the zeros \(\frac12\pm i\gamma_k\) of \(\zeta\) on the critical line, with multiplicity, let \(M_1,M_2,\dots\) be independent symmetric Maxwell variables with density \(x^2e^{-x^2/2}/\sqrt{2\pi}\), and put \(\mathcal M=\sum_kM_k/\gamma_k\), a series that converges almost surely because its terms are independent and centred with summable variances \(3\gamma_k^{-2}\). Then the Riemann Hypothesis holds if and only if \(X+\big(\tfrac12\operatorname{Var}X\big)^{1/2}Z\overset{d}{=}\mathcal M\).

Earlier editions: Theorem 13.5.14, page 561 (edition 1).

Source: ch/fm4.tex, label fm:thm:maxwell.

Total variation distance from the Pólya law to ferromagnetic laws is below two millionths, with no hypothesis

Volume II, Proposition 18.1.17, page 555 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X\) be the Pólya variable, the symmetric random variable with density \(p=\Phi/\xi(\frac12)\), where \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\). A law is ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0Y^2}<\infty\) for some \(b_0>0\), and its moment generating function vanishes only on the imaginary axis. Let \(\delta\) be the infimum of the total variation distances \(\sup_A|P(X\in A)-P(Y\in A)|\) from \(X\) to ferromagnetic variables \(Y\); when \(Y\) has a density \(f\) the distance is \(\operatorname{TV}(f,p)=\frac12\int|f - p|\). Let \(\gamma_1\le\gamma_2\le\cdots\) be the positive ordinates of the zeros \(\frac12\pm i\gamma_k\) of \(\zeta\) on the critical line, with multiplicity, put \(s_K=\sum_{k\le K}\gamma_k^{-2}\) and \(\beta_K=\frac12\operatorname{Var}X - s_K\), and let \(f_K\) be the function with Fourier transform \(\varphi_K(\theta)=e^{-\beta_K\theta^2}\prod_{k\le K}(1-\theta^2/\gamma_k^2)\). With no hypothesis on the zeros: \(f_{16}\) is a ferromagnetic probability density, an explicit sum of a normal variable and Maxwell variables, and \(\operatorname{TV}(f_{16},p)<1.266\times10^{-4}\), so that \(\delta<1.266\times10^{-4}\); and \(f_{150}\) is a ferromagnetic probability density by a computer assisted proof in ball arithmetic (the polynomial in the Hermite basis that multiplies the Gaussian is positive on the real line) with \(\operatorname{TV}(f_{150},p)<1.932\times10^{-6}\), so that \(\delta<1.932\times10^{-6}\); and the law \(q\) with density proportional to \(\cosh(\lambda t)\,e^{-a\cosh\ell t}\), \(\lambda=2.146184\), \(a=5.483952\), \(\ell=2.062866\), is ferromagnetic (by Theorem fm:thm:bessel: the law proportional to \(e^{-a\cosh\ell t}\) is ferromagnetic since \(z\mapsto K_{iz}(a)\) has only real zeros (P'olya), and the tilt by \(\cosh(\lambda t)\) keeps it ferromagnetic; \(L_q(s)=\frac{K_{(s+\lambda)/\ell}(a)+K_{(s-\lambda)/\ell}(a)}{2K_{\lambda/\ell}(a)}\)) and \(\operatorname{TV}(q,p)<6.024\times10^{-6}\), so that \(\delta<6.024\times10^{-6}\). The distances are enclosed in ball arithmetic, with explicit bounds for the discretisation and the tails. The Riemann Hypothesis holds if and only if \(\delta=0\).

Earlier editions: Proposition 13.5.15, page 561 (edition 1).

Source: ch/fm4.tex, label fm:prop:tvcert.

A cusped weight never gives a ferromagnetic tilt nor a GGC tilted clock

Volume II, Theorem 18.1.18, page 556 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, \(\Phi(t)=\sum_{n\ge1}(4\pi^2n^4e^{9t/2}-6\pi n^2e^{5t/2})\exp(-\pi n^2e^{2t})\) and \(p=\Phi/\xi(\frac12)\) the Pólya density, and \(\Phi_{\mathrm{dBN}}(u)=\frac12\Phi(2u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\), the kernel of the de Bruijn Newman flow, with \(H_0(z)=\int_0^\infty\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du=\frac18\xi(\frac12+\frac{iz}2)\). A law is ferromagnetic if it is symmetric, has \(\mathbb{E}e^{b_0X^2}<\infty\) for some \(b_0>0\), and its moment generating function vanishes only on the imaginary axis. Let \(\mu>0\) and \(H^{(\mu)}(z)=\int_0^\infty e^{-\mu u}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\). With \(C\) a standard Cauchy variable, \(H^{(\mu)}(z)=\mathbb{E}\,H_0(z+\mu C)\) and \(z^2H^{(\mu)}(z)\to\mu\,\Phi_{\mathrm{dBN}}(0)=0.4467\,\mu\) as \(z\to\infty\) along the reals. Hence \(H^{(\mu)}\) is positive beyond some height, has finitely many real zeros and infinitely many nonreal ones, the law with density proportional to \(e^{-\mu|t|/2}p(t)\) is not ferromagnetic, and the tilted clock \(R^{(\mu)}(s)=H^{(\mu)}(0)/H^{(\mu)}(2i\sqrt s)\) is not the Laplace transform of a GGC (a generalized gamma convolution, a law on \([0,\infty)\) with Laplace transform \(\exp(-bs-\int_{(0,\infty)}\log(1+s/z)\,U(dz))\) with drift \(b\ge0\) and Thorin measure \(U\ge0\)), for any \(\mu>0\).

Earlier editions: Theorem 13.5.16, page 562 (edition 1).

Source: ch/fm4.tex, label xg:thm:cauchy.

Laplace weighted de Bruijn Newman function H¹ and a sufficient condition for RH

Volume II, Proposition 18.1.19, page 557 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_{\mathrm{dBN}}\) be the de Bruijn Newman kernel, \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)du\) and, for \(a>0\), \(H^{[a]}(z)=\int_0^\infty\frac{\Phi_{\mathrm{dBN}}(u)}{1+a^2u^2}\cos(zu)du\). Let \(L\) be a standard Laplace variable with density \(\frac12e^{-|y|}\). Then \(H^{[a]}(z)=\mathbb{E}H_0(z+aL)\), \(H^{[a]}=\int_0^\infty e^{-v}H_{-va^2}dv\), an exponential mixture of backward heat flows, and \(H_0=(1-a^2\frac{d^2}{dz^2})H^{[a]}\). Consequently, if \(H^{[a]}\) has only real zeros for one \(a>0\), then the Riemann hypothesis holds: \(H^{[a]}\) is even, real on \(\mathbb{R}\) and entire of order one, so it is in the Laguerre Pólya class, and so is \(H_0=(1-a\frac d{dz})(1+a\frac d{dz})H^{[a]}\) by Laguerre’s theorem. The weight has poles at \(u=\pm i/a\), so it is not an entire multiplier; RH is not known to imply that \(H^{[a]}\) has real zeros, and whether it has them for some \(a>0\) is open. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Proposition 13.5.17, page 563 (edition 1).

Source: ch/fm4.tex, label xg:prop:laplace.

If the folded kernels have only real zeros in the disc of radius 2c(N+1)^2 for every N then RH holds

Volume II, Proposition 18.2.1, page 560 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi^F_N=\sum_{n\le N}\phi_n\) be the folded kernel and \(H^F_N\) its cosine transform, \(R_N=\Phi-\Phi^F_N=\sum_{n>N}\phi_n\) on \(u\ge0\) and \(E_N=H-H^F_N\), so that \(E_N(z)=\int_0^\infty R_N(u)\cos(zu)\,du\), where \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\), and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\) with \(\Xi\) the Riemann Xi function. Suppose there is \(c>0\) such that for every \(N\ge1\) all zeros of \(H^F_N\) in the disc \(|z|<2c(N+1)^2\) are real. Then RH holds. (The hypothesis, the height rule, is a heuristic balance of the kink against the size of \(H\) and is verified in the book only for \(N=1,\dots,4\) with \(c=3.6\), by a Rouch'e certificate, and for \(N=5\) in floating point.)

Earlier editions: Proposition 25.7.1, page 929 (edition 1).

Source: ch/ev2.tex, label ev:prop:reduction.

Kink bound: the error of the folded kernel decays like 1/x^2 with an explicit constant

Volume II, Lemma 18.3.1, page 561 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi^F_N=\sum_{n\le N}\phi_n\) be the folded kernel and \(H^F_N\) its cosine transform, \(R_N=\Phi-\Phi^F_N=\sum_{n>N}\phi_n\) on \(u\ge0\) and \(E_N=H-H^F_N\), so that \(E_N(z)=\int_0^\infty R_N(u)\cos(zu)\,du\), where \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\), and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\) with \(\Xi\) the Riemann Xi function. For real \(x\ne0\), \(E_N(x)=-R_N'(0)/x^2-x^{-2}\int_0^\infty R_N''(u)\cos(xu)\,du\) and \(|E_N(x)|\le B_N/x^2\) with \(B_N=|R_N'(0)|+\|R_N''\|_{L^1(0,\infty)}\).

Earlier editions: Lemma 25.8.1, page 930 (edition 1).

Source: ch/ev2.tex, label ev:lem:kink.

Rouche certificate: a circle inequality and a sign change count show that all zeros of the folded kernel below a height are real

Volume II, Proposition 18.3.2, page 561 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi^F_N=\sum_{n\le N}\phi_n\) be the folded kernel and \(H^F_N\) its cosine transform, \(R_N=\Phi-\Phi^F_N=\sum_{n>N}\phi_n\) on \(u\ge0\) and \(E_N=H-H^F_N\), so that \(E_N(z)=\int_0^\infty R_N(u)\cos(zu)\,du\), where \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\), and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\) with \(\Xi\) the Riemann Xi function. Let \(\mathcal N(T)\) be the number of ordinates \(\gamma\in(0,T)\) of nontrivial zeros \(\frac12+i\gamma\) of \(\zeta\) and let \(T\) lie strictly between two consecutive ordinates, and suppose \(\Xi\) has exactly \(2\mathcal N(T)\) zeros in \(|w|<T\), all real and simple. If \(|E_N(2Te^{i\vartheta})|<|H(2Te^{i\vartheta})|\) for \(\vartheta\in[0,\pi/2]\), and \(H^F_N(2x)\) has \(\mathcal N(T)\) sign changes on \((0,T)\), then every zero of \(H^F_N(2x)\) with \(|x|<T\) is real. (In the book the inequality is certified in ball arithmetic for \(N=1,2,3,4\) at \(T=14.4,17.5\) (\(N=1\)), \(31.7,35.3\) (\(N=2\)), \(51.4,57.9\) (\(N=3\)), \(86.1,97.3\) (\(N=4\)), given the count of zeros of \(\Xi\) in the disc, and tested in floating point only for \(N=5\).)

Earlier editions: Proposition 25.8.2, page 930 (edition 1).

Source: ch/ev2.tex, label ev:prop:rouche.

Zeros of Re K_{mu+i nu}(a) for integer mu solve a Liouville problem with a rational spectral boundary condition

Volume II, Theorem 18.3.3, page 564 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\ge0\) be an integer, \(a>0\) and \(K\) the modified Bessel function. By the recurrence \(K_{\nu+1}(x)=K_{\nu-1}(x)+(2\nu/x)K_\nu(x)\), \(\operatorname{Re}K_{\mu+i\nu}(a)=A_\mu(\nu^2)K_{i\nu}(a)+B_\mu(\nu^2)\,aK_{i\nu}'(a)\) with real polynomials \(A_\mu,B_\mu\). Hence at a point with \(B_\mu(\nu^2)\ne0\), \(\nu\) is a zero exactly when \(g(\tau)=K_{i\nu}(ae^\tau)\) solves the Liouville problem \(-g''+a^2e^{2\tau}g=\lambda g\), \(\lambda=\nu^2\), with the spectral boundary condition \(g'(0)=\beta_\mu(\lambda)g(0)\), \(\beta_\mu=-A_\mu/B_\mu\). The first cases are Dirichlet for \(\mu=0\), Neumann (\(\beta_1=0\)) for \(\mu=1\), \(\beta_2=a^2/2-\lambda\) and \(\beta_3=4(a^2-3\lambda)/(a^2+8-4\lambda)\).

Earlier editions: Theorem 25.8.3, page 932 (edition 1).

Source: ch/ev2.tex, label ev:thm:shift.

The boundary problem with a boundary law with a point mass and poles is self adjoint with real eigenvalues, positive under a sign condition

Volume II, Proposition 18.3.4, page 564 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Consider the Liouville problem \(-g''+a^2e^{2\tau}g=\lambda g\) on \((0,\infty)\) with boundary condition \(g'(0)=\beta_\mu(\lambda)g(0)\). Suppose \(\beta_\mu(\lambda)=\alpha-m\lambda-\sum_{j=1}^J\sigma_j/(q_j-\lambda)\) with \(m\ge0\), \(\sigma_j>0\) and \(q_j\) real. Then the eigenvalue problem, away from the poles, is the eigenvalue problem of a self adjoint operator built from the Liouville operator on \((0,\infty)\) with a point mass \(m\) at \(0\) and \(J\) extra coordinates \(w_j\) coupled to \(g(0)\), with stiffness \(q_j\), unit mass and coupling \(-\sqrt{\sigma_j}\). Its eigenvalues are real. If in addition the energy form is nonnegative, for example if every \(q_j>0\) and \(\alpha\ge\sum_j\sigma_j/q_j\), they are positive, so that the corresponding \(\nu\) are real. This does not make the extended system a string: no hitting time is constructed for it.

Earlier editions: Proposition 25.8.4, page 932 (edition 1).

Source: ch/ev2.tex, label ev:prop:ext.

Cotangent law: the boundary values of the shifted Bessel kernels grow like cot(pi mu/2) times sqrt(lambda_k)

Volume II, Proposition 18.3.5, page 565 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the asymptotic \(K_{\mu+i\nu}(a)=\frac12\Gamma(\mu+i\nu)(a/2)^{-\mu-i\nu}(1+O((a/2\nu)^{2\mu}))\) for large \(\nu\), used in a sketch with heuristic error terms (only the leading term, from the phase computation, is derived).

Statement. Let \(a>0\), \(\mu>0\) not an integer, \(K\) the modified Bessel function of the second kind, \(\nu_k\) the positive zeros of \(\nu\mapsto\operatorname{Re}K_{\mu+i\nu}(a)\), \(\lambda_k=\nu_k^2\) and \(\beta_k=g'(0)/g(0)\) for \(g(\tau)=K_{i\nu_k}(ae^\tau)\), that is \(\beta_k=aK_{i\nu_k}'(a)/K_{i\nu_k}(a)\). As \(k\to\infty\), \(\beta_k=\cot(\pi\mu/2)\sqrt{\lambda_k}\,(1+O((a/2\nu_k)^{2\mu}))+O(1)\). The statement is proved by a sketch only for the leading term (the phase computation \(\arg\Gamma(\mu+i\nu)-\nu\log(a/2)=\frac\pi2+k\pi\) with Stirling’s formula); the error terms are heuristic. It matches every integer case of the shift theorem.

Earlier editions: Proposition 25.9.1, page 936 (edition 1).

Source: ch/ev2.tex, label ev:prop:cot.

The mean of the candidate hitting time of the de Bruijn and Newman family increases in the flow time

Volume II, Proposition 18.3.6, page 566 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\), so that \(\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\), \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) and let \(U\) have density proportional to \(e^{tu^2}\Phi(u)\) on \((0,\infty)\). Let \(T_t\) be the candidate hitting time, with Laplace transform \(H_t(0)/H_t(2i\sqrt s)\). The mean of the candidate hitting time is \(\mu_1(t)=\mathbb ET_t=2\mathbb E_t[U^2]\), and \(d\mu_1/dt=2\operatorname{Var}_t(U^2)>0\). Equivalently \(\mu_1(t)=\sum_k\gamma_k(t)^{-2}\), with \(\pm2\gamma_k(t)\) the zeros of \(H_t\) counted with multiplicity.

Earlier editions: Proposition 25.8.5, page 933 (edition 1).

Source: ch/ev2.tex, label ev:prop:dbn.

The coefficients of the tail expansion alternate in sign when the atoms are real and simple

Volume II, Proposition 18.4.1, page 571 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\gamma_1^2<\gamma_2^2<\cdots\) be real and simple atoms and \(c_k=\prod_{j\ne k}(1-\gamma_k^2/\gamma_j^2)^{-1}\). Then \(\operatorname{sign}c_k=(-1)^{k-1}\).

Earlier editions: Proposition 25.10.1, page 941 (edition 1).

Source: ch/ev4.tex, label ev:prop:sign.

RH holds if and only if the reciprocal xi law is a Polya frequency density

Volume II, Corollary 18.4.4, page 572 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi\) be the Riemann Xi function, \(\xi\) the Riemann xi function, \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\) the theta series kernel, and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\). Write \(\pm\gamma_k\) for the zeros of \(\Xi\), so that \(\frac12+i\gamma_k\) are the nontrivial zeros of \(\zeta\), ordered by real part (under the Riemann Hypothesis (RH) they are all real and \(0<\gamma_1\le\gamma_2\le\cdots\)), \(L(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)=H(0)/H(2i\sqrt s)\), which under RH equals \(\prod_k(1+s/\gamma_k^2)^{-1}\), and \(T\) for the random variable with Laplace transform \(L\). Let \(X\) have the law with Laplace transform \(L\) (so \(X=T\)). Then RH holds if and only if \(L(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)\) is the Laplace transform of a P'olya frequency density. Under RH that density is log concave, the hazard rate of \(X\) is increasing, and the mean excess function decreases monotonically, \(e(0)=\sum_k\gamma_k^{-2}=0.0231050\) (computed) down to \(e(\infty)=\gamma_1^{-2}=0.00500524\).

Earlier editions: Corollary 25.10.4, page 942 (edition 1).

Source: ch/ev4.tex, label ev:cor:pf.

Mean excess function: two term expansion with an exponential rate under RH, simple zeros and a lower bound for xi prime

Volume II, Proposition 18.4.5, page 573 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on RH (all zeros of \(\Xi\) are real), all zeros simple, and \(|\xi'(\rho)|\ge e^{-C|\gamma|}\) for some constant \(C\) and all zeros \(\rho=\frac12+i\gamma\).

Statement. Let \(\Xi\) be the Riemann Xi function, \(\xi\) the Riemann xi function, \(\Phi(u)=\sum_{n\ge1}\phi_n(u)\) with \(\phi_n(u)=(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\) the theta series kernel, and \(H(z)=\int_0^\infty\Phi(u)\cos(zu)\,du=\frac18\Xi(z/2)\). Write \(\pm\gamma_k\) for the zeros of \(\Xi\), so that \(\frac12+i\gamma_k\) are the nontrivial zeros of \(\zeta\), ordered by real part (under the Riemann Hypothesis (RH) they are all real and \(0<\gamma_1\le\gamma_2\le\cdots\)), \(L(s)=\xi(\frac12)/\xi(\frac12+\sqrt s)=H(0)/H(2i\sqrt s)\), which under RH equals \(\prod_k(1+s/\gamma_k^2)^{-1}\), and \(T\) for the random variable with Laplace transform \(L\). Assume RH, that the zeros are simple and that \(|\xi'(\rho)|\ge e^{-C|\gamma|}\) for some constant \(C\) and all zeros \(\rho\) (a hypothesis not known to hold), put \(\lambda_k=\gamma_k^2\) and let \(e(u)\) be the mean excess function of \(X\) with \(\mathbb P(X>x)=\sum_kc_ke^{-\lambda_kx}\). Then \(e(u)=\sum_kc_ke^{-\lambda_ku}/\lambda_k\big/\sum_kc_ke^{-\lambda_ku}=\frac1{\lambda_1}+Ae^{-(\lambda_2-\lambda_1)u}+O(e^{-\min(\lambda_3-\lambda_1,\,2(\lambda_2-\lambda_1))u})\) with \(A=\frac{c_2}{c_1}(\frac1{\lambda_2}-\frac1{\lambda_1})>0\). The excess law of \(X\) over \(u\) converges to the exponential law with rate \(\lambda_1\) in total variation at rate \(e^{-(\lambda_2-\lambda_1)u}\). Without simple zeros the two term form follows from the two term tail with \(w_k/\lambda_k\) in place of \(c_k\) for \(k=1,2\).

Earlier editions: Proposition 25.10.5, page 943 (edition 1).

Source: ch/ev4.tex, label ev:prop:meexp.

Volume II, Chapter 19. The Comtet Majumdar family: damped kernels and a solved model

A proof by Spitzer’s identity and Euler Maclaurin summation that the expected maximum of a symmetric stable walk has the constant of Comtet and Majumdar, zeta of s over a sine factor

Volume II, Proposition 19.1.2, page 581 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(S_n\) be a random walk whose steps \(X\) are symmetric stable with \(\mathbb Ee^{ikX}=e^{-|k|^\mu}\), \(1<\mu\le2\), let \(M_n=\max_{0\le j\le n}S_j\) and \(s=1/\mu\). Then \(\mathbb EM_n=\frac{\Gamma(1-s)}{\pi s}n^s+\frac{\Gamma(1-s)\zeta(1-s)}{\pi}+O(n^{s-1})\), where \(\zeta\) is the Riemann zeta function. Hence the constant term, found by Comtet and Majumdar, is \(C_\mu=I(s)=\zeta(s)/\{(2\pi)^s\sin(\pi s/2)\}\), with \(I(s)=\frac1\pi\int_0^\infty x^{-s}(\frac1{e^x-1}-\frac1x)\,dx\).

Earlier editions: Proposition 27.1.2, page 980 (edition 1).

Source: ch/cg1.tex, label cg:prop:spitzer.

The logarithm of the Laplace transform of the uniform law is a regularised integral against Lebesgue minus counting measure

Volume II, Proposition 19.1.3, page 582 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(x>0\), \(\ln\frac{1-e^{-x}}{x}=-\lim_{R\to\infty}\{\int_0^R(1-e^{-xt})\frac{dt}{t}-\sum_{1\le n\le R}\frac{1-e^{-xn}}{n}\}\), a regularised integral against the signed measure \(dt-d\lfloor t\rfloor\) with a common upper cutoff \(R\), where \(\lfloor t\rfloor\) is the integer part. For \(0<\operatorname{Re}z<1\), \(\lim_{R\to\infty}\int_0^Rt^{-z}(d\lfloor t\rfloor-dt)=-z\int_0^\infty\{t\}t^{-z-1}\,dt=\zeta(z)\), with \(\{t\}\) the fractional part and \(\zeta\) the Riemann zeta function. The two parts of the signed measure are mutually singular and \(\int_1^\infty|dt-d\lfloor t\rfloor|/t=\infty\), so the representation is regularised and not a signed L'evy measure.

Earlier editions: Proposition 27.1.3, page 981 (edition 1).

Source: ch/cg1.tex, label cg:prop:signed.

If locally uniform limits of GGC reciprocals approximate Xi then RH holds

Volume II, Corollary 19.2.2, page 583 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi\) be the Riemann Xi function and let \(\Xi_\delta\) be even, real on \(\mathbb R\), entire of order less than two with \(\Xi_\delta(0)>0\), such that \(\Xi_\delta\to\Xi\) locally uniformly as \(\delta\to0\) and each \(w\mapsto\Xi_\delta(0)/\Xi_\delta(i\sqrt w)\) is the Laplace transform of a generalised gamma convolution (GGC). Then all zeros of \(\Xi\) are real, that is the Riemann Hypothesis holds. Conversely, under the Riemann Hypothesis the constant family \(\Xi_\delta=\Xi\) qualifies.

Earlier editions: Corollary 27.2.2, page 982 (edition 1).

Source: ch/cg1.tex, label cg:cor:closure.

The shifted zeta is an explicit series of zeta at s plus k with a tangent factor

Volume II, Theorem 19.3.2, page 583 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g(x)=\frac1{e^x-1}-\frac1x\), \(\delta>0\), \(I_\delta(s)=\frac1\pi\int_0^\infty x^{-s}g(x+\delta)\,dx\) for \(0<\operatorname{Re}s<1\), \(\zeta_\delta(s)=(2\pi)^s\sin(\pi s/2)I_\delta(s)\) (so \(\zeta_0=\zeta\)), \(\operatorname{Li}_\sigma\) the polylogarithm and \((s)_k\) the rising factorial. For \(0<\delta<2\pi\), \(\pi I_\delta(s)=\Gamma(1-s)F_\delta(s)\) with \(F_\delta(s)=\operatorname{Li}_{1-s}(e^{-\delta})-\Gamma(s)\delta^{-s}=\sum_{k\ge0}\zeta(1-s-k)(-\delta)^k/k!\), and \(\zeta_\delta(s)=\sum_{k\ge0}(\delta/2\pi)^k\frac{(s)_k}{k!}\frac{\sin(\pi s/2)}{\sin(\pi(s+k)/2)}\zeta(s+k)=\zeta(s)+\frac{\delta}{2\pi}s\tan\frac{\pi s}2\,\zeta(s+1)-\frac{\delta^2}{8\pi^2}s(s+1)\zeta(s+2)-\cdots\). The ratio of sines is \((-1)^j\) for \(k=2j\) and \((-1)^j\tan(\pi s/2)\) for \(k=2j+1\).

Earlier editions: Theorem 27.3.2, page 982 (edition 1).

Source: ch/cg1.tex, label cg:thm:expansion.

Obstructions: the shifted xi function has a pole at s equal 3, its zeros drift, and its reciprocal is not a GGC transform

Volume II, Proposition 19.3.3, page 584 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0<\delta<2\pi\), \(g(x)=\frac1{e^x-1}-\frac1x\), \(F_\delta(s)=\operatorname{Li}_{1-s}(e^{-\delta})-\Gamma(s)\delta^{-s}\) with \(\operatorname{Li}\) the polylogarithm, \(\zeta_\delta(s)=(2\pi)^s\sin(\pi s/2)\frac1\pi\int_0^\infty x^{-s}g(x+\delta)\,dx\) for \(0<\operatorname{Re}s<1\), continued meromorphically, and \(\xi_\delta(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta_\delta(s)=\frac12s(s-1)\pi^{(s-1)/2}\Gamma(\frac{1-s}2)F_\delta(s)\). (a) \(\xi_\delta\) is meromorphic and not entire. It has a pole at \(s=3\) for every \(\delta\in(0,2\pi)\), because \(F_\delta(3)=4\sum_{k\ge1}\operatorname{Re}(\delta-2\pi ik)^{-3}<0\), and poles at \(s=-1,-2,-3,\dots\) For \(m\ge2\) it has a pole at \(s=2m+1\) exactly when \(F_\delta(2m+1)\ne0\). The values \(F_\delta(5)=0\) at \(\delta=4.7215\) and \(F_\delta(7)=0\) at \(\delta=3.0416\) are computed (not proved), so for those \(\delta\) there is no pole at \(s=5\), respectively \(s=7\). (b) Each simple zero \(\rho=\frac12+i\gamma\) of the Riemann zeta function \(\zeta\) moves to \(\rho_\delta=\rho-\frac\delta{2\pi}\frac{\rho\tan(\pi\rho/2)\zeta(\rho+1)}{\zeta'(\rho)}+O(\delta^2)\), and \(\xi_\delta(s)\ne\xi_\delta(1-s)\). (c) If \(\xi_\delta(\frac12)\ne0\), the function \(w\mapsto\xi_\delta(\frac12)/\xi_\delta(\frac12+\sqrt w)\) is not the Laplace transform of a generalised gamma convolution, nor of any positive measure.

Earlier editions: Proposition 27.3.3, page 983 (edition 1).

Source: ch/cg1.tex, label cg:prop:obstructions.

A signed L'evy measure gives a Thorin density, which is nonnegative if and only if a logarithmic derivative inequality holds

Volume II, Theorem 19.4.1, page 586 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be analytic and without zeros on \(\operatorname{Re}s>\alpha_0\) and positive on \((\alpha_0,\infty)\), and let \(M\) be a signed measure on \((0,\infty)\) with \(\int(1\wedge x)e^{-\alpha x}|M|(dx)<\infty\) for every \(\alpha>\alpha_0\) such that \(\frac{f'}f(s)-\frac{f'}f(\alpha)=\int_0^\infty(e^{-\alpha x}-e^{-sx})M(dx)\) for \(\operatorname{Re}s>\alpha_0\) and \(\alpha>\alpha_0\). Then for every \(\alpha>\alpha_0\) and \(w\ge0\), \(\log\frac{f(\alpha+w)}{f(\alpha)}-w\frac{f'}f(\alpha)=\int_0^\infty\log(1+w^2/y^2)\kappa_\alpha(y)\,dy\) with \(\kappa_\alpha(y)=\frac1\pi\int_0^\infty(1-\cos xy)e^{-\alpha x}M(dx)=\frac1\pi[\operatorname{Re}\frac{f'}f(\alpha+iy)-\frac{f'}f(\alpha)]\). If \(\kappa_\alpha\ge0\), then \(u\mapsto\frac{f(\alpha)}{f(\alpha+\sqrt{2u})}\exp\{\sqrt{2u}\,\frac{f'}f(\alpha)\}\) is the Laplace transform of a generalised gamma convolution (GGC) with Thorin density \(\tau_\alpha(z)=\kappa_\alpha(\sqrt{2z})/\sqrt{2z}\).

Earlier editions: Theorem 27.4.2, page 986 (edition 1).

Source: ch/cg2.tex, label cg:thm:thorinfromlevy.

Zeros on a vertical line give a Thorin density that is a sum of Lorentzians tending to point masses at the ordinates

Volume II, Corollary 19.4.2, page 587 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be analytic and without zeros on \(\operatorname{Re}s>\alpha_0\) and positive on \((\alpha_0,\infty)\), let \(M\) be a signed measure on \((0,\infty)\) with \(\int(1\wedge x)e^{-\alpha x}|M|(dx)<\infty\) and \(\frac{f'}f(s)-\frac{f'}f(\alpha)=\int_0^\infty(e^{-\alpha x}-e^{-sx})M(dx)\) for \(\operatorname{Re}s>\alpha_0\) and \(\alpha>\alpha_0\), let \(\kappa_\alpha(y)=\frac1\pi[\operatorname{Re}\frac{f'}f(\alpha+iy)-\frac{f'}f(\alpha)]\), so that \(\log\frac{f(\alpha+w)}{f(\alpha)}-w\frac{f'}f(\alpha)=\int_0^\infty\log(1+w^2/y^2)\kappa_\alpha(y)\,dy\) for \(w\ge0\), and let \(a\) be real, suppose \(f'/f(s)=a+\sum_k(s-\rho_k)^{-1}\) (symmetric summation) with all \(\rho_k=\beta_0+it_k\) on one vertical line, the set \(\{t_k\}\) symmetric and \(0\notin\{t_k\}\), and write \(\ell_\varepsilon(x)=\varepsilon/(\varepsilon^2+x^2)\) for the Cauchy kernel. Then for \(\alpha=\beta_0+\varepsilon\), \(\varepsilon>0\), \(\kappa_\alpha(y)=\frac1\pi\sum_k[\ell_\varepsilon(y-t_k)-\ell_\varepsilon(t_k)]\). If \(\sum_kt_k^{-2}<\infty\) then \(\kappa_\alpha(y)\,dy\to\sum_{t_k>0}\delta_{t_k}(dy)\) vaguely on \((0,\infty)\) as \(\varepsilon\downarrow0\), and \(\int_0^\infty\log(1+w^2/y^2)\kappa_\alpha(y)\,dy\) tends to \(\sum_{t_k>0}\log(1+w^2/t_k^2)\).

Earlier editions: Corollary 27.4.3, page 986 (edition 1).

Source: ch/cg2.tex, label cg:cor:lorentz.

The tilted uniform law has a strictly positive Thorin density for every positive tilt, a Lorentzian smoothing of the lattice 2 pi m

Volume II, Theorem 19.4.3, page 588 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\varphi_U(s)=(1-e^{-s})/s\) be the Laplace transform of the uniform law on \((0,1)\), \(g=\varphi_U'/\varphi_U\) (so \(g(x)=\frac1{e^x-1}-\frac1x\)), \(\ell_\delta(x)=\delta/(\delta^2+x^2)\) and \(\kappa_\delta(y)=\frac1\pi\int_0^\infty(1-\cos xy)e^{-\delta x}(dx-d\lfloor x\rfloor)=\frac1\pi[\operatorname{Re}g(\delta+iy)-g(\delta)]\) for \(y>0\), with \(g(s)=\frac1{e^s-1}-\frac1s\) and \(\lfloor x\rfloor\) the integer part. For every \(\delta>0\), \(\pi\kappa_\delta(y)=\frac{y^2}{\delta(\delta^2+y^2)}-\frac12\coth\frac\delta2\frac{1-\cos y}{\cosh\delta-\cos y}=\sum_{m\ge1}[\ell_\delta(y-2\pi m)+\ell_\delta(y+2\pi m)-2\ell_\delta(2\pi m)]\), and \(\kappa_\delta(y)>0\) for all \(y>0\). Hence \(u\mapsto\frac{\varphi_U(\delta)}{\varphi_U(\delta+\sqrt{2u})}e^{\sqrt{2u}\,g(\delta)}\) is the Laplace transform of a generalised gamma convolution (GGC) with Thorin density \(\tau_\delta(z)=\kappa_\delta(\sqrt{2z})/\sqrt{2z}\). As \(\delta\downarrow0\), \(\kappa_\delta(y)\,dy\to\sum_{m\ge1}\delta_{2\pi m}\) vaguely, and \(\log\frac{\sinh(w/2)}{w/2}=\sum_{m\ge1}\log(1+\frac{w^2}{4\pi^2m^2})\), so \((w/2)/\sinh(w/2)\) is the Laplace transform in \(u=w^2/2\) of \(\sum_me_m/(2\pi^2m^2)\) with independent standard exponential \(e_m\), a GGC with Thorin measure \(\sum_m\delta_{2\pi^2m^2}\) (a law of Biane, Pitman and Yor type).

Earlier editions: Theorem 27.4.4, page 987 (edition 1).

Source: ch/cg2.tex, label cg:thm:uniform.

The shifted zeta equals (2 pi) to the s times the Mellin transform of the positive Thorin density of the tilted uniform law

Volume II, Theorem 19.4.4, page 589 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(g(s)=\frac1{e^s-1}-\frac1s\), let \(\kappa_\delta(y)=\frac1\pi[\operatorname{Re}g(\delta+iy)-g(\delta)]\) for \(y>0\), the positive Thorin density of the uniform law on \((0,1)\) tilted by \(\delta\), and \(\zeta_\delta(s)=(2\pi)^s\sin(\pi s/2)\frac1\pi\int_0^\infty x^{-s}g(x+\delta)\,dx\) (continued analytically in \(s\); for \(\delta<2\pi\) it equals \(\sum_{k\ge0}(\delta/2\pi)^k\frac{(s)_k}{k!}\frac{\sin(\pi s/2)}{\sin(\pi(s+k)/2)}\zeta(s+k)\), with \((s)_k\) the rising factorial). For \(1<\operatorname{Re}s<3\) and every \(\delta>0\), \(\zeta_\delta(s)=(2\pi)^s\int_0^\infty y^{-s}\kappa_\delta(y)\,dy\). At \(\delta=0\) the right side is formally \((2\pi)^s\sum_{m\ge1}(2\pi m)^{-s}=\zeta(s)\).

Earlier editions: Theorem 27.4.5, page 988 (edition 1).

Source: ch/cg2.tex, label cg:thm:mellinthorin.

Kink theorem: the Gauss damped Riemann kernel has finitely many real zeros and infinitely many nonreal zeros

Volume II, Theorem 19.5.1, page 591 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_n(u)=(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\), \(\Phi=2\sum_{n\ge1}\Phi_n\), so that \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(ut)\,du\) for the Riemann Xi function. For \(\delta>0\) let \(\Phi_\delta=2\sum_{n\ge1}e^{-\pi n^2\delta}\Phi_n\) on \([0,\infty)\) and \(\Xi_\delta(t)=2\int_0^\infty\Phi_\delta(u)\cos(ut)\,du\), an even real entire function of order one with \(\Xi_\delta(0)>0\). Then \(\Xi_\delta\) has only finitely many real zeros and infinitely many nonreal zeros, because \(\Phi_\delta'(0)>0\) (using \(\sum_n\Phi_n'(0)=0\), from the evenness of \(\Phi\), and \(\Phi_n'(0)<0\) for \(n\ge2\)) and \(t^2\Xi_\delta(t)\to-2\Phi_\delta'(0)<0\) as \(t\to\infty\) along \(\mathbb R\). Consequently \(\Xi_\delta(0)/\Xi_\delta(i\sqrt w)\) is not the Laplace transform of a generalised gamma convolution.

Earlier editions: Theorem 27.5.1, page 991 (edition 1).

Source: ch/cg3.tex, label cg:thm:kink.

Modular constraints: a Laguerre P'olya damped kernel must have vanishing odd derivatives of the damped kernel at the origin

Volume II, Corollary 19.5.2, page 592 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Phi_n(u)=(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2})\exp(-\pi n^2e^{2u})\) for \(n\ge1\), the terms of the Riemann kernel \(\Phi=2\sum_n\Phi_n\) for which \(\Xi(t)=2\int_0^\infty\Phi(u)\cos(ut)\,du\) is the Riemann Xi function, let \(\Phi_w=2\sum_{n\ge1}w_n\Phi_n\) with real weights \(w_n\) such that \(\sum_n|w_n|n^Ae^{-\pi n^2}<\infty\) for every \(A\ge0\), and \(\Xi_w(t)=2\int_0^\infty\Phi_w(u)\cos(ut)\,du\). If \(\Xi_w\) lies in the Laguerre P'olya class \(\mathcal{LP}\) (locally uniform limits of real polynomials with only real zeros) and is not of the form \(e^{-at^2}\) times a polynomial, then \(\sum_nw_n\Phi_n^{(2j+1)}(0)=0\) for every \(j\ge0\).

Earlier editions: Corollary 27.5.2, page 992 (edition 1).

Source: ch/cg3.tex, label cg:cor:modular.

Volume II, Chapter 20. Approximating clocks: Jensen, Euler product and boundary hitting times

Jensen polynomial laws mu_d of the reciprocal xi function: convergence, existence if and only if RH, and existence up to degree 2e17

Volume II, Proposition 20.1.1, page 598 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. Write \(G(u)=\sum_{k\ge0}c_ku^k\), so that \(G(u)=\mathbb E\cosh(\sqrt u\,V)\) with \(V\) symmetric with density \(\Phi/\Xi(0)\) (\(\Phi\) Riemann’s kernel, \(\Xi(z)=\xi(\frac12+iz)\)) and \(c_k=\mathbb E V^{2k}/(2k)!>0\). For \(d\ge1\) let \(J_d(u)=\sum_{k=0}^d\binom dk\frac{k!}{d^k}c_ku^k=\sum_{k=0}^dc_k\prod_{j<k}(1-\frac jd)u^k\) (up to scaling the Jensen polynomial of degree \(d\) and shift \(0\) of Griffin, Ono, Rolen and Zagier). (1) \(J_d\to G\) uniformly on compact sets, without any hypothesis. (2) If \(J_d\) has only real zeros, they are negative, \(J_d(u)=\prod_{j\le d}(1+u/\ell_{d,j})\), and \(1/J_d\) is the Laplace transform of the GGC law \(\mu_d\) of \(\sum_{j\le d}e_j/\ell_{d,j}\), the time a pure birth chain with rates \(\ell_{d,1},\dots,\ell_{d,d}\) takes to pass through \(d\) states. (3) RH holds if and only if \(\mu_d\) exists for every \(d\), and then \(\mu_d\Rightarrow\mu\). (4) \(\mu_d\) exists for every \(d\le2\cdot10^{17}\), unconditionally (quoted from Griffin, Ono, Rolen and Zagier, who quote Chass'e). (5) Every \(\mu_d\) has the same mean \(c_1\) as \(\mu\), and its variance exceeds that of \(\mu\) by \(2c_2/d\).

Earlier editions: Proposition 24.6.1, page 866 (edition 1).

Source: ch/ht2.tex, label ht:prop:jensen.

Rolle criterion for real simple zeros of a polynomial with positive coefficients

Volume II, Lemma 20.1.2, page 598 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P\) have degree \(d+1\) and positive coefficients, and let \(P'\) have the simple zeros \(-v_1,\dots,-v_d\) with \(0<v_1<\dots<v_d\). Then \(P\) has \(d+1\) simple real zeros if and only if \((-1)^kP(-v_k)>0\) for \(k=1,\dots,d\).

Source: ch/ht2.tex, label ht:lem:rolle.

Interlacing equivalence for hyperbolicity of Jensen polynomials in the degree

Volume II, Theorem 20.1.3, page 599 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(J^{d,n}\) be the Jensen polynomial of degree \(d\) and shift \(n\) with positive coefficients, \(H(d,n)\) the statement that \(J^{d,n}\) has only real and simple zeros (all negative), and let \(0<u_1<\dots<u_d\) and \(0<v_1<\dots<v_d\) be the magnitudes of the zeros of \(J^{d,n}\) and \(J^{d,n+1}\). Let \(I(d,n)\) be the interlacing \(u_1<v_1<u_2<\dots<u_d<v_d\). Assume \(H(d,n)\) and \(H(d,n+1)\). Then \(H(d+1,n)\) holds if and only if \(I(d,n)\) holds.

Source: ch/ht2.tex, label ht:thm:interlace.

Density of a sum of independent exponentials as a Laplace transform of a B spline, the Dirichlet mean form

Volume II, Proposition 20.2.1, page 605 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(0<\lambda_1<\dots<\lambda_N\), let \(e_k\) be independent standard exponential variables and \(f_N\) the density of \(\sum_{k\le N}e_k/\lambda_k\). Then \(f_N(y)=(\prod_{k\le N}\lambda_k)\frac{y^{N-1}}{(N-1)!}\int M(x;\lambda_1,\dots,\lambda_N)e^{-xy}\,\mathrm dx\), where \(M(\cdot;\lambda_1,\dots,\lambda_N)\) is the Curry and Schoenberg B spline with knots \(\lambda_k\), normalised to integral one. Equivalently \(M\) is the density of the Dirichlet mean \(D_N=\sum_kW_k\lambda_k\) with \((W_k)\) uniform on the simplex, and the formula is the case of Dirichlet parameter \(N\) of the density formula of James, Roynette and Yor.

Earlier editions: Proposition 24.7.1, page 871 (edition 1).

Source: ch/ht2.tex, label ht:prop:bspline.

A sum of exponentials is a linear functional of the Gumbel extremal process

Volume II, Theorem 20.2.2, page 606 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(e_k\) be independent standard exponential variables and \(G_1>G_2>\cdots\) the points of the Gumbel extremal process of the book. For any \(0<\lambda_1\le\lambda_2\le\cdots\) with \(\sum1/\lambda_k<\infty\), and \(b_k=k/\lambda_k\), \(\sum_ke_k/\lambda_k\overset d=\sum_{k\ge1}b_k(G_k-G_{k+1})\), and if moreover \(k\log k/\lambda_k\to0\) the series may be rearranged: \(\sum_{k\ge1}b_k(G_k-G_{k+1})=b_1G_1+\sum_{k\ge2}(b_k-b_{k-1})G_k\). In particular, under RH, the reciprocal xi law \(\mu\) is the law of \(\sum_kk\gamma_k^{-2}(G_k-G_{k+1})\). For finite \(N\) the truncation \(\mu_N\) is the L statistic \(\sum_{k\le N}b_k(e_{(N-k+1)}-e_{(N-k)})\) of \(N\) exponentials, with \(e_{(0)}=0\).

Earlier editions: Theorem 24.7.2, page 872 (edition 1).

Source: ch/ht2.tex, label ht:thm:extremal.

Truncated Euler products give GGC laws Y_P, and first passage to the mean level decomposes as Y_P plus the first passage to the log of N_P

Volume II, Proposition 20.3.1, page 609 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(P\ge2\) let \(\zeta_P(z)=\prod_{p\le P}(1-p^{-z})^{-1}\), fix \(\alpha>0\), and let \(N_P\) be the random integer with \(\Pr(N_P=n)=n^{-\alpha}/\zeta_P(\alpha)\) for \(n\) composed of primes up to \(P\). Let \(L=\log N_P\), \(B_{\alpha,P}=\mathbb EL=\sum_{p\le P}\log p\,p^{-\alpha}/(1-p^{-\alpha})\), and \(T_a\) the first passage time of standard Brownian motion to the level \(a\). For every \(\alpha>0\) and every \(P\) the function \(\Phi_{\alpha,P}(u)=e^{-B_{\alpha,P}\sqrt{2u}}/\mathbb Ee^{-\sqrt{2u}L}=\frac{\zeta_P(\alpha)}{\zeta_P(\alpha+\sqrt{2u})}e^{-B_{\alpha,P}\sqrt{2u}}\) is the Laplace transform of a GGC law \(Y_P\) with zero drift, whose Thorin measure has density in the ordinate \(\tau\) (atom positions \(\tau^2/2\)) equal to \(\frac1\pi\sum_{p\le P}\sum_{k\ge1}\log p\,p^{-k\alpha}(1-\cos(k\tau\log p))\ge0\). In particular this holds on the critical line \(\alpha=\frac12\), unconditionally. If \(Y_P\) is independent of \(N_P\) and of the Brownian motion, then \(T_{\mathbb E\log N_P}\overset d=Y_P+T_{\log N_P}\).

Earlier editions: Proposition 24.8.1, page 875 (edition 1).

Source: ch/ht2.tex, label ht:thm:trunc.

One prime blocks, Möbius and Liouville factors and the Pólya Gamma law

Volume II, Remark 20.3.2, page 610 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Fix a prime \(p\), write \(\ell=\log p\), \(q=p^{-\alpha}\) with \(\alpha>0\), \(w=\sqrt{2u}\) and \(\vartheta=\tau\ell\). For the Möbius (Euler factor) case the Thorin density of the one prime GGC is \(\frac1\pi\sum_{k\ge1}\ell q^k(1-\cos k\vartheta)=\frac1\pi\frac{\ell q(1+q)(1-\cos\vartheta)}{(1-q)(1-2q\cos\vartheta+q^2)}\), which vanishes at \(\tau=2\pi k/\ell\). For the Liouville factor, \(\Phi^{+}_{p,\alpha}(u)=\frac{1+q}{1+qp^{-w}}\exp\big(-\frac{w\ell q}{1+q}\big)\) is, for every \(\alpha>0\), the Laplace transform of a GGC with zero drift and Thorin density in the ordinate \(\tau\) equal to \(\frac1\pi\frac{\ell q(1-q)(1-\cos\vartheta)}{(1+q)(1+2q\cos\vartheta+q^2)}\ge0\), with bumps at \(\tau=\pi(2k-1)/\ell\); the proof applies the sine squared lemma to the signed measure \(\sum_k(-1)^{k+1}\ell q^k\delta_{k\ell}\), whose total variation is that of the Möbius case. As \(\alpha\downarrow0\), \(\Phi^{+}_{p,\alpha}(u)\to1/\cosh(\ell\sqrt{u/2})\), the Laplace transform of \(\ell^2\omega\) with \(\omega\) a Pólya Gamma variable \(\mathrm{PG}(1,0)\), and the Thorin densities converge vaguely to \(\sum_{k\ge1}\delta_{\pi(2k-1)/\ell}\). For \(\alpha>1\) the products over all primes converge and give the GGC Laplace transforms \(\frac{\zeta(\alpha)}{\zeta(\alpha+w)}e^{w\frac{\zeta'}{\zeta}(\alpha)}\) and \(\frac{\zeta(2\alpha+2w)\zeta(\alpha)}{\zeta(\alpha+w)\zeta(2\alpha)}e^{wG_\lambda(\alpha)}\), with \(G_\lambda(s)=\frac{\zeta'}{\zeta}(s)-2\frac{\zeta'}{\zeta}(2s)\), whose Thorin densities are \(\pi^{-1}[\operatorname{Re}\frac{\zeta'}{\zeta}(\alpha+i\tau)-\frac{\zeta'}{\zeta}(\alpha)]\) and \(\pi^{-1}[\operatorname{Re}G_\lambda(\alpha+i\tau)-G_\lambda(\alpha)]\). For \(0<\alpha<1\) neither continued function is the Laplace transform of a probability law, whatever the zeros of \(\zeta\) do: the first vanishes at \(u=(1-\alpha)^2/2\), and the second, defined for \(\alpha\ne\frac12\), vanishes there when \(\frac12<\alpha<1\) and has a pole at \(u=(\frac12-\alpha)^2/2\) when \(0<\alpha<\frac12\). Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Remark 24.8.2, page 876 (edition 1).

Source: ch/ht2.tex, label ht:rem:blocks.

Brownian truncated Euler product identity for xi centred at one half

Volume II, Theorem 20.3.3, page 611 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function, \(\gamma\) the zeros of \(\xi(\frac12+iz)\) counted with multiplicity (so that \(\frac12+i\gamma\) runs over the nontrivial zeros of \(\zeta\)), \(\gamma_1=14.1347\ldots\) the least ordinate of a zero, with \(\operatorname{Re}\gamma^2\ge\gamma_1^2\) for every zero because zeros off the critical line have ordinate above \(3\times10^{12}\), \(T_x\) the first passage time of standard Brownian motion to the level \(x>0\), \(U>0\), and \(s\in\mathbb{C}\) with \(\operatorname{Re}(s^2)>-\gamma_1^2\). Put \(B_U(x,s)=\mathbb{E}[1-e^{-s^2T_x/2};T_x\le U]=\operatorname{erfc}\frac{x}{\sqrt{2U}}-\frac12\big[e^{-xs}\operatorname{erfc}\frac{x-sU}{\sqrt{2U}}+e^{xs}\operatorname{erfc}\frac{x+sU}{\sqrt{2U}}\big]\), \(k_\infty(u)=e^{u/8}-\frac{\log\pi}{2\sqrt{2\pi u}}+\frac1{2\pi}\int_0^\infty e^{-ur^2/2}\operatorname{Re}\frac{\Gamma'}{\Gamma}(\frac14+\frac{ir}2)dr\), \(A_U(s)=\int_0^U(1-e^{-s^2u/2})k_\infty(u)\frac{du}u\), \(R_U(s)=\sum_{\operatorname{Re}\gamma>0}[E_1(\frac{\gamma^2U}2)-E_1(\frac{(\gamma^2+s^2)U}2)]\) with \(E_1(z)=\int_1^\infty e^{-zv}\frac{dv}v\), and \(E_{p,U}(s)=\exp\big(\sum_{m\ge1}\frac{p^{-m/2}}mB_U(m\log p,s)\big)\). Then, with no hypothesis on the zeros, \(\frac{\xi(1/2)}{\xi(1/2+s)}=e^{-A_U(s)}\prod_pE_{p,U}(s)e^{-R_U(s)}\), the product converging absolutely, and \(|R_U(s)|\le\frac2Ue^{-m_sU/2}\sum_{\operatorname{Re}\gamma>0}\big(\frac1{\operatorname{Re}\gamma^2}+\frac1{\operatorname{Re}(\gamma^2+s^2)}\big)\) with \(m_s=\gamma_1^2+\min(0,\operatorname{Re}(s^2))>0\). If moreover \(\operatorname{Re}s>0\) and \(\operatorname{Re}(s^2)\ge0\), then \(E_{p,U}(s)\to(1-p^{-1/2-s})/(1-p^{-1/2})\) as \(U\to\infty\). The proof uses the ordinate bound \(3\times10^{12}\) for zeros off the critical line (Platt and Trudgian 2021). Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Theorem 24.8.3, page 878 (edition 1).

Source: ch/ht2.tex, label ht:thm:euler.

Numerical check of the Brownian truncated Euler product identity

Volume II, Table 20.6, page 613 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. For the Brownian truncated Euler product identity for \(\xi(\frac12)/\xi(\frac12+s)\) with truncation time \(U\), the two sides \(\log\frac{\xi(1/2+s)}{\xi(1/2)}\) and \(A_U-\sum_p\log E_{p,U}\) were computed at \((s,U)=(0.1,0.05)\), \((0.1,1)\), \((0.49,0.05)\), \((0.49,1)\), \((2,1)\) and \((0.3+3i,0.25)\), with primes up to \(10^3\) for \(U\le0.25\) and \(10^6\) for \(U=1\) and the first \(1700\) zeros for the zero remainder \(R_U\) (the rest being bounded by the identity’s own bound). The residual of the identity is below \(10^{-30}\) in every row; at \(U=0.05\) the zero remainder is visible (for example \(3.39\times10^{-7}\) at \(s=0.1\) against a bound of \(1.25\times10^{-2}\)) and at \(U=1\) it is below the working precision of \(30\) digits. This is a computation at these six points and says nothing about other \((s,U)\).

Earlier editions: Table 24.8, page 880 (edition 1).

Source: ch/ht2.tex, label ht:tab:euler.

The Euler hitting times converge after scaling to a spectrally positive stable law of index 3/2

Volume II, Theorem 20.3.4, page 614 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Take \(\alpha=\frac12\) in the truncated Euler product laws: \(Y_P\) is the GGC law with transform \(\zeta_P(\frac12)e^{-B\sqrt{2u}}/\zeta_P(\frac12+\sqrt{2u})\), \(L_P=\log N_P\), \(\kappa_j(P)\) the \(j\)th cumulant of \(L_P\) (so \(K_P(w)=\log\mathbb Ee^{-w(L_P-\mathbb EL_P)}=\sum_{p\le P}\sum_{m\ge1}\frac{p^{-m/2}}m(e^{-wm\log p}-1+wm\log p)\)), and \(\rho_j(P)=\kappa_j(P)/\kappa_3(P)^{j/3}\); \(Y_P\) has mean \(\kappa_2(P)=\operatorname{Var}L_P\). As \(P\to\infty\), \(Z_P=(Y_P-\kappa_2(P))/\kappa_3(P)^{2/3}\Rightarrow Z\), \(\log\mathbb Ee^{-sZ}=(2s)^{3/2}/6\) for \(\operatorname{Re}s\ge0\), a spectrally positive stable law of index \(\frac32\). More precisely, for \(\operatorname{Re}s\ge0\), \(|\log\mathbb Ee^{-sZ_P}-(2s)^{3/2}/6|\le\frac{|s|^2}6\rho_4(P)\), and \(\rho_4(P)\to0\).

Earlier editions: Theorem 24.8.4, page 880 (edition 1).

Source: ch/ht2.tex, label ht:thm:32.

Thorin rigidity: any factorisation of the reciprocal xi law into two GGC laws forces RH and splits the zeros

Volume II, Theorem 20.4.1, page 615 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\rho=\frac12+\delta+i\gamma\) its nontrivial zeros, and for each zero in the upper half plane let \(u_\rho=(\rho-\frac12)^2\) (equal to \(-\gamma^2\) under the Riemann Hypothesis, RH). Let \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) (entire of order one half) and let \(\mu\) be the law on \([0,\infty)\), the law of the clock \(T\) of the book, with Laplace transform \(1/G\) and density \(f\); \(\mu\) is a generalised gamma convolution (GGC) if and only if RH holds, and then \(T=\sum_{\gamma>0}e_\gamma/\gamma^2\) with independent standard exponential variables \(e_\gamma\). Write \(w_\gamma=1/G'(-\gamma^2)\) under RH, and \(\gamma_1<\gamma_2<\gamma_3<\dots\) for the ordinates. If \(\mu=\mu_1*\mu_2\) with \(\mu_1\) and \(\mu_2\) GGC laws, then RH holds and the Thorin measures are \(U_1=\sum_{\gamma>0}a_\gamma\delta_{\gamma^2}\) and \(U_2=\sum_{\gamma>0}(m_\gamma-a_\gamma)\delta_{\gamma^2}\) with \(0\le a_\gamma\le m_\gamma\), where \(m_\gamma\) is the multiplicity of the zero (\(m_\gamma=1\) for simple zeros), and \(\mu_1\) is the law of \(\sum_\gamma g_\gamma/\gamma^2\) with independent gamma variables \(g_\gamma\) of shapes \(a_\gamma\). No GGC law with drift or with a nonzero absolutely continuous Thorin part is a GGC factor of \(\mu\) with a GGC cofactor.

Earlier editions: Theorem 24.9.1, page 881 (edition 1).

Source: ch/ht2.tex, label ht:thm:rigid.

Small time enclosure of the inverse first passage boundary of the reciprocal xi law, independent of RH

Volume II, Proposition 20.7.1, page 618 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\) be the reciprocal xi law of the book, with Laplace transform \(1/G\), \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) over the zeros of the Riemann zeta function, and let \(b\) be the boundary of the inverse first passage problem for \(\mu\): \(b\) is such that \(\tau_b=\inf\{t>0:W_t\ge b(t)\}\) for a standard Brownian motion \(W\) has law \(\mu\). Write \(F(t)=\mu(0,t]\) and \(m(t)=\inf_{0<s\le t}b(s)\), and let \(\Phi_{\mathrm N}\) be the standard normal distribution function, \(\bar\Phi_{\mathrm N}=1-\Phi_{\mathrm N}\). For every \(t>0\), \(\sqrt t\,\bar\Phi_{\mathrm N}^{-1}(F(t))\le b(t)\) and \(m(t)\le\sqrt t\,\bar\Phi_{\mathrm N}^{-1}(\frac12F(t))\). Moreover \(-\log F(t)\sim(\log(1/t))^2/(16t)\) as \(t\downarrow0\), so \(\liminf_{t\downarrow0}b(t)/\log(1/t)\ge\frac1{2\sqrt2}\) and \(\limsup_{t\downarrow0}m(t)/\log(1/t)\le\frac1{2\sqrt2}\), and if \(b\) is nonincreasing near \(0\) then \(b(t)\sim\log(1/t)/(2\sqrt2)\). None of this depends on RH.

Earlier editions: Proposition 24.12.1, page 884 (edition 1).

Source: ch/ht3.tex, label ht:prop:smallb.

Moving a pair of zeros off the line by delta multiplies the Laplace transform by 1+6 delta^2 s/gamma^4 to first order

Volume II, Proposition 20.7.2, page 620 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mu\) be the reciprocal xi law, with Laplace transform \(1/G\), \(G(u)=\prod_{\gamma>0}(1-u/u_\rho)\) over the zeros of the Riemann zeta function (the control being the case of all zeros on the line). Moving the pair of zeros at height \(\gamma\) off the line by \(\delta\) multiplies the Laplace transform, relative to the control, by \(1+6\delta^2s/\gamma^4+O(\delta^2s^2/\gamma^6)\) for \(|s|\ll\gamma^2\). To first order this is the shift \(\tau\mapsto\tau-6\delta^2/\gamma^4\) of the mean \(\tau\).

Earlier editions: Proposition 24.12.2, page 887 (edition 1).

Source: ch/ht3.tex, label ht:prop:resp.

The reciprocal xi law as the reciprocal of an expected cosh under the size biased Biane Pitman Yor measure

Volume II, Proposition 20.8.1, page 623 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(Y=\sqrt{2/\pi}(\max_ub_u-\min_ub_u)\) for a standard Brownian bridge \(b\), so that \(\mathbb EY^s=2\xi(s)\), and let \(\tau\) be the clock with \(\mathbb Ee^{-s\tau}=\xi(\frac12)/\xi(\frac12+\sqrt s)\). Let \(\mathbb P^*\) have density \(Y^{1/2}/\mathbb EY^{1/2}\) with respect to \(\mathbb P\) and let \(U=\log Y\). Under \(\mathbb P^*\) the variable \(U\) is symmetric with density \(\Phi/\Xi(0)\), where \(\Xi(z)=\xi(\frac12+iz)\) and \(\Phi\) is the kernel of Riemann and P'olya; equivalently \(U/2\) has density \(2\Phi(2u)/\Xi(0)\). Moreover \(\mathbb Ee^{-s\tau}=\xi(\frac12)/\xi(\frac12+\sqrt s)=1/\mathbb E^*\cosh(\sqrt sU)=1/\mathbb E^*[\mathbb E[e^{-sT}|U]^{-1}]\), where, given \(U\), \(T\) is the exit time of a standard Brownian motion from \((-|U|/\sqrt2,|U|/\sqrt2)\).

Earlier editions: Proposition 24.13.1, page 890 (edition 1).

Source: ch/ht3.tex, label ht:prop:bpy.

The law of the maximum of a Brownian excursion gives the Biane, Pitman and Yor identity for xi term by term

Volume II, Result 20.8.2, page 623 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(M\) be the maximum of a standard Brownian excursion, whose law is (Chung 1976; Kennedy 1976) \(\mathbb{P}(M\le m)=\sum_{n\in\mathbb{Z}}(1-4n^2m^2)e^{-2n^2m^2}\) for \(m>0\), and \(Y=\sqrt{2/\pi}\,M\), which has the law of \(\sqrt{2/\pi}\) times the range of a Brownian bridge, by the path transformation of Vervaat. Let \(\psi(x)=\sum_{n\ge1}e^{-\pi n^2x}\), \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) and \(\Phi(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\). Then for \(y>0\), \(\mathbb{P}(Y>y)=2\sum_{n\ge1}(2\pi n^2y^2-1)e^{-\pi n^2y^2}\), \(\mathbb{E}(Y-y)^+=2y\,\psi(y^2)\) and the density of \(Y\) is \(f_Y(y)=2\Phi(\log y)\,y^{-3/2}\). Hence \(\mathbb{E}Y^s=2\int_{\mathbb{R}}\Phi(u)e^{(s-\frac12)u}\,du=2\xi(s)\) for every complex \(s\), the identity of Biane, Pitman and Yor derived term by term from the law of \(M\). With \(y=e^u\), the function \(k(u)=e^{-u/2}-2e^{u/2}\psi(e^{2u})\) equals \(e^{-u/2}\,\mathbb{E}\min(Y,e^u)\) for \(u\in\mathbb{R}\); its evenness is the case \(g(y)=\min(1,e^u/y)\) of the size bias identity \(\mathbb{E}g(1/Y)=\mathbb{E}\,Yg(Y)\) of Biane, Pitman and Yor, and its derivative at \(0\) vanishes, which reads \(\mathbb{P}(M>\sqrt{\pi/2})=\mathbb{P}(Y>1)=\frac12-\psi(1)=0.456783\) (computed). The derivation is from a note of Nicholas Polson merged into the book.

Earlier editions: Result 24.13.2, page 890 (edition 1).

Source: ch/ht3.tex, label ht:res:chung.

The Davenport Heilbronn theta function is positive, giving a symmetric variable and a positive Polya kernel for a function violating RH

Volume II, Proposition 20.8.3, page 625 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(a_n\) be the Davenport and Heilbronn coefficients, periodic modulo \(5\) with \((a_0,\dots,a_4)=(0,1,\kappa,-\kappa,-1)\) and \(\kappa=0.2840790438\), let \(\xi_{DH}\) be the completed Davenport and Heilbronn function (a function with a functional equation and an off line zero, as in the book), and \(\theta_{DH}(x)=\sum_{n\ge1}a_nn\,e^{-\pi n^2x/5}\). Then \(\theta_{DH}(x)>0\) for every \(x>0\). Consequently there is a probability law on \((0,\infty)\) whose Mellin transform is a constant multiple of \(\xi_{DH}\), a symmetric \(U_{DH}\) with \(\mathbb E^*e^{wU_{DH}}=\xi_{DH}(\frac12+w)/\xi_{DH}(\frac12)\), and a positive P'olya kernel \(\Phi_{DH}\).

Earlier editions: Proposition 24.13.3, page 891 (edition 1).

Source: ch/ht3.tex, label ht:prop:dhtheta.

Wiener and Gamma representation of the base point laws, and RH as monotonicity of the smoothed zero counting functions N_alpha

Volume II, Proposition 20.8.4, page 626 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(\alpha>\frac12\) let \(\mu^{(\alpha)}\) be the base point law with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt s)\) and \(N_\alpha'(t)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\alpha+it)=\frac1\pi\sum_\rho\frac{\alpha-\beta}{(\alpha-\beta)^2+(t-\gamma)^2}\) (\(\beta=\operatorname{Re}\rho\)), \(N_\alpha(0)=0\), and \(h_\alpha(u)=N_\alpha^{-1}(u)^{-2}\). On one gamma process \(\gamma\), set \(X_\alpha=\int_0^\infty h_\alpha(u)\,\mathrm d\gamma_u\). Then \(X_\alpha\) has law \(\mu^{(\alpha)}\) for every \(\alpha\ge\sup_\rho\operatorname{Re}\rho\), in particular for every \(\alpha\ge1\). RH holds if and only if \(N_\alpha\) is nondecreasing for every \(\alpha>\frac12\).

Earlier editions: Proposition 24.13.4, page 893 (edition 1).

Source: ch/ht3.tex, label ht:prop:wg.

The Taylor coefficients of xi(1/2+sqrt s) are a Polya frequency sequence if and only if RH, and unconditionally PF2

Volume II, Theorem 20.9.1, page 628 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(F(s)=\xi(\frac12+\sqrt s)/\xi(\frac12)=\sum_{n\ge0}c_ns^n\), so that \(c_n=\mathbb EV^{2n}/(2n)!>0\) for the symmetric variable \(V\) with density \(\Phi/\Xi(0)\) (\(\Phi\) Riemann’s kernel), \(c_0=1\), and let \(s_\rho=(\rho-\frac12)^2\) for the zeros \(\rho\) of \(\zeta\) in the upper half plane, with \(\theta_\rho\in[0,\pi)\) the angle between \(s_\rho\) and the negative real axis (\(\theta_\rho=0\) for a zero on the line). A sequence is PF\(_k\) if all minors of its Toeplitz matrix of order at most \(k\) are nonnegative, and a P'olya frequency (PF) sequence if it is PF\(_k\) for every \(k\). (i) RH holds if and only if \((c_n)\) is a P'olya frequency sequence, that is, every minor of the Toeplitz matrix \((c_{i-j})\) is nonnegative. (ii) Unconditionally, \((c_n)\) is PF\(_2\): \(c_n^2\ge c_{n-1}c_{n+1}\) for all \(n\ge1\).

Earlier editions: Theorem 24.16.1, page 902 (edition 1).

Source: ch/ht5.tex, label ht:thm:coef.

A three term quadratic coefficient sequence is PF_k if and only if its angle is at most pi/(k+1)

Volume II, Lemma 20.10.1, page 630 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A sequence is PF\(_k\) if all minors of order at most \(k\) of its Toeplitz matrix are nonnegative. For \(r>0\) and \(0\le\theta<\pi\), the sequence \((1,\,2r^{-1}\cos\theta,\,r^{-2})\), the coefficient sequence of \((1+s/re^{i\theta})(1+s/re^{-i\theta})\), is PF\(_k\) if and only if \(\theta\le\pi/(k+1)\).

Earlier editions: Lemma 24.17.1, page 904 (edition 1).

Source: ch/ht5.tex, label ht:lem:three.

Angle theorem: if every zero has angle at most pi/(k+1) then the coefficient sequence of xi is PF_k

Volume II, Theorem 20.10.2, page 630 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(F(s)=\xi(\frac12+\sqrt s)/\xi(\frac12)=\sum_{n\ge0}c_ns^n\), so that \(c_n=\mathbb EV^{2n}/(2n)!>0\) for the symmetric variable \(V\) with density \(\Phi/\Xi(0)\) (\(\Phi\) Riemann’s kernel), \(c_0=1\), and let \(s_\rho=(\rho-\frac12)^2\) for the zeros \(\rho\) of \(\zeta\) in the upper half plane, with \(\theta_\rho\in[0,\pi)\) the angle between \(s_\rho\) and the negative real axis (\(\theta_\rho=0\) for a zero on the line). A sequence is PF\(_k\) if all minors of its Toeplitz matrix of order at most \(k\) are nonnegative, and a P'olya frequency (PF) sequence if it is PF\(_k\) for every \(k\). If every zero of \(\zeta\) satisfies \(\theta_\rho\le\pi/(k+1)\), then the Taylor coefficients \((c_n)\) of \(F\) form a PF\(_k\) sequence.

Earlier editions: Theorem 24.17.2, page 904 (edition 1).

Source: ch/ht5.tex, label ht:thm:angle.

Unconditional rung count: the coefficient sequence of xi is PF_k for every k up to about 9.4e12

Volume II, Corollary 20.10.3, page 630 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function, \(F(s)=\xi(\frac12+\sqrt s)/\xi(\frac12)=\sum_{n\ge0}c_ns^n\), so that \(c_n=\mathbb EV^{2n}/(2n)!>0\) for the symmetric variable \(V\) with density \(\Phi/\Xi(0)\) (\(\Phi\) Riemann’s kernel), \(c_0=1\), and let \(s_\rho=(\rho-\frac12)^2\) for the zeros \(\rho\) of \(\zeta\) in the upper half plane, with \(\theta_\rho\in[0,\pi)\) the angle between \(s_\rho\) and the negative real axis (\(\theta_\rho=0\) for a zero on the line). A sequence is PF\(_k\) if all minors of its Toeplitz matrix of order at most \(k\) are nonnegative, and a P'olya frequency (PF) sequence if it is PF\(_k\) for every \(k\). Unconditionally, \((c_n)\) is PF\(_k\) for every \(k\le\lfloor\pi H\rfloor-1=9424777960768\), where \(H=3\cdot10^{12}\) is the height to which all zeros are known to lie on the line (Platt and Trudgian). More generally, \((c_n)\) is PF\(_K\) as soon as every zero with \(\gamma\le(K+1)/\pi\) lies on the line.

Earlier editions: Corollary 24.17.3, page 904 (edition 1).

Source: ch/ht5.tex, label ht:cor:rungs.

Volume II, Chapter 21. The quasi Riemann hypothesis and the Thorin programme

Balance of the exponent in Step 2 of the sextic twist method, no half plane left of six sevenths

Volume II, Proposition 21.2.2, page 646 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\nu\) be a finite order Hecke character of \(\mathbb{Q}(\sqrt{-3})\), \(W\) a smooth cutoff, and \(A_u(D)=\sum_{\mathfrak n}\mu(\mathfrak n)\nu(\mathfrak n)\chi_{\mathfrak n}(u)W(N\mathfrak n/D)\) for \(u\in\mathbb{Z}[\omega]\), with \(\chi_{\mathfrak n}(u)=(u/\mathfrak n)_6\) the sextic residue symbol (the rows of Step 2 of the announced zero free half plane argument). Suppose that for some \(h>0\) the mean square bound \(\sum_{0<Nu\le H}|A_u(D)|^2\ll D^{1+\varepsilon}H\) holds with \(H=D^h\), and let \(Y=H^{1/6}\). Then \(|A_1(D)|^2\ll D^{1+\varepsilon}H/Y+D^2/Y^2\), so \(A_1(D)\ll D^{e(h)+\varepsilon}\) with \(e(h)=\max\big(\frac12+\frac{5h}{12},\,1-\frac h6\big)\). For every \(h>0\), \(e(h)\ge\frac67\), with equality only at \(h=\frac67\); \(e(1)=\frac{11}{12}\) and \(e(\frac{11}{10})=\frac{23}{24}\). So, with the trivial bound \(O(D/Y)\) for \(A_1(D)-A_{p^6}(D)\), no mean square bound of this form, fed into this step with rows of the form \(p^6\), gives a half plane to the left of \(\operatorname{Re}s=\frac67\). This is a statement about the bookkeeping of the step as reported, not about the correctness of the announced proofs.

Source: ch/nc_qrh.tex, label qrh:prop:balance.

What a zero free half plane gives the Thorin programme (conditional on no zero right of theta)

Volume II, Corollary 21.3.1, page 647 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on zeta has no zero with real part above theta, for a given theta in [1/2,1).

Statement. Let \(\frac12\le\theta<1\) and assume that \(\zeta\) has no zero with \(\operatorname{Re}s>\theta\). Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), and for a level \(\theta'\) let \(u_{\theta'}(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta'+iv)\) and \(b_{\theta'}=(\xi'/\xi)(\theta')=\pi u_{\theta'}(0)\). Then for every level \(\theta'>\theta\): (i) \(u\mapsto\xi(\theta')/\xi(\theta'+\sqrt u)\) is the Laplace transform of a generalised gamma convolution \(T_{\theta'}\) with Thorin density \(u_{\theta'}\) in the chart \(v=\sqrt z\), and \(u_{\theta'}>0\); it is a sum of Cauchy densities of widths in \([\theta'-\theta,\theta'-1+\theta]\); (ii) \(u_{\theta'}(v)>u_{\theta'}(0)\) for every real \(v\ne0\), so \(T_{\theta'}=S_{\theta'}+T^\circ_{\theta'}\) with \(S_{\theta'}\) positive stable of index \(\frac12\), \(\mathbb{E}e^{-uS_{\theta'}}=e^{-b_{\theta'}\sqrt u}\), and \(T^\circ_{\theta'}\) an independent GGC with no drift; (iii) the mass deficit of the Thorin mass function against the zero count vanishes at every height, and the level mass function is continuous on \((\theta,1]\); (iv) the compensated Levy form of \(\log\xi(\theta'+s)/\xi(\theta')\) converges, with error \(O(X^{\theta-\theta'+\delta})\) for every \(\delta>0\) at prime cutoff \(X\), and the \(2\sin^2\) identity converts it into \(u_{\theta'}\). At the level \(\theta\) itself, (i) holds with an atom at the ordinate of each zero on the line \(\operatorname{Re}s=\theta\), if there is one. Conversely, (i) for every \(\theta'>\theta\), or (ii) for every \(\theta'\in(\theta,1)\), implies that \(\zeta\) has no zero with real part above \(\theta\).

Source: ch/nc_qrh.tex, label qrh:cor:thorin.

The canonical Levy measure of xi at a basepoint is signed, so the 2 sin squared lemma fails alone

Volume II, Proposition 21.5.1, page 659 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and, for \(\alpha>\frac12\), let \(x\Pi_\alpha(dx)\) be the canonical measure of the Levy representation of \(\log\xi(\alpha+s)/\xi(\alpha)\), whose atoms sit at \(\log n\) for \(n\ge2\). For every \(\alpha>\frac12\) its continuous part is \(\big(\frac{e^{-(\alpha+2)x}}{1-e^{-2x}}-e^{-(\alpha-1)x}\big)dx=e^{-(\alpha-1)x}\big(\frac{e^{-3x}}{1-e^{-2x}}-1\big)dx\), which is positive on \((0,x_0)\) and negative on \((x_0,\infty)\), where \(x_0=0.281199574323\ldots\) is the root of \(e^{-3x}+e^{-2x}=1\). Since \(x_0<\log2\), the measure \(x\Pi_\alpha\) is negative on \((x_0,\log2)\) for every \(\alpha\), and the sufficient condition \(xL\ge0\) of the \(2\sin^2\) lemma (positivity of the Levy measure times \(x\)) holds at no basepoint.

Source: ch/nc_qrh2.tex, label qrh:prop:signed.

Open problem, minimum at the origin for chi 4 and chi 12 under the half plane

Volume II, Problem 21.1, page 663 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Assume that \(L(s,\chi_4)\) and \(L(s,\chi_{12})\) have no zero with real part above \(\frac78\). Prove that for every level \(\alpha\in(\frac78,1)\) the function \(v_\alpha(\theta)=\operatorname{Re}(\Lambda'/\Lambda)(\alpha+i\theta)\) (\(\pi\) times the level \(\alpha\) Thorin density) of the completed \(L\) function \(\Lambda\) satisfies \(v_\alpha(\theta)>b_\alpha=(\Lambda'/\Lambda)(\alpha)\) for every real \(\theta\ne0\) (the strict minimum at the origin). A window argument with a zero counting formula should apply above the verified heights, below which the zeros are known to lie on the critical line, and a convexity argument would need a margin at levels below one.

Source: ch/nc_qrh2.tex, label qrh:prob:chi4.

Open problem, a Thorin reading of the smoothed Mobius signal

Volume II, Problem 21.2, page 663 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The signal of the continuation argument for the announced zero free half plane is the inverse Mellin transform of \(e^{(s-5/6)^2}H_\eta(s)/L_F^S(s,\eta)\), where \(L_F^S\) is a Hecke \(L\) function of the field \(\mathbb{Q}(\sqrt{-3})\) with the Euler factors at a finite set \(S\) of primes removed and \(H_\eta\) is a smoothing weight; its zeta analogue is a smoothed Mobius sum. Is there a choice of the weight for which the signal is the density of a law whose infinite divisibility, or generalised gamma convolution property, is equivalent to the absence of zeros to the right of a given line, as the Poisson theorem for the shifted clock \(\xi(\frac12)/\xi(\frac12+\sqrt s)\) is? Such a reading would place the estimates of the proof inside the Thorin programme.

Source: ch/nc_qrh2.tex, label qrh:prob:signal.

Open problem, lower bound on the exponent of the cubic theta method

Volume II, Problem 21.3, page 663 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The step from a mean square bound to the Mobius bound, with the trivial bound for the difference of the rows, cannot give an exponent below \(\frac67\) (the balance proposition for sixth power rows). Is there a corresponding lower bound for the full method of the announced zero free half plane argument, independent of Liu’s parametrisation of the optimisation problem (arXiv:2610.12234), and is it above \(\frac34\)? In particular, can any argument that bounds the Mobius function through a mean square over a family of twists of fixed order \(k\) give an exponent below \(\frac{2k-1}{2k}\)?

Source: ch/nc_qrh2.tex, label qrh:prob:exponent.

Open problem, independent certified check of the zero free half plane beyond the verified height

Volume II, Problem 21.4, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. A certified computation in the book gives positivity of the level \(\frac78\) Thorin density \(u_{7/8}(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\frac78+iv)\) up to height \(1420\) with no hypothesis, and excludes zeros of \(\zeta\) with real part above \(\frac34\) up to height \(T^*=1419.9\). Combine the mass deficit (the difference between the Thorin mass up to height \(T\) at level \(\frac78\) and the zero count) with the zero detector of the announced argument to give an independent certified check of the absence of zeros with real part above \(\frac78\) up to a height beyond the verification of the Riemann hypothesis. This would test the announced manuscripts, not prove them.

Source: ch/nc_qrh2.tex, label qrh:prob:cert.

Open problem, does positivity of the Thorin density on one line imply no zero right of it

Volume II, Problem 21.5, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(u_\theta(v)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\theta+iv)\) with \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\). Does positivity of \(u_\theta\) on the single line \(\operatorname{Re}s=\theta\), for one \(\theta\in(\frac12,1)\), imply that \(\zeta\) has no zero with real part above \(\theta\)? The book shows that the analogous implication fails for general entire functions of the shape of \(\xi\). A positive answer for \(\zeta\) would make the descent in level a question about one line at a time.

Source: ch/nc_qrh2.tex, label qrh:prob:oneline.

Open problem, the level one Thorin measure from the Euler product with half widths one half

Volume II, Problem 21.6, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Give a representation of the Thorin measure of \(\xi(1)/\xi(1+\sqrt s)\) (\(\xi\) the completed Riemann zeta function) directly from the Euler product and the \(2\sin^2\) identity, without using the zeros, and prove that its Lorentzian (Cauchy) decomposition into densities centred at the ordinates of the zeros has all half widths equal to \(\frac12\). This is the Riemann hypothesis, stated at a level where the measure is known to be positive and the question is the shape of a known measure.

Source: ch/nc_qrh2.tex, label qrh:prob:levelone.

Open problem, the Thorin measures between one half and seven eighths in a weak metric (under RH)

Volume II, Problem 21.7, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Assume the Riemann hypothesis. The Thorin measure at level \(\alpha\) is absolutely continuous and its limit as \(\alpha\downarrow\frac12\), the sum of point masses at \(\gamma^2\) over the positive ordinates of the zeros, is atomic, so total variation is useless. Quantify the approach in a weak metric for \(\frac12<\alpha<\frac78\), tracking the atom at \(-(1-\alpha)^2\) that the pole of \(\zeta\) detaches and the sharpening of the peaks.

Source: ch/nc_qrh2.tex, label qrh:prob:weak.

Open problem, rate and positivity for truncated Thorin measures from prime powers

Volume II, Problem 21.8, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. The truncated Thorin measures express the zeros as the distributional limit of a centred sum of \(\sin^2\) terms over prime powers. Find the rate of that limit, decide whether a Gaussian or Fejer smoothing in \(t\) gives a positivity statement testable from the primes (name the functional), and prove the damped Weil inequality of the earlier chapters for \(\alpha\) in some interval \((1-\eta,1]\) from the \(2\sin^2\) structure and not from the zeros. A proof for an interval \((1-\eta,1]\) is a zero free half plane, so this is open for every \(\eta>0\).

Source: ch/nc_qrh2.tex, label qrh:prob:trunc.

Open problem, the regularised prime side between seven eighths and one (conditional on the half plane)

Volume II, Problem 21.9, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(\alpha\in(\frac78,1)\) the Euler side of the level \(\alpha\) explicit formula is a divergent Dirichlet series, and if the announced half plane holds (no zero of \(\zeta\) with real part above \(\frac78\)) the zero side is a positive measure. The identity continued to this range then says that the regularised prime side (the stable part minus the \(\sin^2\) defects plus the pole and archimedean kernels) is a positive measure. Make the regularisation explicit in terms of the Mobius sums over the field \(K=\mathbb{Q}(\sqrt{-3})\) that the announced manuscripts control, and extract a lower bound for the Thorin density away from the zeros.

Source: ch/nc_qrh2.tex, label qrh:prob:reg.

Open problem, is the reciprocal of xi a positive mixture of toy reciprocal Xi functions

Volume II, Problem 21.10, page 664 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Determine whether \(1/\xi(\frac12+\sqrt\lambda)\), with \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), is a positive mixture of toy reciprocal Xi functions that include their own gamma factor, that is of reciprocals of completed \(L\) functions of a single prime with super exponential decay; a positive answer would force the Riemann hypothesis. Separately, the hyperbolic secant decomposition expresses the damped Weil functional at damping \(a\) as a Mobius series of damped functionals; at \(a=\frac12\) the first term has prime weights \(\Lambda(n)/(n+1)\) (\(\Lambda\) the von Mangoldt function) and the pole flattened to a constant. Study its positivity alone as \(a\) decreases below \(\frac12\), where \(\operatorname{sech}(a\omega)\cosh(\omega/2)\) grows like \(e^{(1/2-a)|\omega|}\), and whether the terms of the Mobius series with index \(m\ge2\) can be controlled from the primes.

Source: ch/nc_qrh2.tex, label qrh:prob:mixture.

Volume II, Chapter 22. Iwaniec’s approach and the level of the Thorin measure

Consequences of a zero free half plane beyond 7/8: GGC basepoint and even part laws, imaginary zeros of the even and odd parts of xi, and a bound on the de Bruijn Newman constant

Volume II, Theorem 21.4.1, page 649 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on \(\zeta(s)\ne0\) for \(\operatorname{Re}s>\frac78\) (announced in October 2026, not refereed).

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) be Riemann’s xi function, \(\sigma_{\max}=\sup\operatorname{Re}\rho\) over its zeros \(\rho\), \(\Delta=\sigma_{\max}-\frac12\), \(b_\alpha=\frac{\xi'}\xi(\alpha)\), \(U_\alpha(\tau)\) the clock, \(\pi^{-1}\) times the continuous variation of \(\arg\xi(\alpha+it)\) over \(0\le t\le\tau\), and \(A_\alpha(w)=\frac12(\xi(\alpha+w)+\xi(\alpha-w))\), \(B_\alpha(w)=\frac1{2i}(\xi(\alpha+w)-\xi(\alpha-w))\) the even and odd parts of \(w\mapsto\xi(\alpha+w)\). Assume that \(\zeta(s)\ne0\) for \(\operatorname{Re}s>\frac78\), so that \(\Delta\le\frac38\) and every zero has real part in \([\frac18,\frac78]\). (i) For every \(\alpha\ge\frac78\) the law \(\mu^{(\alpha)}\) with Laplace transform \(\xi(\alpha)/\xi(\alpha+\sqrt s)\) is a generalized gamma convolution (GGC). For \(\alpha>\frac78\) its Thorin measure is absolutely continuous, with density \(\pi^{-1}\operatorname{Re}\frac{\xi'}\xi(\alpha+i\tau)\) in the ordinate variable \(\tau\), a sum of Lorentzians of widths at least \(\alpha-\frac78\), and \(\mu^{(\alpha)}((y,\infty))\sim b_\alpha/\sqrt{\pi y}\) as \(y\to\infty\). (ii) For \(\alpha>\frac78\) the clock \(U_\alpha\) is strictly increasing, and \(\operatorname{Re}\frac{\xi'}\xi>0\) on \(\operatorname{Re}s>\frac78\); in terms of \(\zeta\), \(-\operatorname{Re}\frac{\zeta'}\zeta(\alpha+i\tau)<\operatorname{Re}\mathcal{A}(\alpha+i\tau)=\frac12\log\frac\tau{2\pi}+O(\tau^{-2})\) at every height \(\tau\) and for every \(\alpha>\frac78\), where \(\mathcal{A}(s)=\frac1s+\frac1{s-1}-\frac12\log\pi+\frac12\frac{\Gamma'}\Gamma(\frac s2)\). (iii) For \(\alpha\ge\frac78\) the functions \(A_\alpha\) and \(B_\alpha\) have only purely imaginary zeros, simple for \(\alpha>\frac78\); for every \(\alpha>\frac78\) the positive zeros of \(t\mapsto\operatorname{Re}\xi(\alpha+it)\) are the points where \(U_\alpha\) crosses the levels \(k-\frac12\), \(k=1,2,\dots\), and the even part law with Laplace transform \(\xi(\alpha)/A_\alpha(\sqrt s)\) is a GGC. (iv) The de Bruijn Newman constant \(\Lambda\), the infimum of the \(t\) for which \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du\) has only real zeros, where \(\Phi(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u})\) and \(H_0(z)=\frac18\xi(\frac{1+iz}2)\), satisfies \(\Lambda\le2\Delta^2\le\frac9{32}=0.28125\), by de Bruijn’s theorem.

Earlier editions: Theorem 16.12.1, page 668 (edition 1).

Source: ch/polya9.tex, label pol:thm:qrh.

Under (Q_{7/8}) the poles of 1/xi(1/2+sqrt w) lie in a parabola with vertex at most 9/64, and the sliver of the local theorem becomes thinner

Volume II, Proposition 21.4.2, page 650 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The quasi Riemann hypothesis \((\mathrm Q_{7/8})\) of the book, which rests on the unrefereed half plane of OpenAI (2026).

Statement. Assume \((\mathrm Q_{7/8})\), which rests on the unrefereed half plane of OpenAI (2026). Every pole \((\rho-\frac12)^2\) of \(1/\xi(\frac12+\sqrt w)\) lies in the closed region \((\operatorname{Im}w)^2\le4\Delta^2(\Delta^2-\operatorname{Re}w)\le\frac9{16}(\frac9{64}-\operatorname{Re}w)\), bounded by a parabola with focus \(0\) and vertex \(\Delta^2\le\frac9{64}\), against the vertex \(\frac14\) with no hypothesis. Theorem bs:thm:local holds with the sliver \(\mathcal E_{T_0}\) replaced by the thinner set \(\mathcal E'_{T_0}=\{x+iy:\ x<-T_0^2+\frac34T_0+\frac9{64},\ y^2\le\frac9{16}(\frac9{32}-x)\}\), whose half width at the abscissa \(-X^2\) is about \(\frac34X\), against about \(X\) for \(\mathcal E_{T_0}\).

Earlier editions: Proposition 16.12.2, page 670 (edition 1).

Source: ch/polya9.tex, label tx5:prop:sliver.

Gaussian smoothed Möbius sum: explicit formula over the zeros, with RH and the quasi Riemann hypothesis as growth exponents, and the trivial zero term as a finite range leading term only

Volume II, Proposition 21.4.3, page 652 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(x>0\) let \(M_{\rm g}(x)=\frac1{2\pi i}\int_{(2)}x^{v-2/3}\frac{e^{(v-5/6)^2}}{\zeta(v)}dv=\frac{x^{-2/3}}{2\sqrt\pi}\sum_{n\ge1}\mu(n)(\frac xn)^{5/6}\exp(-\frac14\log^2\frac xn)\), and \(\sigma_{\max}=\sup\operatorname{Re}\rho\). (i) For every \(K\ge1\) there are heights \(T_m\in[m,m+1]\) such that \(M_{\rm g}(x)=\lim_m\sum_{|\gamma|<T_m}\operatorname{Res}_{v=\rho}\frac{x^{v-2/3}e^{(v-5/6)^2}}{\zeta(v)}+\sum_{k=1}^K\frac{e^{(2k+5/6)^2}}{\zeta'(-2k)}x^{-2k-2/3}+I_K(x)\), with \(I_K(x)=O(x^{-2K-5/3})\); at a simple zero the residue is \(e^{(\rho-5/6)^2}x^{\rho-2/3}/\zeta'(\rho)\), and as \(K\to\infty\) the trivial zero series diverges. (ii) \(\sigma_{\max}\le\sigma\) if and only if \(M_{\rm g}(x)=O(x^{\sigma-2/3+\varepsilon})\) as \(x\to\infty\) for every \(\varepsilon>0\); RH is the exponent \(-\frac16\) and \((\mathrm Q_{7/8})\) the exponent \(\frac5{24}\). (iii) \(M_{\rm g}(x)=c_1x^{-8/3}+E(x)+O(x^{-11/3})\) with \(c_1=e^{(17/6)^2}/\zeta'(-2)=-1.00659\times10^5\) and \(E\) the sum over the zeros, and \(E(x)\) is not \(O(x^{\sigma-2/3})\) for any \(\sigma<\sigma_{\max}\); so \(c_1x^{-8/3}\) is the leading term on a finite range of \(x\) only. Computed three ways, \(M_{\rm g}(10^3)=-5.53094\times10^{-4}\), and \(M_{\rm g}(x)/(c_1x^{-8/3})=0.5495,0.9470,0.99856,0.999983\) at \(x=10^3,\dots,10^6\).

Earlier editions: Proposition 16.12.3, page 671 (edition 1).

Source: ch/polya9.tex, label tx3:prop:signal.

Under (Q_{7/8}), M(x) is O(x^{7/8+epsilon}), the Möbius series converges to 1/zeta on Re s above 7/8, and the Möbius series of 1/zeta(1/2+sqrt s) converges on the parabola region H_{3/8}

Volume II, Proposition 21.4.4, page 653 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on The quasi Riemann hypothesis \((\mathrm Q_{7/8})\) of the book, which rests on the unrefereed half plane of OpenAI (2026).

Statement. Assume \((\mathrm Q_{7/8})\), which rests on the unrefereed half plane of OpenAI (2026). Then \(M(x)=\sum_{n\le x}\mu(n)=O(x^{7/8+\varepsilon})\) for every \(\varepsilon>0\), and \(\sum_n\mu(n)n^{-s}\) converges to \(1/\zeta(s)\) on \(\operatorname{Re}s>\frac78\). So the Möbius series of \(1/\zeta(\frac12+\sqrt s)\) converges on \(H_{3/8}=\{\operatorname{Re}\sqrt s>\frac38\}=\{u+iy:\ u>\frac9{64}-\frac{16}9y^2\}\), which contains the region \(H_{1/2}\) of absolute convergence of Section ch:rz1. The proof follows Titchmarsh’s argument for the Riemann hypothesis, with \(\frac78\) in place of \(\frac12\): \(1/\zeta(s)=O(|t|^\varepsilon)\) on \(\operatorname{Re}s\ge\frac78+\varepsilon\), then Perron’s formula and partial summation.

Earlier editions: Proposition 16.12.4, page 672 (edition 1).

Source: ch/polya9.tex, label tx5:prop:mertens.

Least prime in a progression under the half plane 7/8 for every Dirichlet L function: psi(x;q,a) = x/phi(q) + O(x^{7/8} log^2(qx)) and the least prime is O(q^{8+epsilon})

Volume II, Remark 21.4.5, page 654 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on That no Dirichlet \(L\) function vanishes in \(\operatorname{Re}s>\frac78\), which is claimed by OpenAI (2026) and not refereed.

Statement. Assume, as OpenAI (2026) claims and without a refereed proof, that no Dirichlet \(L\) function vanishes in \(\operatorname{Re}s>\frac78\), so that every nontrivial zero has \(\frac18\le\beta\le\frac78\) and \(|\rho|\ge\frac18\). The explicit formula for \(\psi(x,\chi)\) with \(T=x\) (Davenport, Chapters 19 and 20) and \(\sum_{|\gamma|\le x}|\rho|^{-1}=O(\log^2(qx))\) give \(\psi(x;q,a)=x/\varphi(q)+O(x^{7/8}\log^2(qx))\) uniformly in \((a,q)=1\), so the least prime \(p\equiv a\pmod q\) is \(O(q^{8+\varepsilon})\). This is weaker than the unconditional exponent \(5\) of Xylouris (2011), and it is not the Linnik constant \(K\) of Chapter gl.

Earlier editions: Remark 16.12.5, page 673 (edition 1).

Source: ch/polya9.tex, label tx5:rem:leastprime.

Compensated clock positivity on every line to the right of alpha zero under a zero free half plane, by pairing each zero with its mirror image

Volume II, Theorem 21.4.8, page 656 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on \(\zeta(s)\ne0\) for \(\operatorname{Re}s>\alpha_0\), a fixed \(\alpha_0\) with \(\frac12\le\alpha_0<1\) (RH when \(\alpha_0=\frac12\)).

Statement. Let \(\xi\) be Riemann’s xi function and \(D_\alpha(r)=\operatorname{Re}\frac{\xi'}\xi(\alpha+ir)-\frac{\xi'}\xi(\alpha)\). Let \(\frac12\le\alpha_0<1\) and assume that \(\zeta\) has no zero with \(\operatorname{Re}s>\alpha_0\). Then \(D_\alpha(r)>0\) for every \(\alpha>\alpha_0\) and every real \(r\ne0\) (for \(\alpha_0<\alpha\le1\) by the pairing argument, and for \(\alpha\ge1\) with no hypothesis).

Earlier editions: Theorem 16.12.8, page 675 (edition 1).

Source: ch/polya9.tex, label pol:thm:qrhpair.

A quasi Riemann hypothesis at alpha zero is equivalent to the compensated clock condition on every line to its right

Volume II, Corollary 21.4.10, page 657 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and, for \(\alpha>\frac12\) and real \(r\), \(D_\alpha(r)=\operatorname{Re}\frac{\xi'}\xi(\alpha+ir)-\frac{\xi'}\xi(\alpha)\). For \(\frac12\le\alpha_0<1\) the following are equivalent. (a) The zeta function has no zero with \(\operatorname{Re}s>\alpha_0\). (b) \(D_\alpha(r)\ge0\) for every \(\alpha\in(\alpha_0,1)\) and every real \(r\) at which \(D_\alpha\) is defined. When they hold, \(D_\alpha(r)>0\) for every \(\alpha\in(\alpha_0,1)\) and every real \(r\ne0\).

Earlier editions: Corollary 16.12.10, page 676 (edition 1).

Source: ch/polya9.tex, label pol:cor:equiv.

One block for each zero, the Hadamard blocks of the compensated reciprocal of xi

Volume II, Remark 21.4.9, page 657 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and let \(\frac12\le\alpha_0<1\), suppose that \(\zeta\) has no zero with \(\operatorname{Re}s>\alpha_0\) and let \(\alpha>\alpha_0\), so that every width \(c=\alpha-\beta_\rho\) of a zero \(\rho=\beta_\rho+i\gamma_\rho\) from the basepoint \(\alpha\) is positive. The Hadamard product splits the compensated reciprocal of \(\xi\) into one factor for each pair \(\rho,\bar\rho\) with \(\gamma=\gamma_\rho>0\), \(\Phi_{\alpha,\rho}(\frac{w^2}2)=\frac{c^2+\gamma^2}{(c+w)^2+\gamma^2}e^{2cw/(c^2+\gamma^2)}\). By Frullani’s integral \(-\log\Phi_{\alpha,\rho}(w^2/2)=\int_0^\infty(e^{-wy}+wy-1)\varrho_\rho(dy)\) with the signed measure \(\varrho_\rho(dy)=-2y^{-1}e^{-cy}\cos(\gamma y)dy\), and the sine squared lemma gives the density \(\pi^{-1}[f(c,r-\gamma)+f(c,r+\gamma)-2f(c,\gamma)]\) in the ordinate \(r\), with \(f(c,x)=c/(c^2+x^2)\) (Cauchy kernels of width \(c\) recentred at \(\pm\gamma\)), which sums over the zeros to \(\pi^{-1}D_\alpha(r)\), \(D_\alpha(r)=\operatorname{Re}\frac{\xi'}\xi(\alpha+ir)-\frac{\xi'}\xi(\alpha)\). No single block is a GGC, nor a Laplace transform, since \(\Phi_{\alpha,\rho}(w^2/2)\to\infty\) as \(w\to\infty\). As \(c\downarrow0\) the block tends to \((1+w^2/\gamma^2)^{-1}\), an exponential law in \(u=w^2/2\) whose Thorin measure is the unit atom at \(r=\gamma\). Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Remark 16.12.9, page 676 (edition 1).

Source: ch/polya9.tex, label pol:rem:blocks.

GGC basepoints of the compensated reciprocal of xi form the closed half line from the largest real part of a zero

Volume II, Result 21.4.4, page 657 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be Riemann’s xi function and \(\Theta=\sup\operatorname{Re}\rho\) over the nontrivial zeros \(\rho\). The set of \(\alpha>0\) at which \(e^{\sqrt{2u}\,b_\alpha}\xi(\alpha)/\xi(\alpha+\sqrt{2u})\), with \(b_\alpha=\frac{\xi'}\xi(\alpha)\), is the Laplace transform of a generalized gamma convolution is exactly \([\Theta,\infty)\). For \(\alpha<\Theta\) a zero with \(\operatorname{Re}\rho>\alpha\) gives the transform a pole at \(u=(\rho-\alpha)^2/2\) off the negative axis, so it is not a GGC. For \(\alpha>\Theta\) it is one, and at \(\alpha=\Theta\) it is the weak limit of GGC laws as \(\alpha\downarrow\Theta\), hence a GGC. For \(\Theta=\frac12\) this is the Riemann hypothesis. A zero on the line \(\operatorname{Re}s=\Theta\) gives the Thorin measure at \(\Theta\) an atom at its ordinate, of mass its multiplicity; if \(\Theta\) is not attained there is no atom. Credited to a note of Nicholas Polson merged into the book.

Earlier editions: Result 16.12.4, page 676 (edition 1).

Source: ch/polya9.tex, label pol:res:halfline.

The compensated reciprocal of xi is a GGC with positive Thorin density at every basepoint above 7/8 under a zero free half plane

Volume II, Corollary 21.4.11, page 658 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on \(\zeta(s)\ne0\) for \(\operatorname{Re}s>\frac78\) (announced in October 2026, not refereed).

Statement. Let \(\xi\) be Riemann’s xi function and \(b_\alpha=\frac{\xi'}\xi(\alpha)\). Assume that \(\zeta(s)\ne0\) for \(\operatorname{Re}s>\frac78\), and let \(\alpha>\frac78\). Then the compensated reciprocal \(e^{wb_\alpha}\xi(\alpha)/\xi(\alpha+w)\) is the Laplace transform in \(u=w^2/2\) of a generalized gamma convolution, whose Thorin density \(\omega_\alpha(z)=\frac1{\pi\sqrt{2z}}\big[\operatorname{Re}\frac{\xi'}\xi(\alpha+i\sqrt{2z})-\frac{\xi'}\xi(\alpha)\big]\) is positive at every height \(z>0\). For \(\alpha\ge1\) this holds with no hypothesis.

Earlier editions: Corollary 16.12.11, page 677 (edition 1).

Source: ch/polya9.tex, label pol:cor:qrhggc.

Factor lemma, Laplace transform of a Cauchy shifted square root as a GGC

Volume II, Lemma 22.2.2, page 669 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A probability law on \([0,\infty)\) is a generalised gamma convolution (GGC) when its Laplace transform is \(\exp\{-as-\int_{(0,\infty)}\log(1+s/t)\,U(dt)\}\) with \(a\ge0\) and a nonnegative Thorin measure \(U\) with \(\int_{(0,1]}|\log t|\,U(dt)+\int_{(1,\infty)}t^{-1}U(dt)<\infty\). Let \(\gamma\ne0\) and \(\delta\) be real, and \(f(s)=(\delta^2+\gamma^2)/((\sqrt s+\delta)^2+\gamma^2)\) on \(\mathbb{C}\setminus(-\infty,0]\) with the principal root. (i) If \(\delta\ge0\), \(f\) is the Laplace transform of a GGC with \(a=0\) and a Thorin measure of total mass one. For \(\delta>0\) its density is \(u(t)=\frac{\delta(\delta^2+\gamma^2+t)}{\pi\sqrt t\,[(\delta^2+\gamma^2-t)^2+4\delta^2t]}=\frac1{2\pi\sqrt t}\big[\frac{\delta}{\delta^2+(\sqrt t-\gamma)^2}+\frac{\delta}{\delta^2+(\sqrt t+\gamma)^2}\big]\), and for \(\delta=0\) the Thorin measure is the unit atom at \(\gamma^2\), the weak limit of \(u\) as \(\delta\downarrow0\). (ii) If \(\delta<0\), \(f\) has poles at \(s=(-\delta\pm i\gamma)^2\in\mathbb{C}\setminus(-\infty,0]\) and is not a GGC transform. (iii) For \(\delta>0\), \(g(s)=\delta/(\delta+\sqrt s)\) is a GGC transform with \(a=0\) and Thorin density \(\delta/(2\pi\sqrt t(\delta^2+t))\), of total mass \(\frac12\). For \(\delta<0\), \(g\) has a pole at \(s=\delta^2>0\) and is not a GGC transform.

Source: ch/nc_iw.tex, label iw:lem:factor.

Level alpha dictionary, no zero right of alpha is equivalent to a GGC Thorin transform

Volume II, Theorem 22.3.1, page 670 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Call \(\Lambda\) admissible if it is one of: \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\); \(\Lambda(s,\chi)=(q/\pi)^{(s+\mathfrak a)/2}\Gamma(\frac{s+\mathfrak a}2)L(s,\chi)\) for a real primitive Dirichlet character \(\chi\) modulo \(q>1\) with \(\chi(-1)=(-1)^{\mathfrak a}\); the product \(\Lambda(s,\chi)\Lambda(s,\bar\chi)\) for a primitive complex character \(\chi\); or \(\Lambda_E(s)=N^{s/2}(2\pi)^{-s-1/2}\Gamma(s+\frac12)L(E,s+\frac12)\) for an elliptic curve \(E\) over \(\mathbb{Q}\) of conductor \(N\). Each is entire of order one, real on \(\mathbb{R}\), satisfies \(\Lambda(1-s)=\pm\Lambda(s)\), and has all its zeros \(\rho=\beta+i\gamma\) in the strip \(0<\beta<1\). Let \(\alpha\ge\frac12\) with \(\Lambda(\alpha)\ne0\), let \(\varphi_\alpha(s)=\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) on \(\mathbb{C}\setminus(-\infty,0]\) with the principal root, and let \(\mathrm{QRH}(\alpha)\) be the statement that \(\Lambda\) has no zero with real part above \(\alpha\). The following are equivalent: (a) \(\mathrm{QRH}(\alpha)\); (b) \(\varphi_\alpha\) is the Laplace transform of a generalised gamma convolution (GGC); (c) \(\operatorname{Re}(\Lambda'/\Lambda)(s)>0\) whenever \(\operatorname{Re}s>\alpha\). When they hold, the drift is \(a=0\) and the Thorin measure \(U_\alpha\) is the sum of the factor measures with \(\delta=\alpha-\beta\) and \(\gamma\) the ordinate (the Thorin measure of the transform \((\delta^2+\gamma^2)/((\sqrt s+\delta)^2+\gamma^2)\) for a pair, and of \(\delta/(\delta+\sqrt s)\) for a real zero), one for each conjugate pair of zeros (mass one) and one for each real zero (mass \(\frac12\)). Off the atoms at \(\gamma^2\) contributed by zeros on the line \(\operatorname{Re}s=\alpha\), it has density \(u_\alpha(t)=\frac1{2\pi\sqrt t}\operatorname{Re}\frac{\Lambda'}{\Lambda}(\alpha+i\sqrt t)\).

Source: ch/nc_iw.tex, label iw:thm:dictionary.

Quasi Riemann hypothesis at level beta and the Thorin level, with the Riemann hypothesis as beta one half

Volume II, Corollary 22.3.3, page 671 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda\) be admissible (one of \(\xi\), the completed \(L\) function of a real primitive Dirichlet character, the product \(\Lambda(s,\chi)\Lambda(s,\bar\chi)\) for a complex primitive character, or the completed \(L\) function of an elliptic curve over \(\mathbb{Q}\) centred at \(s=\frac12\); each is entire of order one, real on \(\mathbb{R}\), with \(\Lambda(1-s)=\pm\Lambda(s)\) and zeros in \(0<\operatorname{Re}s<1\)), let \(\varphi_\alpha(s)=\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) and let \(\mathrm{QRH}(\beta)\) mean that \(\Lambda\) has no zero with real part above \(\beta\). Let \(\beta\in[\frac12,1)\) with \(\Lambda(\beta)\ne0\). Then \(\mathrm{QRH}(\beta)\) holds if and only if \(\varphi_\beta\) is the Laplace transform of a generalised gamma convolution, and then so is \(\varphi_\alpha\) for every \(\alpha\ge\beta\). The Riemann hypothesis is the case \(\beta=\frac12\) (for \(\Lambda(\frac12)\ne0\)), where the Thorin measure is \(U_{1/2}=\sum_{\gamma>0}\delta_{\gamma^2}\) over the positive ordinates of the zeros. At \(\alpha=1\) the hypothesis \(\Lambda(1)\ne0\) holds for \(\xi\), for Dirichlet \(L\) functions and for elliptic curve \(L\) functions (at \(L(E,\frac32)\)), \(\mathrm{QRH}(1)\) is automatic, and the nonvanishing on \(\operatorname{Re}s=1\) (Hadamard and de la Vallee Poussin) says that \(U_1\) has no atoms.

Source: ch/nc_iw.tex, label iw:cor:qrh.

Cauchy semigroup of the level densities with converse, for every admissible L function

Volume II, Theorem 22.4.1, page 672 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda\) be admissible (one of \(\xi\), the completed \(L\) function of a real primitive Dirichlet character, the product \(\Lambda(s,\chi)\Lambda(s,\bar\chi)\) for a complex primitive character, or the completed \(L\) function of an elliptic curve over \(\mathbb{Q}\) centred at \(s=\frac12\); each is entire of order one, real on \(\mathbb{R}\), with \(\Lambda(1-s)=\pm\Lambda(s)\) and zeros in \(0<\operatorname{Re}s<1\)), let \(\mathrm{QRH}(\alpha)\) mean that \(\Lambda\) has no zero with real part above \(\alpha\), and let \(D_\alpha(dy)=\pi^{-1}\operatorname{Re}(\Lambda'/\Lambda)(\alpha+iy)\,dy\) on \(\mathbb{R}\) (at ordinates of zeros on \(\operatorname{Re}s=\alpha\) the boundary measure with atoms). Let \(P_h(y)=h/(\pi(h^2+y^2))\). If \(\mathrm{QRH}(\alpha)\) holds, then \(D_{\alpha+h}=P_h*D_\alpha\) for every \(h>0\). Conversely, if \(D_{\alpha+h}=P_h*D\) for some nonnegative measure \(D\) and all \(h>0\), then \(\operatorname{Re}\Lambda'/\Lambda>0\) in \(\operatorname{Re}s>\alpha\) and \(\mathrm{QRH}(\alpha)\) holds.

Source: ch/nc_iw.tex, label iw:thm:semigroup.

Thorin mass up to T squared is N(T)+O(log T) at every level alpha in [1/2,1] (conditional on QRH(alpha))

Volume II, Theorem 22.4.3, page 673 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on QRH(alpha) for xi, that is no zero of xi with real part above alpha.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), let \(N(T)\) be the number of zeros \(\rho=\beta+i\gamma\) of \(\xi\) with \(0<\gamma\le T\), counted with multiplicity, and assume \(\mathrm{QRH}(\alpha)\) for \(\xi\) (no zero with real part above \(\alpha\)). Let \(U_\alpha\) be the Thorin measure of \(\xi(\alpha)/\xi(\alpha+\sqrt s)\) and \(D_\alpha(dy)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\alpha+iy)\,dy\), and let \(T\ge2\) not be an ordinate of a zero. If zeros lie on \(\operatorname{Re}s=\alpha\), read \(\arg\xi(\alpha+iy)\) as the limit from \(\alpha'\downarrow\alpha\), so that each such zero contributes a jump of \(\pi\) per unit multiplicity. Then, uniformly in \(\alpha\in[\frac12,1]\), \(U_\alpha([0,T^2])=D_\alpha((0,T])=\frac1\pi\Delta_{0\le y\le T}\arg\xi(\alpha+iy)=N(T)+O(\log T)\), where the argument varies continuously from \(\arg\xi(\alpha)=0\).

Source: ch/nc_iw.tex, label iw:thm:mass.

Level alpha explicit formula with the continuation formula for a single zero

Volume II, Theorem 22.5.1, page 673 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\phi\) be real and even with \(\widehat\phi(u)=\int\phi(y)e^{-iyu}dy\) continuous, piecewise twice continuously differentiable and supported in \([-X,X]\), and put \(L(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\), an entire function. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(\psi\) the digamma function, \(\Lambda\) the von Mangoldt function, and \(\rho=\beta+i\gamma\) the nontrivial zeros of \(\zeta\) with multiplicity. For \(\alpha>0\) define the finite prime sum \(W_\alpha(\phi)=\operatorname{Re}L(\alpha-1)+\frac1\pi\int_{\mathbb{R}}\phi(y)\operatorname{Re}\big[\frac1s-\frac{\log\pi}2+\frac12\psi(\frac s2)\big]_{s=\alpha+iy}dy-\frac1\pi\sum_{n\le e^X}\Lambda(n)n^{-\alpha}\widehat\phi(\log n)\). Then \(W_\alpha(\phi)\) is real analytic in \(\alpha>0\) and \(W_\alpha(\phi)=\sum_\rho\operatorname{Re}L(\alpha-\rho)\), the sum converging absolutely, where, with \(P_h(y)=h/(\pi(h^2+y^2))\) and \(\phi(z)=(2\pi)^{-1}\int\widehat\phi(u)e^{izu}du\) the entire extension of \(\phi\), \(\operatorname{Re}L(\alpha-\rho)\) equals \((P_{\alpha-\beta}*\phi)(\gamma)\) if \(\alpha>\beta\), \(\phi(\gamma)\) if \(\alpha=\beta\), and \(-(P_{\beta-\alpha}*\phi)(\gamma)+2\operatorname{Re}\phi(\gamma+i(\alpha-\beta))\) if \(\alpha<\beta\). In particular, under \(\mathrm{QRH}(\alpha)\) for \(\xi\) (no zero with real part above \(\alpha\)), \(W_\alpha(\phi)=\int\phi\,dD_\alpha\ge0\) for every \(\phi\ge0\), where \(D_\alpha(dy)=\pi^{-1}\operatorname{Re}(\xi'/\xi)(\alpha+iy)\,dy\). The analogue for other admissible \(L\) functions is not proved.

Source: ch/nc_iw.tex, label iw:thm:weil.

Residue identity, prime side functional minus zero side density

Volume II, Corollary 22.5.2, page 675 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. In the notation of the level \(\alpha\) explicit formula (\(\xi\) the completed Riemann zeta function, \(\phi\) a real even test function with \(\widehat\phi\) continuous, piecewise twice continuously differentiable and supported in \([-X,X]\), \(\phi(z)\) its entire extension, and \(W_\alpha(\phi)\) the functional computed from a finite prime sum of von Mangoldt weights up to \(e^X\) together with the archimedean term, equal to \(\sum_\rho\operatorname{Re}L(\alpha-\rho)\) over the zeros \(\rho=\beta+i\gamma\), where \(L(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\)), for every \(\alpha>0\) that is not the real part of a zero, \(W_\alpha(\phi)-\frac1\pi\int\phi(y)\operatorname{Re}\frac{\xi'}\xi(\alpha+iy)\,dy=2\sum_{\beta>\alpha}\operatorname{Re}\phi\big(\gamma+i(\alpha-\beta)\big)\). The prime side functional and the zero side density differ exactly by the entire extension of the test function at the zeros to the right of the level.

Source: ch/nc_iw.tex, label iw:cor:residue.

Positivity of the level alpha functional on nonnegative test functions is equivalent to QRH(alpha)

Volume II, Proposition 22.5.3, page 675 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\alpha>0\) and let \(\mathcal C\) be the class of real even test functions \(\phi\) with \(\widehat\phi(u)=\int\phi(y)e^{-iyu}dy\) continuous, piecewise twice continuously differentiable and compactly supported. Let \(W_\alpha(\phi)\) be the level \(\alpha\) functional computed from a finite prime sum (von Mangoldt weights up to \(e^X\) with \(X\) the support radius of \(\widehat\phi\), plus the archimedean term and \(\operatorname{Re}L(\alpha-1)\)), which equals \(\sum_\rho\operatorname{Re}L(\alpha-\rho)\) over the nontrivial zeros \(\rho\) of \(\zeta\), where \(L(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\). Let \(\phi_0(y)=(\sin y/y)^4\) and let \(\mathrm{QRH}(\alpha)\) be the statement that \(\zeta\) has no zero with real part above \(\alpha\). Then \(\mathrm{QRH}(\alpha)\) holds if and only if \(W_\alpha(\phi)\ge0\) for every \(\phi\in\mathcal C\) with \(\phi\ge0\), and already if \(W_\alpha\big(\phi_0(1\pm\cos a\,\cdot)\big)\ge0\) for both signs and every \(a\ge5\). Consequently the set of levels at which \(W_\alpha\) is nonnegative on nonnegative test functions is exactly \([\Theta,\infty)\), with \(\Theta\) the supremum of the real parts of the zeros.

Source: ch/nc_iw.tex, label iw:prop:converse.

The Jackson family of nonnegative test functions detects the quasi Riemann level

Volume II, Proposition 22.5.4, page 677 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(R_\Omega(x)=\frac{3\Omega}{8\pi}\big(\frac{\sin(\Omega x/4)}{\Omega x/4}\big)^4\ge0\) be the Jackson kernel, with \(\widehat{R_\Omega}(u)=B(u/\Omega)\) for the normalised cubic B spline \(B\), and \(\phi_{\tau,\Omega}(y)=\frac12(R_\Omega(y-\tau)+R_\Omega(y+\tau))\). Let \(\frac12\le c<1\), \(T^{(c)}_\Omega(\tau)=W_c(\phi_{\tau,\Omega})\) the finite prime sum of the Weil form at level \(c\), and \(K_{1-c,\Omega}\) the function with transform \(e^{(1-c)|u|}B(u/\Omega)\). Let \(D_c\) be the even distribution of the level \(c\) density and \(\mathrm{QRH}(c)\) the statement that \(\xi\) has no zero with real part above \(c\). (a) \(T^{(c)}_\Omega=K_{1-c,\Omega}*D_1\) and \(P_{1-c}*T^{(c)}_\Omega=R_\Omega*D_1\). (b) If \(\mathrm{QRH}(c)\) holds, then \(T^{(c)}_\Omega=R_\Omega*D_c\ge0\) for every \(\Omega\). (c) Conversely, if \(T^{(c)}_{\Omega_n}\ge0\) on \(\mathbb R\) for a sequence \(\Omega_n\to\infty\), then \(\mathrm{QRH}(c)\) holds. In particular RH holds if and only if \(T^{(1/2)}_{\Omega_n}\ge0\) for some sequence \(\Omega_n\to\infty\).

Source: ch/nc_iw.tex, label iw:prop:jackson.

Weil functional as the average of the level alpha and level one minus alpha functionals

Volume II, Proposition 22.5.5, page 678 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. With \(\phi\) as in the level \(\alpha\) explicit formula (real, even, \(\widehat\phi\) continuous, piecewise twice continuously differentiable and compactly supported; \(\phi(z)\) its entire extension) and \(W_\alpha(\phi)=\sum_\rho\operatorname{Re}L(\alpha-\rho)\), with \(L(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\) and \([-X,X]\) the support of \(\widehat\phi\), the level \(\alpha\) functional over the nontrivial zeros \(\rho=\beta+i\gamma\) of \(\zeta\), for every \(\alpha\in(0,1)\), \(\sum_\rho\operatorname{Re}\phi\big(\gamma+i(\alpha-\beta)\big)=\frac12[W_\alpha(\phi)+W_{1-\alpha}(\phi)]\). At \(\alpha=\frac12\) the left side is Weil’s functional. By Weil’s criterion, the Riemann hypothesis holds if and only if \(W_{1/2}(\phi_{\mathrm{ev}})\ge0\) for every \(\phi(z)=f(z)\overline{f(\bar z)}\) with \(f\) band limited and \(\widehat f\) continuous, piecewise twice continuously differentiable and compactly supported, where \(\phi_{\mathrm{ev}}(z)=\frac12(\phi(z)+\phi(-z))\) is the even part. Every integrable nonnegative band limited \(\phi\) has this form (Akhiezer’s factorisation) and the functional sees only the even part, which is again nonnegative, so the Riemann hypothesis is equivalent to \(W_{1/2}(\phi)\ge0\) for every nonnegative band limited \(\phi\).

Source: ch/nc_iw.tex, label iw:prop:weilavg.

Li’s coefficients at level alpha as an integral of an explicit kernel against the level alpha zero measure

Volume II, Theorem 22.6.1, page 679 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) with zeros \(\rho=\beta+i\gamma\) (with multiplicity), \(L=\xi'/\xi\), and for \(n\ge1\) put \(g_n(s)=1-\big(\frac{s-1}{s}\big)^n\), so that Li’s coefficients are \(\lambda_n=\sum_\rho g_n(\rho)\), with \(\varphi_n(\rho)=g_n(\rho)+g_n(1-\rho)\). For \(a>0\) let \(P_a(y)=\frac{a}{\pi(a^2+y^2)}\) be the Cauchy kernel. For \(\alpha>\frac12\) let \(\varrho_\alpha(y)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\alpha+iy)\) and let \(D_\alpha\) be the measure on \(\mathbb{R}\) with density \(\varrho_\alpha\), with an atom at the ordinate of each zero on \(\operatorname{Re}s=\alpha\) (the boundary measure). Let \(\mathrm{QRH}(\alpha)\) be the statement that no zero of \(\xi\) has real part above \(\alpha\). Put \(f_n^{(\alpha)}(y)=\operatorname{Re}g_n(1-\alpha+iy)=1-\operatorname{Re}\big[\big(\frac{iy-\alpha}{iy+1-\alpha}\big)^n\big]\). Then: (a) if \(\frac12<\alpha<1\) and \(\mathrm{QRH}(\alpha)\) holds, \(\lambda_n=\int_{\mathbb{R}}f_n^{(\alpha)}\,dD_\alpha\), the integral converging absolutely because \(f_n^{(\alpha)}(y)=O(y^{-2})\) and \(D_\alpha([T,T+1])=O(\log(2+|T|))\). (b) Let \(f\) be bounded and continuous on \(\mathbb{R}\) and suppose that for some open interval \(I\) of real parts, \(\int f(y)\big(P_{\alpha-\beta}+P_{\alpha-1+\beta}\big)(y-\gamma)\,dy=\operatorname{Re}\varphi_n(\beta+i\gamma)\) for all \(\beta\in I\) and all \(\gamma\in\mathbb{R}\). If \(\frac12<\alpha<1\) then \(f=f_n^{(\alpha)}\); if \(\alpha>1\) no such \(f\) exists. (c) If \(\alpha>1\) (no hypothesis) and \(\tau=\alpha-1\), then \(\int f_n^{(\alpha)}\,dD_\alpha=\sum_\rho\operatorname{Re}g_n(\rho+1-2\alpha)=\sum_{k=1}^n(-1)^k\binom nk\frac{L^{(k-1)}(2\alpha-1)}{(k-1)!}\), which in general differs from \(\lambda_n\), and \(\lambda_n=\int f_n^{(\alpha)}\,dD_\alpha+2\pi\sum_{k=1}^n\binom nk\frac{\varrho_\alpha^{(k-1)}(i\tau)}{i^{k-1}(k-1)!}\), where \(\varrho_\alpha\) is continued analytically from \(\mathbb{R}\) by \(2\pi\varrho_\alpha(y)=L(\alpha+iy)+L(\alpha-iy)\), analytic on a neighbourhood of the closed strip \(|\operatorname{Im}y|\le\alpha-1\) because \(\xi\) has no zero with real part at least one. (d) At \(\alpha=1\), with \(c_j=L^{(j)}(1)/j!\), \(\lambda_n=\sum_{j=0}^{n-1}\binom n{j+1}c_j\), with \(c_j=(-1)^{j/2}\pi\varrho_1^{(j)}(0)/j!\) for even \(j\) and \(c_j=-(-1)^{(j+1)/2}\,\mathrm{f.p.}\!\int_{\mathbb{R}}\varrho_1(y)y^{-j-1}\,dy\) for odd \(j\), where \(\mathrm{f.p.}\) is the Hadamard finite part. In particular \(c_0=\lambda_1=\pi\varrho_1(0)\), \(c_1=\int_{\mathbb{R}}(\varrho_1(y)-\varrho_1(0))y^{-2}\,dy\) and \(c_2=-\frac\pi2\varrho_1''(0)\).

Source: ch/nc_iw.tex, label iw:thm:li.

The level alpha Li kernel is positive for n equal to one but takes negative values for every n at least two

Volume II, Proposition 22.6.2, page 680 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12<\alpha<1\) and, for \(n\ge1\), let \(f_n^{(\alpha)}(y)=1-\operatorname{Re}\big[\big(\frac{iy-\alpha}{iy+1-\alpha}\big)^n\big]\), the kernel with which \(\lambda_n=\sum_\rho\big[1-\big(\frac{\rho-1}{\rho}\big)^n\big]\) (Li’s coefficient, the sum over the zeros \(\rho\) of the completed zeta function \(\xi\)) equals \(\int f_n^{(\alpha)}\,dD_\alpha\) whenever no zero has real part above \(\alpha\), where \(D_\alpha\) is the level \(\alpha\) measure with density \(\varrho_\alpha(y)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\alpha+iy)\) and an atom at the ordinate of each zero on \(\operatorname{Re}s=\alpha\). Let \(P_a(y)=\frac{a}{\pi(a^2+y^2)}\) for \(a>0\) and let \(P_0\) be the unit point mass at \(0\). Then \(f_1^{(\alpha)}(y)=\frac{1-\alpha}{(1-\alpha)^2+y^2}=\pi P_{1-\alpha}(y)>0\), and the Cauchy semigroup \(D_1=P_{1-\alpha}*D_\alpha\) together with \(\lambda_1=\int f_1^{(\alpha)}\,dD_\alpha\) gives \(\lambda_1=\pi\varrho_1(0)\) (under the hypothesis that no zero has real part above \(\alpha\); the identity \(\lambda_1=\pi\varrho_1(0)\) itself holds unconditionally). For every \(n\ge2\), \(f_n^{(\alpha)}\) takes negative values. Moreover, for every \(n\ge2\) there is \(\gamma>0\) such that the four points \(Z=\{\rho,\bar\rho,1-\rho,1-\bar\rho\}\) with \(\rho=\alpha+i\gamma\) have real parts in \([1-\alpha,\alpha]\) and no real element, the measure \(D_\sigma^Z=\sum_{\rho'\in Z}P_{\sigma-\operatorname{Re}\rho'}(\cdot-\operatorname{Im}\rho')\) at \(\sigma=\alpha\) is a positive measure obeying the Cauchy semigroup \(D_{\alpha+\delta}^Z=P_\delta*D_\alpha^Z\) for \(\delta>0\), and \(\lambda_n(Z)=\sum_{\rho'\in Z}\big[1-\big(\frac{\rho'-1}{\rho'}\big)^n\big]=2\operatorname{Re}\big(g_n(\rho)+g_n(1-\rho)\big)<0\), where \(g_n(s)=1-\big(\frac{s-1}{s}\big)^n\).

Source: ch/nc_iw.tex, label iw:prop:lisign.

Origin coefficient of the level density bounds the analytic rank of an elliptic curve

Volume II, Proposition 22.7.1, page 681 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on QRH(alpha) for the elliptic curve L function for the bound at alpha below one (unconditional at alpha one); GRH for the identity.

Statement. Let \(E\) be an elliptic curve of analytic rank \(r\) with completed \(L\) function \(\Lambda_E\) centred at \(s=\frac12\) (admissible: entire of order one, real on \(\mathbb{R}\), functional equation \(\Lambda_E(1-s)=\pm\Lambda_E(s)\), zeros in \(0<\operatorname{Re}s<1\)). Let \(\alpha>\frac12\) with \(\Lambda_E(\alpha)\ne0\), \(\kappa_\alpha=(\Lambda_E'/\Lambda_E)(\alpha)\), and let \(\mathrm{QRH}(\alpha)\) mean that \(\Lambda_E\) has no zero with real part above \(\alpha\). If \(\mathrm{QRH}(\alpha)\) holds, then \(r\le(\alpha-\frac12)\kappa_\alpha\), where \(\kappa_\alpha=\lim_{t\to0}2\pi\sqrt t\,u_\alpha(t)\) for the level \(\alpha\) Thorin density \(u_\alpha(t)=\frac1{2\pi\sqrt t}\operatorname{Re}\frac{\Lambda_E'}{\Lambda_E}(\alpha+i\sqrt t)\); the bound is unconditional at \(\alpha=1\), where it reads \(r\le\frac12\kappa_1\). Under the generalised Riemann hypothesis for \(L(E,s)\), \((\alpha-\frac12)\kappa_\alpha=r+(\alpha-\frac12)^2\sum_{\gamma\ne0}\frac1{(\alpha-\frac12)^2+\gamma^2}=r+O((\alpha-\frac12)^2)\), the sum over the ordinates \(\gamma\ne0\) of the zeros \(\frac12+i\gamma\).

Source: ch/nc_iw.tex, label iw:prop:rank.

Curves over finite fields at every level, Weil functional and Toeplitz positivity equivalent to the root bound

Volume II, Theorem 22.8.1, page 683 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(q\) be a prime power, \(L=\log q\), and let \(\{\alpha_j\}_{j\le2g}\) be a multiset of complex numbers closed under \(\alpha\mapsto q/\alpha\) and under complex conjugation (for a curve of genus \(g\) over \(\mathbb{F}_q\), the inverse roots of the numerator \(P(T)=\prod_j(1-\alpha_jT)\) of its zeta function). Put \(S_m=\sum_j\alpha_j^m\), \(\Lambda_P(s)=q^{gs}P(q^{-s})\) (so \(\Lambda_P(1-s)=\Lambda_P(s)\)), and for real \(\sigma\) put \(t_m=q^{-\sigma|m|}S_{|m|}\) with \(t_0=2g\), and for \(\sigma\ge\frac12\) put \(\varphi_\sigma(s)=\Lambda_P(\sigma)/\Lambda_P(\sigma+\sqrt s)\). Let \(\mathcal{C}\) be the class of real even functions \(\phi\) on \(\mathbb{R}\) whose Fourier transform \(\widehat\phi(u)=\int\phi(y)e^{-iyu}dy\) is continuous, piecewise twice continuously differentiable and supported in some \([-X,X]\), and for \(\phi\in\mathcal{C}\) put \(L_\phi(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\) and \(W_\sigma(\phi)=\sum_\rho\operatorname{Re}L_\phi(\sigma-\rho)\) over the zeros \(\rho\) of \(\Lambda_P\) with multiplicity. Then: (i) for every real \(\sigma\) and every \(\phi\in\mathcal{C}\), \(W_\sigma(\phi)=\frac{L}{2\pi}\sum_{m\in\mathbb{Z}}t_m\widehat\phi(mL)\), a finite sum with no archimedean term. (ii) For \(\sigma\ge\frac12\) the following are equivalent: (a) \(|\alpha_j|\le q^\sigma\) for every \(j\); (d) every Toeplitz matrix \((t_{k-l})_{0\le k,l<n}\) is positive semidefinite; (e) \(W_\sigma(\phi)\ge0\) for every nonnegative \(\phi\in\mathcal{C}\); (f) \(|t_m|\le2g\) for every \(m\ge1\), that is, \(W_\sigma\big(\psi(1\pm\cos mL\,\cdot)\big)\ge0\) for every \(m\ge1\), where \(\widehat\psi(u)=(1-|u|/L)_+\). If moreover \(P(q^{-\sigma})\ne0\), each of these is equivalent to (b) \(\varphi_\sigma\) is the Laplace transform of a generalised gamma convolution, and to (c) \(\operatorname{Re}\Lambda_P'/\Lambda_P>0\) on \(\operatorname{Re}s>\sigma\). (iii) If \(P(q^{-\sigma})\ne0\) and (a) holds, then \(\varphi_\sigma(s)=e^{-gL\sqrt s}P(q^{-\sigma})/P(q^{-\sigma}e^{-L\sqrt s})\), and the Thorin measure of \(\varphi_\sigma\) has zero drift and equals the half stable part \(\frac{gL}{2\pi}t^{-1/2}dt\) (the Thorin measure of the factor \(e^{-gL\sqrt s}\)) plus a signed oscillating part. If all \(|\alpha_j|=\sqrt q\), \(\alpha_j=\sqrt q\,e^{i\theta_j}\), and \(\delta=\sigma-\frac12>0\), then the density of this Thorin measure in the variable \(y=\sqrt t\) is the sum of wrapped Cauchy kernels \(\varrho_\sigma(y)=\frac1\pi\operatorname{Re}\frac{\Lambda_P'}{\Lambda_P}(\sigma+iy)=\sum_{j=1}^{2g}\frac{L}{2\pi}\frac{\sinh L\delta}{\cosh L\delta-\cos(Ly-\theta_j)}\), periodic of period \(2\pi/L\) with mean \(gL/\pi\).

Source: ch/nc_iw.tex, label iw:thm:curves.

Open problem, family deconvolution test and the rate of the recovered atom at the origin

Volume II, Problem 22.1, page 689 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For a family of elliptic curves along the double zero progressions with parameters \(A\approx X^{1/3}\) and \(B\approx X^{1/4}\), let \(\overline D_\alpha\) be the family average of the level \(\alpha\) densities \(\pi^{-1}\operatorname{Re}(\Lambda_E'/\Lambda_E)(\alpha+iy)\,dy\) of the completed \(L\) functions, computed at a level \(\alpha>1\) from averages of the Fourier coefficients \(a_E(p)\) (the Dirichlet series of \(\Lambda_E'/\Lambda_E\) is not absolutely convergent at \(\alpha=1\)). Invert the Cauchy semigroup \(D_{\alpha+h}=P_h*D_\alpha\) down to level \(\frac12\). Under the generalised Riemann hypothesis for the family, the recovered measure has an atom at the origin of mass \(\overline r\ge2\), the family average of the analytic ranks, on the progressions. An analogue of the level \(\alpha\) explicit formula for \(\Lambda_E\), not proved, would say that the prime side functional at level \(\frac12\) reproduces \(\int\phi\,d\overline D_{1/2}\) for band limited \(\phi\) and that the atom appears as \(\phi(0)\overline r\). The open question is the rate at which the recovered atom converges to \(\overline r\) as the support \(X\) grows.

Source: ch/nc_iw.tex, label iw:prob:family.

Volume II, Chapter 23. Ramanujan’s Delta in Thorin form

Ramanujan Delta L function is admissible for the level alpha Thorin dictionary

Volume II, Proposition 23.1.1, page 692 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(L(s,\Delta)=\sum_{n\ge1}\tau(n)n^{-s}\) be the L function of Ramanujan’s discriminant form \(\Delta\), \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\), which is entire with \(\Lambda(s)=\Lambda(12-s)\) and real on \(\mathbb{R}\). Then \(\Lambda^\circ(s)=\Lambda(s+\frac{11}2)\) is entire of order one, real on \(\mathbb{R}\), satisfies \(\Lambda^\circ(1-s)=\Lambda^\circ(s)\), and has all its zeros in the open strip \(0<\operatorname{Re}s<1\). The level \(\alpha\) dictionary between the Thorin condition and the zero free half planes (the Thorin density of \(\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) against the zeros of \(\Lambda\)) therefore holds for \(\Lambda^\circ\) verbatim.

Source: ch/nc_delta.tex, label dl:prop:admissible.

Level alpha dictionary for Delta, GGC and zero free half plane

Volume II, Theorem 23.1.2, page 692 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) for Ramanujan’s \(\Delta\), with zeros \(\rho=\beta_\rho+i\gamma_\rho\) in \(\frac{11}2<\beta_\rho<\frac{13}2\), and \(\Theta_\Delta=\sup_\rho\beta_\rho\). Let \(\alpha\ge6\) and \(a_\rho=\alpha-\beta_\rho\). The following are equivalent: (a) \(\Lambda\) has no zero with real part \(>\alpha\); (b) \(\varphi_\alpha(s)=\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) is the Laplace transform of a generalised gamma convolution (GGC); (c) \(\operatorname{Re}(\Lambda'/\Lambda)(s)>0\) for \(\operatorname{Re}s>\alpha\). When they hold the drift vanishes and the Thorin measure is \(U_\alpha(dx)=\tilde u_\alpha(r)\,dr\) in the variable \(r=\sqrt x\) on the part off the atoms, with \(\tilde u_\alpha(r)=\frac1\pi\operatorname{Re}\frac{\Lambda'}\Lambda(\alpha+ir)=\frac1\pi\sum_\rho\frac{a_\rho}{a_\rho^2+(r-\gamma_\rho)^2}\), the sum over all zeros. For \(\alpha>\Theta_\Delta\) the Thorin measure is absolutely continuous, and under the Riemann hypothesis for \(\Delta\) (that is \(\Theta_\Delta=6\)) all heights equal \(\alpha-6\). At \(\alpha=\Theta_\Delta\) each zero with \(\beta_\rho=\alpha\) gives an atom at \(\gamma_\rho^2\) instead.

Source: ch/nc_delta.tex, label dl:thm:level.

Riemann hypothesis for Delta equivalent to the Thorin condition at level 6

Volume II, Corollary 23.1.3, page 693 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) and let \(\mathrm{RH}_\Delta\) be the statement that every zero of \(\Lambda\) has real part \(6\). The following are equivalent: \(\mathrm{RH}_\Delta\); \(\Lambda(6)/\Lambda(6+\sqrt s)\) is the Laplace transform of a generalised gamma convolution (GGC); \(\operatorname{Re}(\Lambda'/\Lambda)(s)\ge0\) for \(\operatorname{Re}s>6\) (the Lagarias and Hinkkanen criterion for \(\Delta\)). Under \(\mathrm{RH}_\Delta\) the Thorin measure at level \(6\) is \(\sum_{\gamma>0}\delta_{\gamma^2}\) over the ordinates \(\gamma\) of the zeros \(6+i\gamma\), and for \(\alpha>6\) the Thorin density \(\tilde u_\alpha\) in \(r=\sqrt x\) is the Poisson integral of \(\sum_\rho\delta_{\gamma_\rho}\) at height \(\alpha-6\). Unconditionally \(\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) is a GGC transform for every \(\alpha\ge\frac{13}2\).

Source: ch/nc_delta.tex, label dl:cor:rh.

Levy density of the Thorin GGC of Delta as an erfc sum over the zeros

Volume II, Theorem 23.2.1, page 693 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) with zeros \(\rho\), \(\Theta_\Delta=\sup_\rho\operatorname{Re}\rho\), and let \(\alpha>\Theta_\Delta\). The generalised gamma convolution with Laplace transform \(\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) has Levy density \(\ell_\alpha(x)=\frac1x\int_0^\infty e^{-xr^2}\tilde u_\alpha(r)\,dr=\frac1{2x}\sum_\rho\operatorname{Re}\bigl[e^{x(\alpha-\rho)^2}\operatorname{erfc}\bigl((\alpha-\rho)\sqrt x\bigr)\bigr]\ge0\), where \(\tilde u_\alpha(r)=\frac1\pi\operatorname{Re}\frac{\Lambda'}\Lambda(\alpha+ir)\) is its Thorin density in \(r=\sqrt x\) and the sum runs over all zeros. Equivalently \(\ell_\alpha(x)=\frac{x^{-3/2}}{2\sqrt\pi}\sum_\rho\operatorname{Re}\int_0^\infty e^{-(\alpha-\rho)y-y^2/(4x)}\,dy\).

Source: ch/nc_delta.tex, label dl:thm:levy.

Deligne bound as compound Poisson law for zeta squared over L of Delta

Volume II, Proposition 23.2.3, page 694 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Write \(\tau(p)=2p^{11/2}\cos\theta_p\) for primes \(p\). For \(\operatorname{Re}w>1\), \(\zeta(w)^2/L(w+\frac{11}2,\Delta)=\exp\bigl(\sum_p\sum_{k\ge1}\frac{4\sin^2(k\theta_p/2)}{k}p^{-kw}\bigr)\). All coefficients \(4\sin^2(k\theta_p/2)/k\) are nonnegative if and only if \(|\tau(p)|\le2p^{11/2}\) for every prime \(p\). Hence Ramanujan’s bound (Deligne’s theorem) is equivalent to the statement that for each \(\sigma>1\) the function \(t\mapsto\zeta(\sigma+it)^2L(\sigma+\frac{11}2,\Delta)/\bigl(\zeta(\sigma)^2L(\sigma+it+\frac{11}2,\Delta)\bigr)\) is the characteristic function of a compound Poisson law on \(\{-k\log p\}\) with Levy measure \(\sum_{p,k}\frac{4\sin^2(k\theta_p/2)}{k}p^{-k\sigma}\delta_{-k\log p}\).

Source: ch/nc_delta.tex, label dl:prop:deligne.

No level one Hecke eigenform has a GGC Euler side

Volume II, Theorem 23.2.5, page 696 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be a holomorphic Hecke eigenform of level one with \(L(s,f)=\sum\lambda_f(n)n^{-s}\) normalised so that the critical line is \(\operatorname{Re}s=\frac12\), and \(\lambda_f(p)=2\cos\theta_p\) with \(\theta_p\in[0,\pi]\) (Deligne). For \(\beta>1\) and \(t\in\mathbb{R}\) put \(S_{f,\beta}(t)=\sum_p\sum_{m\ge1}2\cos(m\theta_p)\log p\;p^{-m\beta}\bigl(1-\cos(mt\log p)\bigr)\), the prime part of \(\operatorname{Re}(L'/L)(\beta+it)-(L'/L)(\beta)\), and \(S_{\zeta,\beta}\) the same with \(2\cos(m\theta_p)\) replaced by \(1\). Then for every such \(f\) and every \(\beta>1\), \(S_{f,\beta}(t)<0\) on an interval of \(t\). Hence the compensated reciprocal of \(L(s,f)\) at level \(\beta\), whose Thorin density would be \(S_{f,\beta}/\pi\), is never a generalised gamma convolution, whereas \(S_{\zeta,\beta}\ge0\). The proof uses a prime \(p_0\) with \(\lambda_f(p_0)<0\) and Kronecker’s theorem.

Source: ch/nc_delta.tex, label dl:thm:never.

Both compensated reciprocals of zeta squared times L of an eigenform are GGC on the Euler side

Volume II, Proposition 23.2.6, page 696 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be a holomorphic Hecke eigenform of level one with \(\lambda_f(p)=2\cos\theta_p\) (Deligne), and for \(\beta>1\) let \(S_{f,\beta}(t)=\sum_p\sum_{m\ge1}2\cos(m\theta_p)\log p\;p^{-m\beta}(1-\cos(mt\log p))\) and \(S_{\zeta,\beta}(t)=\sum_p\sum_{m\ge1}\log p\;p^{-m\beta}(1-\cos(mt\log p))\). Then for every \(\beta>1\) and every \(t\), \(2S_{\zeta,\beta}(t)-S_{f,\beta}(t)\ge0\) and \(2S_{\zeta,\beta}(t)+S_{f,\beta}(t)\ge0\), that is \(|S_{f,\beta}|\le2S_{\zeta,\beta}\). Equivalently the compensated reciprocals of \(\zeta^2L(s,f)^{\mp1}\) are generalised gamma convolutions on the Euler side. The weights are \(4\sin^2(m\theta_p/2)\) and \(4\cos^2(m\theta_p/2)\) times \(\log p\;p^{-m\beta}(1-\cos(mt\log p))\).

Source: ch/nc_delta.tex, label dl:prop:hecke2sin.

Hecke cone of symmetric power L function products with GGC Euler side

Volume II, Theorem 23.2.7, page 697 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f\) be a level one Hecke eigenform with \(\lambda_f(p)=2\cos\theta_p\). For a finitely supported real vector \(c=(c_0,c_1,\dots)\) let \(F_c=\prod_nL(s,\mathrm{sym}^nf)^{c_n}\) (\(\mathrm{sym}^0\) gives \(\zeta\)) and \(\mathcal T_c(\varphi)=\sum_nc_nU_n(\cos\varphi)\) with \(U_n\) the Chebyshev polynomial of the second kind. The local weights of \(F_c\) at \(p^m\) are \(\mathcal T_c(m\theta_p)\log p\), and \(S_{F_c,\beta}(t)=\sum_p\sum_{m\ge1}\mathcal T_c(m\theta_p)\log p\;p^{-m\beta}(1-\cos(mt\log p))\). Let the Hecke cone be \(K_{\rm Hecke}=\{c:\mathcal T_c\ge0\text{ on }[0,\pi]\}\). If \(c\in K_{\rm Hecke}\) then \(S_{F_c,\beta}(t)\ge0\) for every eigenform \(f\), every \(\beta>1\) and every \(t\). By the Fejer and Riesz theorem \(c\in K_{\rm Hecke}\) exactly when \(\mathcal T_c(\varphi)=|Q(e^{i\varphi})|^2\) for a polynomial \(Q\). In particular \(L(s,\mathrm{sym}^nf\times\mathrm{sym}^nf)=\prod_{j=0}^nL(s,\mathrm{sym}^{2j}f)\), for which \(\mathcal T_c=U_n^2\) by the Clebsch and Gordan rule, always has \(S_{F_c,\beta}\ge0\).

Source: ch/nc_delta.tex, label dl:thm:cone.

Thorin density of Delta positive on a window given Keiper’s zero verification

Volume II, Proposition 23.3.1, page 699 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Keiper’s computation that every zero of the completed L function of Delta with absolute ordinate at most 6397 lies on the line with real part 6.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) and \(\tilde u_\alpha(r)=\frac1\pi\operatorname{Re}\frac{\Lambda'}\Lambda(\alpha+ir)\). Assume Keiper’s computation that every zero of \(\Lambda\) with \(|\gamma|\le6397\) lies on \(\operatorname{Re}s=6\). Then \(\tilde u_\alpha(r)>0\) for every \(\alpha>6\) and every \(|r|\le6396\), whatever the real parts of the zeros above height \(6397\). The proof groups each off line zero with its partner under \(\rho\mapsto12-\bar\rho\).

Source: ch/nc_delta.tex, label dl:prop:window.

Compensated Thorin density of Delta and the minimum at the origin

Volume II, Proposition 23.3.2, page 700 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on for part (c): the constants of the Delta column of the minimum at the origin table, computed in floating point with certification in ball arithmetic pending, and for the range up to height 6397 also Keiper’s computation of the zeros of the completed L function of Delta.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\), \(\mathrm{RH}_\Delta\) the statement that all zeros have real part \(6\), and for \(\alpha>6\) put \(b_\alpha=(\Lambda'/\Lambda)(\alpha)>0\) and \(\tau_\alpha(r)=\tilde u_\alpha(r)-\tilde u_\alpha(0)\) where \(\tilde u_\alpha(r)=\frac1\pi\operatorname{Re}\frac{\Lambda'}\Lambda(\alpha+ir)\). (a) Whenever the Thorin representation of \(\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\) holds at level \(\alpha\) (in particular for \(\alpha\ge\frac{13}2\) and under \(\mathrm{RH}_\Delta\)), \(\log\bigl(e^{\sqrt s\,b_\alpha}\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\bigr)=-\int_0^\infty\log(1+s/r^2)\,\tau_\alpha(r)\,dr\), so the compensated function is the Laplace transform of a generalised gamma convolution with no drift and Thorin density \(\tau_\alpha\) exactly when \(\tau_\alpha\ge0\) (proved). (b) If \(\tau_\alpha(r)>0\) for every \(r\ne0\) and every \(\alpha>6\), then \(\mathrm{RH}_\Delta\) holds (proved). (c) \(\mathrm{RH}_\Delta\) holds if and only if \(\tau_\alpha(r)>0\) for every \(r\ne0\) on every line \(6<\alpha<\frac{13}2\), with no gap condition on the zeros. Moreover \(\tau_\alpha(r)>0\) for \(0<|r|\le6397-\frac{1+\sqrt2}2\) when \(6<\alpha\le12.4\), and for every \(r\ne0\) when \(6.7071\le\alpha\le12.4\), with no hypothesis on the zeros. Part (c) is conditional on constants computed in floating point, and the range in \(r\) also on Keiper’s verification of the zeros up to height \(6397\).

Source: ch/nc_delta.tex, label dl:prop:minorigin.

Strengthened Laguerre inequality for Delta under GRH

Volume II, Corollary 23.3.3, page 701 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the generalised Riemann hypothesis for Delta (all zeros of the completed L function on the line with real part 6); one inequality between three special values evaluated in floating point.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\), \(Z(t)=\Lambda(6+it)\) (real and even) with ordinates \(\gamma\) of both signs, and \(c=\sum_\gamma\gamma^{-2}=\operatorname{Var}V_6=0.1123518\), where \(V_6\) has characteristic function \(Z(t)/Z(0)\). Assume the generalised Riemann hypothesis for \(\Delta\). Then \(E=Z'^2-ZZ''-cZ^2\ge0\) on all of \(\mathbb{R}\). The proof uses the computed value \(A(7)-S-3c=0.02327>0\), with \(A(t)=-\log2\pi+\operatorname{Re}\psi(9+it)\) and \(S=-2\zeta'/\zeta(\frac72)\), evaluated in floating point.

Source: ch/nc_delta.tex, label dl:cor:laguerre.

GHS inequality for Delta, proved for h at least 6.574

Volume II, Theorem 23.3.4, page 702 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) and \(K(h)=\log(\Lambda(6+h)/\Lambda(6))\), the cumulant generating function of the even random variable \(V_6\). Then \(K'''(h)<0\) for every \(h\ge6.574\) (proved, using Deligne’s bound \(|\Lambda_\Delta(n)|\le2n^{11/2}\Lambda(n)\), where \(-L'/L(s,\Delta)=\sum_n\Lambda_\Delta(n)n^{-s}\) and \(\Lambda(n)\) is the von Mangoldt function,, where \(-L'/L(s,\Delta)=\sum_n\Lambda_\Delta(n)n^{-s}\) and \(\Lambda(n)\) is the von Mangoldt function, and the crossing of \((6+h)^{-2}\) with \(-2(\zeta'/\zeta)''(h+\frac12)\) at \(h=6.57388\)). On \((0,6.6]\) the following is a computation only: on a grid of step \(0.02\) over \([0.02,6.6]\) the maximum of \(K'''\) is \(-4.794\times10^{-5}\) at \(h=0.02\), \(K'''(1)=-0.0023251\), and near the origin \(K'''(h)=\kappa_4h+O(h^3)\) with \(\kappa_4=m_4-3m_2^2=-0.00240\) (\(m_{2k}=\mathbb{E}V_6^{2k}\)). Between grid points \(K'''\) has not been bounded, so the range \((0,6.574)\) is a numerical check and not a proof.

Source: ch/nc_delta.tex, label dl:thm:ghs.

Eta block single spin measures are not Lee Yang (given Dobner’s theorem)

Volume II, Proposition 23.3.5, page 703 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on Dobner’s theorem that the de Bruijn Newman constant of an L function in the extended Selberg class is at least 0.

Statement. Let \(\eta\) be the Dedekind eta function, \(\Phi_\eta(u)=e^{u/4}\eta(ie^u)\) (positive and even, with \(\Phi_\Delta=\Phi_\eta^{24}\)) and \(f=\Phi_\eta/\int\Phi_\eta\). A finite measure on \(\mathbb{R}\) is Lee Yang if its Fourier transform has only real zeros. Assume Dobner’s theorem that the de Bruijn Newman constant of an L function in the extended Selberg class is at least \(0\). Then for every \(\beta>0\) the measure \(f(u)e^{-\beta u^2}\,du\) is not Lee Yang. The proof identifies its Fourier transform, after the change of variable \(s\mapsto2s\), with the de Bruijn Newman deformation at negative time of the completed \(L(2s,\chi_{12})\).

Source: ch/nc_delta.tex, label dl:prop:eta.

Putnam type theorem for Delta, no zeros on an arithmetic progression

Volume II, Theorem 23.3.6, page 703 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the bound S(y)=O(y^{1/3+epsilon}) for the partial sums of the normalised coefficients of Delta, for which no reference is yet verified.

Statement. Let \(\lambda(n)=\tau(n)n^{-11/2}\) and \(L(s)=\sum\lambda(n)n^{-s}=L(s+\frac{11}2,\Delta)\), with critical line \(\operatorname{Re}s=\frac12\). Let \(d>0\) and \(r=e^{2\pi/d}\). Assume that the partial sums \(S(y)=\sum_{n\le y}\lambda(n)\) satisfy \(S(y)=O(y^{1/3+\varepsilon})\) for every \(\varepsilon>0\). If no positive integer power of \(r\) is an integer, or if \(r^q=a\) is an integer with \(q\ge1\) minimal and \(H(a)=\sum_{j\ge0}\lambda(a^j)a^{-j/2}\ne0\), then \(L(\frac12+idn)\ne0\) for some \(n\ge1\).

Source: ch/nc_delta.tex, label dl:thm:putnam.

The law of V_6 has no independent uniform factor (conditional)

Volume II, Corollary 23.3.8, page 704 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: conditional on the bound S(y)=O(y^{1/3+epsilon}) for the partial sums of the normalised coefficients of Delta, as in the Putnam type theorem for Delta.

Statement. Let \(V_6\) be the random variable with characteristic function \(\Xi_\Delta(t)/\Xi_\Delta(0)\) where \(\Xi_\Delta(t)=\Lambda(6+it)\), \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\). Assume the partial sum bound \(\sum_{n\le y}\tau(n)n^{-11/2}=O(y^{1/3+\varepsilon})\). Then the law of \(V_6\) is not the law of \(aU+W\) with \(a>0\), \(U\) uniform on \([-1,1]\) and \(W\) independent of \(U\), unless \(r=e^{2a}\) has a power \(r^q=a'\) that is an integer with at least two distinct prime factors and \(H(a')=\sum_{j\ge0}\tau(a'^j)a'^{-11j/2}a'^{-j/2}=0\).

Source: ch/nc_delta.tex, label dl:cor:putnam.

Delta as the eighth power of the antipodal heat kernel of the sphere

Volume II, Theorem 23.4.1, page 705 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(p_t(x,y)\) be the transition density, with respect to surface measure, of Brownian motion on the unit sphere \(S^2\) with generator \(\frac12\Delta_{S^2}\), and \(-x\) the antipode of \(x\). For \(t>0\), \(\prod_{n\ge1}(1-e^{-nt})^3=\sum_{k\ge0}(-1)^k(2k+1)e^{-k(k+1)t/2}=4\pi\,p_t(x,-x)=\mathbb{P}(M_3\le t)\), where \(M_3\) is the maximum of three independent copies of \(M=\sup_nE_n/n\) with \(E_n\) independent standard exponentials. Consequently \(\Delta(iw)=e^{-2\pi w}[4\pi\,p_{2\pi w}(x,-x)]^8\) and \(\Lambda(s)=(2\pi)^{-s}\int_0^\infty t^{s-1}e^{-t}[4\pi\,p_t(x,-x)]^8dt\), where \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) and \(\Delta(\tau)=\eta(\tau)^{24}\), and the variable \(M^*\) with \(L(\alpha,\Delta)=\mathbb{P}(M^*\le G_\alpha)\), for \(G_\alpha\) a gamma variable of shape \(\alpha\) and scale one independent of \(M^*\), is the maximum of eight independent copies of \(M_3\). The first equality is Jacobi’s identity.

Source: ch/nc_delta.tex, label dl:thm:sphere.

Subordination of the Delta hitting time to Brownian first passage

Volume II, Proposition 23.5.1, page 708 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\) and assume the Riemann hypothesis for \(\Delta\) (all zeros on \(\operatorname{Re}s=6\)). Let \(a>0\), \(\alpha=6+a\), \(T_\alpha\) a random variable with Laplace transform \(\Lambda(\alpha)/\Lambda(\alpha+\sqrt s)\), \(T_6\) one with transform \(\Lambda(6)/\Lambda(6+\sqrt s)\), \(\zeta_a\sim\mathrm{Exp}(a^2)\) independent of \(T_6\), and \(\widehat T\) distributed as \(T_6\) conditioned on \(T_6<\zeta_a\). Then \(\mathbb{P}(T_6<\zeta_a)=\Lambda(6)/\Lambda(6+a)\) and \(T_\alpha\overset d=\widehat T+H_{\sqrt2\,a\widehat T}\), where \(H_\ell\) is the first passage time of a standard Brownian motion to level \(\ell\), independent of \(\widehat T\). Without the Riemann hypothesis, for \(\alpha\ge\Theta_\Delta=\sup_\rho\operatorname{Re}\rho\) and each conjugate pair \(\{\rho,\bar\rho\}\) of zeros \(\rho=\beta_\rho+i\gamma_\rho\) with \(a_\rho=\alpha-\beta_\rho\), let \(Y_\rho\sim\mathrm{Exp}(a_\rho^2+\gamma_\rho^2)\) and \(H^{(\rho)}\) be independent first passage processes; then \(T_\alpha\overset d=\sum_{\{\rho,\bar\rho\}}[Y_\rho+H^{(\rho)}_{\sqrt2a_\rho Y_\rho}]\).

Source: ch/nc_delta.tex, label dl:prop:sub.

Symmetrised truncations of the Delta kernel have complex zeros for N at most 2

Volume II, Proposition 23.6.3, page 710 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: computed.

Statement. Truncate the kernel of \(\Delta\) to \(e^{6u-2\pi e^u}\prod_{n\le N}(1-e^{-2\pi ne^u})^{24}\) and let \(F_N(t)=\Lambda_N(6+it)+\Lambda_N(6-it)\) with \(\Lambda_N(s)=(2\pi)^{-s}\Gamma(s)D_N(s)\) and \(D_N\) the Dirichlet polynomial of the coefficients of \(\prod_{n\le N}(1-q^n)^{24}\). Then \(F_N\) is real and even, is the Fourier transform of the even part of the truncated kernel, and converges to \(2\Xi_\Delta\) with \(\Xi_\Delta(t)=\Lambda(6+it)\). By computation (argument principle for the counts, Newton’s method for the complex zero) its zeros do not all lie on the real axis for \(N\le2\); for \(N=2\) there is a zero at \(28.79373902+7.55852217i\) with relative residual \(4\times10^{-37}\). For \(N=3,4\) no complex zero was found in the rectangle \([0,40]\times[-8,8]\), and real rootedness for larger \(N\) is not settled.

Source: ch/nc_delta.tex, label dl:prop:r4.

No mixing variable makes the Mobius mixture positive

Volume II, Proposition 23.6.4, page 710 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(V>0\) be a random variable with \(\mathbb{E}V^{-1/4-\varepsilon}<\infty\) and \(\mathbb{E}V^a<\infty\) for some \(\varepsilon,a>0\) (every generalised gamma convolution with total Thorin mass above \(\frac14\) has the first moment), and with \(\mathbb{E}V^{-s}\ne0\) at some \(s=\frac12(1-\rho)\) where \(\rho\) is a nontrivial zero of \(\zeta\) with \(\operatorname{Re}\rho=\Theta=\sup\operatorname{Re}\rho\) over the nontrivial zeros. Put \(g(y)=\mathbb{E}e^{-yV}\) and \(P_g(x)=\sum_{n\ge1}\frac{\mu(n)}ng(x/n^2)\) with \(\mu\) the Mobius function. Then \(P_g(x)=\mathbb{E}\,P(xV)\) with \(P(x)=\sum_n\frac{\mu(n)}ne^{-x/n^2}\), its Mellin transform is \(\Gamma(s)\,\mathbb{E}V^{-s}/\zeta(1-2s)\), and \(P_g\) changes sign infinitely often on \((0,\infty)\).

Source: ch/nc_delta.tex, label dl:prop:r5.

Archimedean slope of the Thorin density of completed symmetric power quotients

Volume II, Theorem 23.6.6, page 711 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(f=\Delta\), \(c=(c_0,c_1,\dots)\) a finitely supported real vector and \(\Lambda_{F_c}=\xi^{c_0}\prod_{n\ge1}\Lambda(\mathrm{sym}^n\Delta,s)^{c_n}\) the completion of \(F_c=\prod_nL(s,\mathrm{sym}^n\Delta)^{c_n}\), each completed symmetric power carrying its archimedean factor of degree \(n+1\) and \(\xi\) carrying its factor \(s(s-1)\). For \(\beta=\alpha-\frac{11}2>1\) let \(D_{c,\beta}(t)\) be the density of \(\Lambda_{F_c}\), the sum of the Euler side prime part \(S_{F_c,\beta}(t)=\sum_p\sum_{m\ge1}\mathcal T_c(m\theta_p)\log p\;p^{-m\beta}(1-\cos(mt\log p))\), where \(\tau(p)=2p^{11/2}\cos\theta_p\), \(\mathcal T_c(\varphi)=\sum_nc_nU_n(\cos\varphi)\) and \(U_n\) is the Chebyshev polynomial of the second kind, and the archimedean and polynomial parts. Then \(D_{c,\beta}(t)=\frac12\mathcal T_c(0)\log t+O_{\beta,c}(1)\) as \(t\to\infty\), with \(\mathcal T_c(0)=\sum_nc_n(n+1)\). So \(\mathcal T_c(0)<0\) forces negative values, \(\mathcal T_c(0)>0\) gives positivity for large \(t\), and when \(\mathcal T_c(0)=0\) the sign is decided by the bounded terms.

Source: ch/nc_delta.tex, label dl:thm:slope.

Ramanujan’s lost notebook formula as a density with an RH criterion

Volume II, Theorem 23.7.1, page 714 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\Xi\) be the Riemann Xi function on the critical line, \(\phi(x)=\psi(x)+\frac1{2x}-\log x\), \(\gamma\) Euler’s constant. Let \(U_1,U_2\) be independent with the Polya law (characteristic function \(\Xi(t)/\Xi(0)\)), \(G_1,G_2\) independent \(\mathrm{Gamma}(\frac34)\), \(L_1,L_2\) independent standard Laplace, all independent, and \(Z=\frac12(U_1+U_2)+E\), \(E=\frac14\log(G_1/G_2)+L_1+L_2\). Then \(E\) is an extended generalised gamma convolution (EGGC) with Thorin measure \(2(\delta_1+\delta_{-1})+\sum_{k\ge0}(\delta_{3+4k}+\delta_{-3-4k})\), \(\mathbb{E}e^{itZ}=\bigl(\frac{\Xi(t/2)}{\Xi(0)}\bigr)^2\frac{|\Gamma(\frac{3+it}4)|^2}{\Gamma(\frac34)^2(1+t^2)^2}\), and for \(\alpha>0\), \(f_Z(\frac12\log\alpha)=-\frac{\sqrt\pi\sqrt\alpha}{16\,\Xi(0)^2\Gamma(\frac34)^2}\bigl(\frac{\gamma-\log(2\pi\alpha)}{2\alpha}+\sum_{k\ge1}\phi(k\alpha)\bigr)\) is the density of \(Z\) (Ramanujan’s formula, proved by Berndt and Dixit, read as a Fourier inversion). Moreover \(-\phi(x)=\int_0^\infty w(t)e^{-xt}dt\) with \(w(t)=\frac1{e^t-1}-\frac1t+\frac12\ge0\) (Binet), so \(S(\alpha)=-\sum_k\phi(k\alpha)\) is completely monotone with Bernstein density \(\rho(v)=\sum_kw(v/k)/k\), and \(f_Z(x)=c\bigl(e^xS(e^{2x})+\frac12e^{-x}(\log2\pi+2x-\gamma)\bigr)\) with \(c=\sqrt\pi/(16\,\Xi(0)^2\Gamma(\frac34)^2)\). Since \(f_Z>0\), \(\frac{\gamma-\log2\pi\alpha}{2\alpha}+\sum_k\phi(k\alpha)<0\) for all \(\alpha>0\). The Riemann hypothesis holds if and only if every zero of \(\mathbb{E}e^{itZ}\) in the strip \(|\operatorname{Im}t|<1\) is real.

Source: ch/nc_delta.tex, label dl:thm:Z.

Open problem, smoothing of the prime formula for Delta at level 13/2

Volume II, Problem 23.1, page 715 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\tilde u_{13/2}\) be the Thorin density of \(\Lambda(\frac{13}2)/\Lambda(\frac{13}2+\sqrt s)\) for \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\), given at \(\alpha>\frac{13}2\) by the prime formula \(\tilde u_\alpha(r)=\frac1\pi\bigl[\operatorname{Re}\psi(\alpha+ir)-\log2\pi-\sum_p\sum_{k\ge1}2\cos(k\theta_p)\log p\;p^{-k(\alpha-11/2)}\cos(kr\log p)\bigr]\) with \(\tau(p)=2p^{11/2}\cos\theta_p\), which converges only conditionally at \(\alpha=\frac{13}2\). Open problem: compute \(\tilde u_{13/2}\) from this formula by a smoothing that does not blur the Poisson kernel of height \(\frac12\), and compare it in total variation with the Poisson smoothing of Keiper’s zeros of \(\Lambda\).

Source: ch/nc_delta.tex, label dl:prob:primes.

Open problem, real rootedness of truncated Delta kernels for N at least 3

Volume II, Problem 23.2, page 715 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. For \(N\ge1\) consider the transform of the even part of the truncated kernel \(e^{6u-2\pi e^u}\prod_{n\le N}(1-e^{-2\pi ne^u})^{24}\) of Ramanujan’s \(\Delta\). Open problem: for which \(N\ge3\) does this transform have only real zeros? The computation reported in the book shows complex zeros for \(N\le2\) and none in the rectangle \([0,40]\times[-8,8]\) for \(N=3,4\).

Source: ch/nc_delta.tex, label dl:prob:lee.

Yoshida analogue for Ramanujan’s Delta, positivity of the archimedean Weil form below length log 2

Volume II, Problem 23.3, page 715 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\Delta\) be Ramanujan’s discriminant form, \(\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s,\Delta)\), and \(A_6(t)=\frac1\pi\big(-\log2\pi+\operatorname{Re}\psi(6+it)\big)\) with \(\psi\) the digamma function. For real \(f\in L^2[-\frac L2,\frac L2]\) put \(F(t)=\int f(\nu)e^{i\nu t}\,d\nu\) and define the archimedean Weil form \(Q^{\mathrm{arch}}_L(f)=\int_{\mathbb R}A_6(t)|F(t)|^2\,dt\). For \(L<\log2\) no prime power enters the explicit formula, so \(Q^{\mathrm{arch}}_L(f)=\sum_\rho F(\tilde\gamma_\rho)\overline{F(\overline{\tilde\gamma_\rho})}\) over the zeros \(\rho=\beta_\rho+i\gamma_\rho\) of \(\Lambda\), with \(\tilde\gamma_\rho=\gamma_\rho+i(\beta_\rho-6)\); under the Riemann hypothesis for \(\Delta\) this is nonnegative. Problem: prove unconditionally that \(Q^{\mathrm{arch}}_L\) is positive definite for \(L<\log2\) (the analogue of Yoshida’s theorem for \(\zeta\)), and determine the length \(L_0\) at which the Weil form of \(\Delta\) becomes indefinite; a computation gives \(L_0\le0.8485\), and \(L_0\ge\log2\) under the Riemann hypothesis for \(\Delta\).

Source: ch/nc_delta.tex, label dl:prob:yoshida.

Volume II, Chapter 24. What the Euler product and the functional equation force

One sign suffices for the level alpha positivity criterion of the quasi Riemann hypothesis

Volume II, Proposition 24.1.1, page 718 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\beta+i\gamma\), and for \(\alpha>0\) let \(\mathrm{QRH}(\alpha)\) be the statement that no zero has \(\beta>\alpha\). Let \(\phi_0(y)=(\sin y/y)^4\), let \(W_\alpha\) be the level \(\alpha\) explicit formula functional (the real part of the sum over zeros, the prime sum and the archimedean term, acting on even test functions with compactly supported Fourier transform), \(M_\alpha=W_\alpha(\phi_0)\) and \(g_\alpha(a)=W_\alpha(\phi_0\cos(a\,\cdot))\), so that \(W_\alpha(\phi_0(1\pm\cos a\,\cdot))=M_\alpha\pm g_\alpha(a)\). For \(\alpha>0\) the following are equivalent: (a) \(\mathrm{QRH}(\alpha)\); (b) \(M_\alpha+g_\alpha(a)\ge0\) for every \(a\ge5\); (c) \(M_\alpha-g_\alpha(a)\ge0\) for every \(a\ge5\); (d) for every \(\varepsilon>0\) there is \(C_\varepsilon\) with \(g_\alpha(a)\ge-C_\varepsilon e^{\varepsilon a}\) on \([5,\infty)\), or symmetrically \(g_\alpha(a)\le C_\varepsilon e^{\varepsilon a}\). The proof is a smoothed form of Landau’s oscillation theorem applied to the Laplace transform of \(g_\alpha+C_\varepsilon e^{\varepsilon a}\). Equivalently, under \(\mathrm{QRH}(\alpha)\) the Toeplitz matrices \((g_\alpha((j-k)\eta))\) with \(g_\alpha\) extended evenly and \(g_\alpha(0)=M_\alpha\) are positive semidefinite, and semidefiniteness for all sizes and all \(\eta\) in a nonempty interval gives \(|g_\alpha(k\eta)|\le M_\alpha\) on a half line and hence \(\mathrm{QRH}(\alpha)\).

Source: ch/nc_rigid.tex, label rg:prop:onesign.

The Toeplitz form of the Weil functional at level one has a continuous nonnegative symbol and smallest eigenvalue of order N to the minus four

Volume II, Proposition 24.1.2, page 718 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function with zeros \(\rho=\beta+i\gamma\), let \(\phi_0(y)=(\sin y/y)^4\), and for \(\alpha>0\) let \(W_\alpha\) be the level \(\alpha\) Weil functional, \(W_\alpha(\phi)=\sum_\rho\operatorname{Re}L_\phi(\alpha-\rho)\) with \(L_\phi(w)=\pi^{-1}\int_0^\infty e^{-wu}\widehat\phi(u)\,du\) and \(\widehat\phi(u)=\int\phi(y)e^{-iyu}dy\) (for \(\phi\) real and even, with \(\widehat\phi\) continuous, piecewise twice continuously differentiable and compactly supported). Put \(M_\alpha=W_\alpha(\phi_0)\), \(g_\alpha(a)=W_\alpha(\phi_0\cos a\,\cdot)\) for real \(a\), and \(\varrho_\alpha(y)=\frac1\pi\operatorname{Re}\frac{\xi'}{\xi}(\alpha+iy)\). Let \(\alpha>0\) be such that no zero of \(\xi\) has real part above \(\alpha\), no zero lies on \(\operatorname{Re}s=\alpha\), and \(\varrho_\alpha(y)=O(\log(2+|y|))\) (all three hold at \(\alpha=1\)). For \(h>0\) let \(M_N(\alpha,h)=\big(g_\alpha(|j-k|h)\big)_{0\le j,k<N}\) with \(g_\alpha(0)=M_\alpha\). Then \(M_N(\alpha,h)\) is the Toeplitz matrix of the continuous symbol \(F_{\alpha,h}(\theta)=\frac{2\pi}{h}\sum_{m\in\mathbb{Z}}(\phi_0\varrho_\alpha)\big(\frac{\theta+2\pi m}{h}\big)\ge0\); the smallest eigenvalue \(\lambda_{\min}(M_N(\alpha,h))\) decreases to \(\min F_{\alpha,h}\) as \(N\to\infty\); and \(\min F_{\alpha,h}=0\) if and only if \(h=2/j\) for an integer \(j\ge2\). At \(\alpha=h=1\), for every \(N\ge1\), \(\frac{4k_1}{(N+1)^2(N+2)^2}\le\lambda_{\min}(M_N(1,1))<\frac{126K_1}{N^4}\), where \(k_1\) and \(K_1\) are the minimum and the maximum of \(F_{1,1}(\theta)/(1+\cos\theta)^2\) on the circle, both positive and finite.

Source: ch/nc_rigid.tex, label rg:prop:toeplitz1.

Beurling system with positive prime measure, polynomial growth and a zero off the critical line

Volume II, Theorem 24.2.1, page 719 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Fix \(\beta\in(0,1)\), \(\gamma>0\), \(\rho=\beta+i\gamma\), \(T_a=\log 4/(1-\beta)\), \(T_b>T_a\) and a nonincreasing \(\chi\in C^2\) with \(\chi=1\) on \([0,T_a]\) and \(\chi=0\) on \([T_b,\infty)\). Put \(q(t)=1-e^{-t}-2e^{-(1-\beta)t}\cos(\gamma t)(1-\chi(t))\), \(d\Pi_B=q(\log u)\,du/\log u\) and \(\Lambda_B=q(\log u)\,du\) on \((1,\infty)\), with generalised integers \(N_B=\exp_*\Pi_B\), zeta function \(\zeta_B(s)=\int u^{-s}dN_B=\exp\int u^{-s}d\Pi_B\) and \(\mu_B=\exp_*(-\Pi_B)\). Then (a) \(q\ge0\) and \(\frac14\le q\le2\) for \(t\ge T_a\), so \(\Lambda_B\) has density at most \(2\), \(N_B\) is a positive measure with density \(\ll_\varepsilon u^\varepsilon\) and \(|\mu_B|\le N_B\). (b) \(\zeta_B(s)=\frac{s(s-\rho)(s-\bar\rho)}{s-1}e^{-E(s-\rho)-E(s-\bar\rho)}\) with \(E\) entire, so \(\zeta_B\) is meromorphic, its only pole is a simple pole at \(1\) with positive residue, its zeros are exactly \(0,\rho,\bar\rho\), all zeros have real part at most \(\beta\), and the statement that no zero has real part above \(\beta-\varepsilon\) fails for every \(\varepsilon>0\). (c) \(\zeta_B\) is bounded in vertical strips for \(|t|\ge1\); \(\int w(u/x)\,dN_B=c\,x\,\mathcal{M}w(1)+O_{K,w}(x^{-K})\) for \(w\in C_c^2\) and every \(K\), with \(\mathcal{M}\) the Mellin transform; and \(\psi_B(x)-x=-2\operatorname{Re}(x^\rho/\rho)+O(\log x)\). (d) For \(a\ge T_b+4\) the criterion function of the book’s level \(\alpha\) positivity criterion for \(B\) is \(g_{B,\alpha}(a)=(\sinh\alpha/\alpha)^4e^{-\alpha a}+2\operatorname{Re}\big[(\sinh(\rho-\alpha)/(\rho-\alpha))^4e^{(\rho-\alpha)a}\big]\); at \(\alpha=\beta\) it stays below \(M_{B,\beta}\) and for \(\alpha<\beta\) it grows like \(e^{(\beta-\alpha)a}\).

Source: ch/nc_rigid.tex, label rg:thm:beurling.

Barrier for arguments about the integers only

Volume II, Corollary 24.2.2, page 720 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Consider an argument for the quasi Riemann hypothesis at level \(\alpha\) for \(\zeta\) (no zero with real part above \(\alpha\)) that uses about the integers only: positivity of the prime measure and \(|\mu|\le N\) for the generalised integers \(N\) and the Mobius type measure \(\mu\); growth of the zeta function of at most polynomial order in vertical strips; and type I asymptotics with a power saving error. It applies word for word to the Beurling system \(B_{\beta,\gamma}\), which has all three properties a fortiori, and the quasi Riemann hypothesis at level \(\alpha\) fails for \(B_{\beta,\gamma}\) for every \(\alpha<\beta\). So no such argument proves the quasi Riemann hypothesis at any level \(\alpha<1\). The statement about the model is proved; as a statement about arguments it is an informal obstruction whose content depends on what the word “uses” admits.

Source: ch/nc_rigid.tex, label rg:cor:barrier.

The functional equation term of the explicit formula vanishes for zeta, L of chi minus 4 and Davenport Heilbronn

Volume II, Proposition 24.3.1, page 721 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be \(\zeta\), \(L(s,\chi_{-4})\) or the Davenport and Heilbronn function \(f_{\mathrm{DH}}\) of conductor \(5\) (odd, with a zero at \(0.808517182457+85.699348485378i\)), each with a completed function \(\Xi_F(s)=\Xi_F(1-s)\). For every \(\alpha>0\) and \(a>4\) the criterion function of \(F\) (the level \(\alpha\) functional applied to \(\phi_0\cos(a\,\cdot)\), \(\phi_0(y)=(\sin y/y)^4\)) is the sum over the nontrivial zeros of \(F\) of \(\operatorname{Re}[(\sinh(\rho-\alpha)/(\rho-\alpha))^4e^{(\rho-\alpha)a}]\), and the term produced by the functional equation in the explicit formula, a Dirichlet series at the reflected point evaluated on a test function supported on positive logarithms, vanishes identically. This follows from Weil’s explicit formula for this test function. The identity was also confirmed numerically to \(10^{-8}\) for all three functions (a computation, not part of the proof).

Source: ch/nc_rigid.tex, label rg:prop:empty.

Level alpha criterion for Dirichlet L functions and the sign that cannot exclude a Landau Siegel zero

Volume II, Proposition 24.3.3, page 721 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi\) be a primitive Dirichlet character, \(g^\chi_\alpha\) the criterion function of \(L(s,\chi)\) (the analogue of \(g_\alpha\) built from \(\phi_0(y)=(\sin y/y)^4\), with the kernel \(-e^{-(\alpha+\mathfrak a)u}/(1-e^{-2u})\) in place of the pole and gamma kernels, \(\mathfrak a\in\{0,1\}\) the parity of \(\chi\)) and \(M^\chi_\alpha=W^\chi_\alpha(\phi_0)\). (a) The quasi Riemann hypothesis at level \(\alpha\) for \(L(s,\chi)\) (no zero with real part above \(\alpha\)) holds if and only if \(g^\chi_\alpha\le M^\chi_\alpha\) on \([5,\infty)\). (b) If only \(g^\chi_\alpha\ge-M^\chi_\alpha\) is assumed, then either the hypothesis at level \(\alpha\) holds or \(\sup\operatorname{Re}\rho>\alpha\) is attained at a real zero \(\beta_0\) of multiplicity \(m\), which adds \(+m(\sinh\delta_0/\delta_0)^4e^{\delta_0a}\) with \(\delta_0=\beta_0-\alpha\); this sign cannot exclude a Landau Siegel zero. (c) For \(\chi\in\{\chi_{-4},\chi_{-3},\chi_5\}\), \(L(\sigma,\chi)>0\) for all \(\sigma>0\) and either sign alone is equivalent to the hypothesis at level \(\alpha\). (d) By the classification of degree one functions (Kaczorowski and Perelli), the hypothesis at level \(\alpha\) for every function of degree one is equivalent to (a) for all primitive \(\chi\).

Source: ch/nc_rigid.tex, label rg:prop:dirichlet.

Near misses at conductors 5, 15 and 20, real zeros to the right of one

Volume II, Theorem 24.3.4, page 721 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi_5,\chi_{-3},\chi_{-4},\chi_{-15},\chi_{-20}\) be the real primitive characters of the indicated conductors and put \(F^{(5)}_\varepsilon=L(s,\chi_5)-\varepsilon\zeta(s)(1+5^{\frac12-s})\), \(F^{(20)}_\varepsilon=(1+5^{\frac12-s})L(s,\chi_{-4})-\varepsilon L(s,\chi_{-20})\) and \(F^{(15)}_\varepsilon=(1+5^{\frac12-s})L(s,\chi_{-3})-\varepsilon L(s,\chi_{-15})\). Each has real periodic coefficients and the functional equation of the type of \(\chi_5\), \(\chi_{-20}\), \(\chi_{-15}\) respectively, and \(\sup_n|a_\varepsilon(n)-\chi_5(n)|=\varepsilon(1+\sqrt5)\) for the first. For \(\varepsilon\in(0,1)\), \(F^{(5)}_\varepsilon\) has a real zero \(\sigma_\varepsilon>1\) with \(\sigma_\varepsilon-1\sim(1+5^{-1/2})\varepsilon/L(1,\chi_5)\), where \(L(1,\chi_5)=0.4304\). \(F^{(20)}_\varepsilon\) has a real zero in \((1,\infty)\) for \(\frac{1+\sqrt5}4<\varepsilon<1\) and \(F^{(15)}_\varepsilon\) for \(\frac{1+\sqrt5}6<\varepsilon<1\). As \(\varepsilon\to0\) the criterion function of \(F^{(5)}_\varepsilon\) does not converge to that of \(L(s,\chi_5)\), because the zero \(\sigma_\varepsilon\) merges with the pole. The zeros of \(F^{(20)}_\varepsilon\) and \(F^{(15)}_\varepsilon\) at \(\varepsilon=0.81\) and \(0.54\) (at \(1.018\) and \(1.009\)) are isolated by sign changes in ball arithmetic (certified).

Source: ch/nc_rigid.tex, label rg:thm:nearmiss.

Conductor 7 function with a positive even theta kernel and no Euler product

Volume II, Theorem 24.3.5, page 722 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\chi\) be the even primitive nonreal character modulo \(7\) with \(\chi(3)=e^{2\pi i/3}\), Gauss sum \(\tau=2.37047-1.17510i\), \(\kappa=\tan(\frac12\arg\tau)=-0.234261\), \(a(n)=\operatorname{Re}\chi(n)+\kappa\operatorname{Im}\chi(n)\) (so \(a(n)=1,-0.297124,-0.702876,-0.702876,-0.297124,1,0\) for \(n\equiv1,\dots,6,0\pmod 7\)) and \(f(s)=\sum a(n)n^{-s}\). Then \(f\) is entire, \(\Xi_f(s)=(7/\pi)^{s/2}\Gamma(\frac s2)f(s)\) satisfies \(\Xi_f(s)=\Xi_f(1-s)\) (the gamma factor of \(\zeta\)), and \(\Xi_f(\frac12+z)=\int K_f(u)e^{zu}du\) with the even kernel \(K_f(u)=2e^{u/2}\sum_na(n)e^{-\pi n^2e^{2u}/7}\ge0.9024\cdot2e^{|u|/2}e^{-\pi e^{2|u|}/7}>0\). There is no Euler product (\(a(2)a(3)=0.20884\ne a(6)=1\)) and \(f\) has no zero with \(\operatorname{Re}s\ge1.47\) (dominance, \(\sum_{n\ge2}|a(n)|n^{-1.47}=0.92378<1\)). The kernel bound uses that the symmetry holds exactly when \((1-i\kappa)\tau/\sqrt7=1+i\kappa\) and that \(\sum_{n\ge2}|a(n)|e^{-\pi(n^2-1)/7}=0.0975394<1\). Separately, a computation by the argument principle with two independent codes, not certified and not part of the proved claim, finds exactly seven zeros in \(0.505\le\operatorname{Re}s\le1.6\), \(0<\operatorname{Im}s\le400\), the rightmost at \(0.8727263+131.2069379i\).

Source: ch/nc_rigid.tex, label rg:thm:cond7.

The damped Polya kernel has a zero to the right of the critical line for every positive damping

Volume II, Proposition 24.3.6, page 722 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi\) be the completed Riemann zeta function and \(\Xi(t)=\xi(\frac12+it)\). In the normalisation of Rodgers and Tao let \(\Phi_{\mathrm{dBN}}(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)\exp(-\pi n^2e^{4u})\), which is positive for real \(u\), and for real \(t\) let \(H_t(z)=\int_0^\infty e^{tu^2}\Phi_{\mathrm{dBN}}(u)\cos(zu)\,du\) be the de Bruijn and Newman function, so that \(H_0(z)=\frac18\Xi(z/2)\). The de Bruijn and Newman constant \(\Lambda\) is the number such that \(H_t\) has only real zeros exactly for \(t\ge\Lambda\). For \(\varepsilon>0\) let \(F_\varepsilon(s)=H_{-\varepsilon}(-i(2s-1))\), whose kernel \(e^{-\varepsilon u^2}\Phi_{\mathrm{dBN}}(u)\) is positive (at \(\varepsilon=0\) this would be \(\frac18\xi(s)\)). Then \(F_\varepsilon\) is entire of order at most one, real and positive on \(\mathbb{R}\), \(F_\varepsilon(1-s)=F_\varepsilon(s)\), and \(F_\varepsilon(\frac12+\theta)=\int_0^\infty e^{-\varepsilon u^2}\Phi_{\mathrm{dBN}}(u)\cosh(2\theta u)\,du\) is entire. Let \(\Theta_\varepsilon=\sup\operatorname{Re}\rho\) over the zeros \(\rho\) of \(F_\varepsilon\). For \(\alpha\ge\frac12\) the function \(F_\varepsilon(\alpha)/F_\varepsilon(\alpha+\sqrt s)\) is the Laplace transform of a generalised gamma convolution if and only if \(\alpha\ge\Theta_\varepsilon\), if and only if \(\operatorname{Re}F_\varepsilon'/F_\varepsilon>0\) on \(\operatorname{Re}s>\alpha\); and \(\Theta_\varepsilon>\frac12\) for every \(\varepsilon>0\). The last claim uses the theorem of Rodgers and Tao that \(\Lambda\ge0\), together with de Bruijn’s theorem that real zeros persist forward in time (all zeros of \(H_{-\varepsilon}\) real would force \(\Lambda\le-\varepsilon\)).

Source: ch/nc_rigid.tex, label rg:prop:damped.

Functional equation and positive integer measure force the Riemann zeta function

Volume II, Theorem 24.4.1, page 723 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(N\) be a positive Borel measure on \([1,\infty)\) with \(\int x^{-\sigma}dN<\infty\) for \(\sigma>1\) and \(Z_N(s)=\int_{[1,\infty)}x^{-s}dN(x)\), such that \((s-1)Z_N(s)\) extends to an entire function of finite order and \(\pi^{-s/2}\Gamma(\frac s2)Z_N(s)=\pi^{-(1-s)/2}\Gamma(\frac{1-s}2)Z_N(1-s)\). Then \(N=R\sum_{n\ge1}\delta_n\) with \(R\) the residue of \(Z_N\) at \(1\), and \(Z_N=R\zeta\). In particular a Beurling system (positive prime measure, \(N=\exp_*\Pi\), \(N(\{1\})=1\)) with the functional equation of \(\zeta\) and, with finite order, no pole other than \(s=1\) is \(\zeta\). The ingredients are Hamburger’s theorem and the one dimensional linear programming bound of Cohn and Elkies; the authors have not checked the literature for the combination.

Source: ch/nc_rigid.tex, label rg:thm:A.

Rigidity of zeta under finite surgery with a positive Euler product

Volume II, Theorem 24.4.2, page 724 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F\) be meromorphic and real on \(\mathbb{R}\) with: (FE) the completed functional equation of \(\zeta\); (EP) \(F=\exp\int_{(0,\infty)}e^{-su}d\Pi_F(u)\) for \(\operatorname{Re}s>\sigma_1\) with a positive measure \(\Pi_F\); (S) \(F\) of finite order with finitely many poles, none of them real except a simple pole at \(1\), and \(F/\zeta\) with finitely many poles. Then \(F=\zeta\). When the only pole of \(F\) is a simple pole at \(1\) this already follows from the rigidity of positive integer measures, since \(\Pi_F\ge0\) makes \(N=\exp_*\Pi_F\) positive; the nonreal poles are what (S) adds. Each hypothesis is needed: without positivity the points \(\rho_1,\bar\rho_1,\rho_2,\bar\rho_2\) on the critical line can be moved to \(\rho,\bar\rho,1-\rho,1-\bar\rho\) keeping (FE); without (FE) there is the Beurling system \(B_{\beta,\gamma}\); without (EP) the Davenport and Heilbronn function; and without the restriction on real poles there are examples with a real pole. With infinitely many extra poles rigidity fails: for \(T>0\), \(\zeta(s)\coth^2(\pi(s-\frac12)/T)\) has the functional equation of \(\zeta\), finite order, a simple pole at \(1\), a positive prime measure and double zeros at \(\frac12+i(k+\frac12)T\) beyond those of \(\zeta\), and poles at the points \(\frac12+ikT\), \(k\in\mathbb{Z}\), among them a double pole at \(\frac12\).

Source: ch/nc_rigid.tex, label rg:thm:surgery.

Comparison principle for a ratio of Euler products with positive prime difference

Volume II, Proposition 24.4.3, page 725 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(G=F/F_0\) be meromorphic with \(G(\sigma)\to1\) as \(\sigma\to\infty\) and \(-G'/G=\int_{[0,\infty)}e^{-su}d\rho(u)\) for large \(\operatorname{Re}s\), with a positive measure \(\rho\ne0\) (that is, the prime measures satisfy \(\Lambda_F-\Lambda_{F_0}=\rho\ge0\)), and let \(\sigma_\rho\) be the abscissa of convergence of this transform. (a) If \(\sigma_\rho>-\infty\) then \(\sigma_\rho\) is the largest real pole \(p^*\) of \(G\) and \(G\) has no zero or pole in \(\operatorname{Re}s>\sigma_\rho\), so \(F\) and \(F_0\) have the same zeros and poles there; if \(\sigma_\rho=-\infty\) then \(G\) has none at all. No functional equation is used. (b) If \(F\) and \(F_0\) have the same completed functional equation, then \(G(s)=G(1-s)\), \(\sigma_\rho>-\infty\), \(p^*\ge\frac12\), and all zeros and poles of \(G\) lie in \(1-p^*\le\operatorname{Re}s\le p^*\); in particular if \(G\) has no real pole, \(G\equiv1\). (c) If \(G\) is real on \(\mathbb{R}\) with no nonreal pole on \(\operatorname{Re}s=p^*\), and \(m\) is the order of the pole at \(p^*\), then a zero of \(G\) of multiplicity \(n\) on \(\operatorname{Re}s=p^*\) forces \(m\ge2n\).

Source: ch/nc_rigid.tex, label rg:prop:comparison.

Residue rigidity for functions with prime measure below that of zeta

Volume II, Proposition 24.4.5, page 726 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F_0=\exp\int e^{-su}u^{-1}d\Lambda_{F_0}(u)\) for \(\operatorname{Re}s>1\) with \(0\le\Lambda_{F_0}\le\Lambda_\zeta\) (the von Mangoldt measure of \(\zeta\), as measures), and suppose \(F_0\) has a pole at \(1\). Then the pole is simple and \(\operatorname{Res}_{s=1}F_0\le1\), with equality if and only if \(F_0=\zeta\). No functional equation is used.

Source: ch/nc_rigid.tex, label rg:prop:residue.

Positive primes and a conductor larger than one, finitely many primes in the atoms

Volume II, Theorem 24.4.6, page 726 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. A Beurling zeta function \(X\) has the functional equation of \(\zeta\) with conductor \(Q\) if \(Q^{s/2}\pi^{-s/2}\Gamma(\frac s2)X(s)\) is invariant under \(s\mapsto1-s\) and \((s-1)X\) is entire of finite order; it has a positive prime measure and \(N_X=\exp_*\Pi_X\ge0\). Let \(X\) be such a function vanishing at every nontrivial zero of \(\zeta\) with multiplicity, so that \(G=X/\zeta\) is entire. Then \(Q\ge1\) and \(G(s)=\int_{[0,\log Q]}e^{-su}dM(u)\) for a finite real measure \(M\) with \(M(\{0\})=1\) and \(M(\{\log Q\})=\sqrt Q\). Let \(\mathfrak M\) be the monoid generated by the points \(e^u\), \(u\) an atom of \(M\). If only finitely many primes have a power in \(\mathfrak M\) (for instance if \(M\) has finitely many atoms, its continuous part being arbitrary), then \(Q=1\) and \(X=\zeta\).

Source: ch/nc_rigid.tex, label rg:thm:B.

Quantitative form, a zero added to zeta costs negative prime coefficients

Volume II, Theorem 24.4.7, page 727 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(X=\zeta G\) be a Beurling type function with the functional equation of \(\zeta\) of conductor \(Q>1\), \(G=\int e^{-su}dM\) with \(M\) a finite real measure on \([0,\log Q]\), \(M(\{0\})=1\), \(M(\{\log Q\})=\sqrt Q\), as in the positive primes and conductor theorem, but with \(\Pi_X\) only a real measure and the atoms of \(M\) involving finitely many primes. Let \(G_a\) be the atomic part of \(G\), \(n\) the negative part of the atomic part of \(\Pi_X\) on \(\log\mathfrak M\) (with \(\mathfrak M\) the monoid generated by the points \(e^u\), \(u\) an atom of \(M\)), \(\sigma_-\) the abscissa of convergence of \(\int e^{-su}dn(u)\) and \(\sigma_*=\max(\sigma_-,0)\). Then every zero of \(G_a\) lies in \(1-\sigma_*\le\operatorname{Re}s\le\sigma_*\), and \(\sigma_-\ge\frac12\): the negative part of the prime measure is not \(O(x^\theta)\) for any \(\theta<\frac12\), so \(\Pi_X\ge-C\Pi_\zeta\) is impossible. A zero added to \(\zeta\) at real part \(\beta>\frac12\) by a finite sum of point masses costs negative prime coefficients of order at least \(x^\beta\); this is sharp for \(X=\zeta(s)(1+Aq^{-s}+q^{1-2s})\).

Source: ch/nc_rigid.tex, label rg:thm:Bq.

Nonnegative Euler products over F5 with zeros off the line

Volume II, Proposition 24.4.8, page 727 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(P_\pm(T)=1\mp5T+5T^2\) and \(Z_\pm(T)=P_\pm(T)/((1-T)(1-5T))\). Then \(Z_\pm\) is rational, satisfies the functional equation \(Z_\pm(1/(5T))=Z_\pm(T)\) of genus one, and has an Euler product \(Z_\pm(T)=\prod_{d\ge1}(1-T^d)^{-M_d}\) with integers \(M_d\ge0\) for every \(d\); hence the point counts \(N_m=\sum_{d\mid m}dM_d\) and the divisor counts \(A_n\) (the coefficients of the power series of \(Z_\pm\)) are nonnegative. The inverse roots of \(P_\pm\) are \(\frac{\pm5\pm\sqrt5}{2}\), with \(|\alpha|/\sqrt5=0.618034\) and \(1.618034\), so \(Z_\pm(5^{-s})\) has zeros at real parts \(\log_5\frac{5+\sqrt5}{2}=0.798994\) and \(0.201006\). (The nonnegativity of \(M_d\) is checked on exact integer Newton sums for \(d\le60\) and follows for \(d\ge4\) from \(dM_d\ge5^d-|S_d|-\sum_{e\le d/2}(5^e+1+|S_e|)>0\) with \(S_d\) the sum of the \(d\)th powers of the two inverse roots and \(|S_e|\le2(3.62)^e\). Computed: for \(P_+\) (trace \(5\)) \(M_d=1,5,25,110,500,2215,10000\) and for \(P_-\) (trace \(-5\)) \(M_d=11,0,55,110,748,2200,12320\), for \(d=1,\dots,7\).)

Source: ch/nc_rigid.tex, label rg:prop:fakes.

A positive minorant of zeta is transparent to the zeros of zeta

Volume II, Theorem 24.5.1, page 728 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(F_0\) be meromorphic with the completed functional equation of \(\zeta\), a pole at \(1\) and prime measure \(0\le\Lambda_{F_0}\le\Lambda_\zeta\), and let \(G=\zeta/F_0\), \(\rho=\Lambda_\zeta-\Lambda_{F_0}\ge0\) with abscissa of convergence \(\sigma_\rho\). Then either \(F_0=\zeta\), or \(\frac12\le\sigma_\rho<1\), \(G\) is holomorphic and zero free in \(\operatorname{Re}s>\sigma_\rho\) and \(\operatorname{Re}s<1-\sigma_\rho\), and \(\sigma_\rho\) is a pole of \(G\) and a zero of \(F_0\) (of even order at least \(2\) if \(\sigma_\rho=\frac12\)). So if \(\sigma_\rho>\frac12\), \(F_0\) has the real zero \(\sigma_\rho\in(\frac12,1)\) whatever the zeros of \(\zeta\) are; if \(\sigma_\rho=\frac12\), the zeros of \(F_0\) off the line are exactly those of \(\zeta\), with multiplicity. Hence the statement that every zero of \(F_0\) in \(0<\operatorname{Re}s<1\) has real part \(\frac12\) holds if and only if the Riemann Hypothesis holds and \(\sigma_\rho\in\{\frac12,-\infty\}\).

Source: ch/nc_rigid.tex, label rg:thm:minorant.

Explicit obstruction, zeros off the line with a positive minorant need real poles

Volume II, Theorem 24.5.2, page 728 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. For \(\frac12<\beta<1\) and \(\gamma>0\) let \(H_{\beta,\gamma}(s)=1+\gamma^2/(s-\beta)^2\) and \(F_{\beta,\gamma}=\zeta H_{\beta,\gamma}H_{1-\beta,\gamma}\). It has exactly the completed functional equation of \(\zeta\), prime measure \(\Lambda_\zeta+r(u)\,du\) with \(r(u)=2(e^{\beta u}+e^{(1-\beta)u})(1-\cos\gamma u)\ge0\), zeros at \(\beta\pm i\gamma\) and \(1-\beta\pm i\gamma\), a simple pole at \(1\) and double poles at \(\beta\) and \(1-\beta\); the abscissa \(\sigma_\rho\) of \(r\) and the largest real pole \(p^*\) of \(F_{\beta,\gamma}/\zeta\) equal \(\beta\), and the pole order \(m\) is twice the zero multiplicity \(n\), the equality case of the comparison principle. For \(\frac12<\beta<p<1\) and every \(\gamma>0\) there is also a symmetric \(G=G(1-s)\) with simple real poles, largest \(p\), and a zero at \(\beta+i\gamma\): zeros \(\frac12\pm c\pm i\gamma\), poles \(\frac12\pm a\) and \(\frac12\pm b\) with \(c=\beta-\frac12\), \(a=p-\frac12\), \(b=\max(2c^2-a^2,a^2/4)^{1/2}\), and density \(2e^{u/2}[\cosh au+\cosh bu-2\cosh cu\cos\gamma u]\ge0\) by the power mean inequality. The case \(\beta=p\) occurs only with a pole of order at least \(2\), and \(\beta>p\) never occurs.

Source: ch/nc_rigid.tex, label rg:thm:Fbg.

The level shift operator carries the Weil functional from level alpha minus epsilon to level alpha and leaves the nonnegative cone

Volume II, Lemma 24.6.1, page 730 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function. Let \(\mathcal{C}\) be the class of real even functions \(\phi\) on \(\mathbb{R}\) whose Fourier transform \(\widehat\phi(u)=\int\phi(y)e^{-iyu}\,dy\) is continuous, piecewise twice continuously differentiable and supported in some interval \([-X,X]\); each \(\phi\in\mathcal{C}\) extends to an entire function. For \(\phi\in\mathcal{C}\) put \(L_\phi(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\) and, for \(\alpha>0\), define the Weil functional of level \(\alpha\) by \(W_\alpha(\phi)=\operatorname{Re}L_\phi(\alpha-1)+\frac1\pi\int_{\mathbb{R}}\phi(y)\operatorname{Re}\big[\frac1s-\frac{\log\pi}2+\frac12\psi(\frac s2)\big]_{s=\alpha+iy}\,dy-\frac1\pi\sum_{n\le e^X}\Lambda(n)n^{-\alpha}\widehat\phi(\log n)\), where \(\psi=\Gamma'/\Gamma\) and \(\Lambda\) is the von Mangoldt function; equivalently \(W_\alpha(\phi)=\sum_\rho\operatorname{Re}L_\phi(\alpha-\rho)\), summed over the zeros \(\rho\) of \(\xi\) with multiplicity. Let \(\phi_0(y)=(\sin y/y)^4\), an element of \(\mathcal{C}\) with \(\widehat{\phi_0}\) supported in \([-4,4]\). For \(0<\varepsilon<\alpha\) and \(\phi\in\mathcal{C}\) define \(T_\varepsilon\phi\) by \(\widehat{T_\varepsilon\phi}(u)=e^{\varepsilon|u|}\widehat\phi(u)\). Then \(T_\varepsilon\phi\in\mathcal{C}\) and \(W_{\alpha-\varepsilon}(\phi)=W_\alpha(T_\varepsilon\phi)\). For \(a>4\), \(T_\varepsilon(\phi_0\cos a\,\cdot)(y)=e^{\varepsilon a}\operatorname{Re}\big[e^{iay}\phi_0(y-i\varepsilon)\big]\), with \(\phi_0(z)=(\sin z/z)^4\) the entire extension and \(|\sin(y-i\varepsilon)|^2=\sin^2y+\sinh^2\varepsilon\), and \(T_\varepsilon\phi_0(y)=-\frac{2\varepsilon}{3}y^{-2}+O(y^{-4})\) as \(|y|\to\infty\). In particular, for every \(\varepsilon\in(0,\alpha)\) and \(a\ge5\), neither \(T_\varepsilon\big(\phi_0(1+\cos a\,\cdot)\big)\) nor \(T_\varepsilon\big(\phi_0(1-\cos a\,\cdot)\big)\) is a nonnegative function.

Source: ch/nc_rigid.tex, label rg:lem:shift.

The level step from alpha to alpha minus epsilon is equivalent to a one sided exponential bound on one inequality

Volume II, Proposition 24.6.2, page 730 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, and for \(\sigma>0\) write \(\mathrm{QRH}(\sigma)\) for the statement that no zero of \(\xi\) has real part greater than \(\sigma\). Let \(\mathcal{C}\) be the class of real even functions \(\phi\) on \(\mathbb{R}\) whose Fourier transform \(\widehat\phi(u)=\int\phi(y)e^{-iyu}\,dy\) is continuous, piecewise twice continuously differentiable and supported in some interval \([-X,X]\). For \(\phi\in\mathcal{C}\) put \(L_\phi(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\) and, for \(\sigma>0\), define the Weil functional of level \(\sigma\) by \(W_\sigma(\phi)=\operatorname{Re}L_\phi(\sigma-1)+\frac1\pi\int_{\mathbb{R}}\phi(y)\operatorname{Re}\big[\frac1s-\frac{\log\pi}2+\frac12\psi(\frac s2)\big]_{s=\sigma+iy}\,dy-\frac1\pi\sum_{n\le e^X}\Lambda(n)n^{-\sigma}\widehat\phi(\log n)\), where \(\psi=\Gamma'/\Gamma\) and \(\Lambda\) is the von Mangoldt function; equivalently \(W_\sigma(\phi)=\sum_\rho\operatorname{Re}L_\phi(\sigma-\rho)\) over the zeros \(\rho\) of \(\xi\) with multiplicity. Let \(\phi_0(y)=(\sin y/y)^4\), with entire extension \(\phi_0(z)=(\sin z/z)^4\), and \(g_\sigma(a)=W_\sigma(\phi_0\cos a\,\cdot)\). Let \(0<\varepsilon<\alpha\), let \(g^{(2)}_\alpha(a)=W_\alpha\big(\phi_0(\cdot/\sqrt2)\cos a\,\cdot\big)\), and let \(F_{\alpha,\varepsilon}(a)=W_\alpha\big(\operatorname{Re}[e^{ia\,\cdot}\phi_0(\cdot-i\varepsilon)]\big)\), so that \(F_{\alpha,\varepsilon}(a)=e^{-\varepsilon a}g_{\alpha-\varepsilon}(a)\) for \(a>4\). The following are equivalent: (a) \(\mathrm{QRH}(\alpha-\varepsilon)\); (b) there is \(C\) with \(F_{\alpha,\varepsilon}(a)\le Ce^{-\varepsilon a}\) for all \(a\ge5\) (equivalently, there is \(C\) with \(F_{\alpha,\varepsilon}(a)\ge-Ce^{-\varepsilon a}\) for all \(a\ge5\)); (c) there is \(C\) with \(g_\alpha(a)\le Ce^{-\varepsilon a}\) and \(g^{(2)}_\alpha(a)\le Ce^{-\varepsilon a}\) for all \(a\ge5\). Under (a), \(|g_\alpha(a)|\le e^{-\varepsilon a}\sum_\rho\big|\big(\sinh(\rho-\alpha)/(\rho-\alpha)\big)^4\big|\) for \(a\ge5\), the sum being over the zeros \(\rho\) of \(\xi\) with multiplicity.

Source: ch/nc_rigid.tex, label rg:prop:step.

What positivity at level alpha gives, and why no positive certificate reaches level alpha minus epsilon

Volume II, Theorem 24.6.4, page 731 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function. Let \(\mathcal{C}\) be the class of real even functions \(\phi\) on \(\mathbb{R}\) whose Fourier transform \(\widehat\phi(u)=\int\phi(y)e^{-iyu}\,dy\) is continuous, piecewise twice continuously differentiable and supported in some interval \([-X,X]\). For \(\phi\in\mathcal{C}\) put \(L_\phi(w)=\pi^{-1}\int_0^Xe^{-wu}\widehat\phi(u)\,du\) and, for \(\sigma>0\), define the Weil functional of level \(\sigma\) by \(W_\sigma(\phi)=\operatorname{Re}L_\phi(\sigma-1)+\frac1\pi\int_{\mathbb{R}}\phi(y)\operatorname{Re}\big[\frac1s-\frac{\log\pi}2+\frac12\psi(\frac s2)\big]_{s=\sigma+iy}\,dy-\frac1\pi\sum_{n\le e^X}\Lambda(n)n^{-\sigma}\widehat\phi(\log n)\), where \(\psi=\Gamma'/\Gamma\) and \(\Lambda\) is the von Mangoldt function. Let \(\frac12<\alpha\le1\) and \(0<\varepsilon<\alpha-\frac12\), and assume that no zero of \(\xi\) has real part greater than \(\alpha\) (equivalently, \(W_\alpha(\phi)\ge0\) for every nonnegative \(\phi\in\mathcal{C}\)). Let \(\phi_0(z)=(\sin z/z)^4\), \(\phi_a^-(y)=\phi_0(y)(1-\cos ay)\), \(g_\sigma(a)=W_\sigma(\phi_0\cos a\,\cdot)\), and let \(T_\varepsilon\phi\) be defined by \(\widehat{T_\varepsilon\phi}(u)=e^{\varepsilon|u|}\widehat\phi(u)\). (a) With \(\phi_1(z)=(\sin z/z)^2\) and \(\Psi_\varepsilon(y)=|\phi_1(y-i\varepsilon)|^2\), the function \(\Psi_\varepsilon\) is a nonnegative element of \(\mathcal{C}\) and \(|g_{\alpha-\varepsilon}(a)|\le e^{\varepsilon a}W_\alpha(\Psi_\varepsilon)\) for \(a>4\). (b) For \(a>4\), nonnegative \(\psi_j\in\mathcal{C}\) and \(c_j\ge0\), the function \(h=T_\varepsilon\phi_a^--\sum_jc_j\psi_j\) satisfies \(h(0)\le e^{4\varepsilon}-e^{\varepsilon a}(\sinh\varepsilon/\varepsilon)^4<0\). (c) For any \(f_1,\dots,f_J\in\mathcal{C}\) with Fourier transforms supported in \([-U,U]\) there is a positive measure \(D\) on \(\mathbb{R}\) with \(\int f_j\,dD=W_\alpha(f_j)\) for every \(j\) and \(\int T_\varepsilon\phi_a^-\,dD<0\) for arbitrarily large \(a\).

Source: ch/nc_rigid.tex, label rg:thm:certificate.

A function with the functional equation of zeta, a positive prime measure, a positive level measure and a zero to the left of the level

Volume II, Theorem 24.6.7, page 732 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12<\alpha\le1\) and suppose that the Riemann zeta function has no zero with real part at least \(\alpha\) (true for \(\alpha=1\)). Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed zeta function, \(D_\alpha(y)=\pi^{-1}\operatorname{Re}\,(\xi'/\xi)(\alpha+iy)\) its level \(\alpha\) density and \(m_\alpha=\min_{y\in\mathbb{R}}D_\alpha(y)>0\). Let \(\frac12<\beta<\alpha\), \(w=\alpha-\beta\), \(w'=\alpha+\beta-1\), \(P_v(y)=v/(\pi(v^2+y^2))\) the Cauchy kernel, and \(0<\gamma<\gamma_0(\alpha,\beta)=\big(\pi m_\alpha/(2(w^{-3}+w'^{-3}))\big)^{1/2}\). Let \(H_{\beta,\gamma}(s)=1+\gamma^2/(s-\beta)^2\), which has zeros at \(\beta\pm i\gamma\) and a double pole at \(\beta\), and \(\xi_F=\xi H_{\beta,\gamma}H_{1-\beta,\gamma}\), the completion of \(F=\zeta H_{\beta,\gamma}H_{1-\beta,\gamma}\). Then \(\xi_F(s)=\xi_F(1-s)\); \(F\) has the positive prime measure \(\Lambda_\zeta+2(e^{\beta u}+e^{(1-\beta)u})(1-\cos\gamma u)\,du\) on \((0,\infty)\), that is \(-F'/F(s)=\int_0^\infty e^{-su}\big(\Lambda_\zeta(du)+2(e^{\beta u}+e^{(1-\beta)u})(1-\cos\gamma u)\,du\big)\) for \(\operatorname{Re}s>1\), where \(\Lambda_\zeta=\sum_{n\ge2}\Lambda(n)\delta_{\log n}\) with \(\Lambda\) the von Mangoldt function; \(\xi_F\) has zeros at \(\beta\pm i\gamma\) and \(1-\beta\pm i\gamma\); and its level \(\alpha\) density, with the poles counted, is \(D^F_\alpha(y)=\pi^{-1}\operatorname{Re}\,(\xi_F'/\xi_F)(\alpha+iy)=D_\alpha(y)+\sum_{v\in\{w,w'\}}[P_v(y-\gamma)+P_v(y+\gamma)-2P_v(y)]\ge m_\alpha-\frac{2\gamma^2}{\pi}(w^{-3}+w'^{-3})>0\). So the level \(\alpha\) Weil functional of \(\xi_F\), \(\phi\mapsto\int\phi\,D^F_\alpha\,dy\), is nonnegative on nonnegative test functions, while \(\xi_F\) has a zero with real part greater than \(\alpha'\) for every \(\alpha'<\beta\).

Source: ch/nc_rigid.tex, label rg:thm:passes.

The functional equation, a positive prime measure and a positive level measure give the quasi Riemann hypothesis at no lower level

Volume II, Corollary 24.6.8, page 733 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12<\alpha\le1\) and suppose that the Riemann zeta function has no zero with real part at least \(\alpha\) (true for \(\alpha=1\)). Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(m_\alpha=\min_{y\in\mathbb{R}}\pi^{-1}\operatorname{Re}\,(\xi'/\xi)(\alpha+iy)>0\). For every \(\varepsilon>0\) and every \(\beta\in(\max(\frac12,\alpha-\varepsilon),\alpha)\), the function \(F=\zeta H_{\beta,\gamma}H_{1-\beta,\gamma}\), with \(H_{\beta,\gamma}(s)=1+\gamma^2/(s-\beta)^2\) and \(0<\gamma<\big(\pi m_\alpha/(2((\alpha-\beta)^{-3}+(\alpha+\beta-1)^{-3}))\big)^{1/2}\), has (a) the gamma factor of \(\zeta\) and the completed functional equation \(\xi_F(s)=\xi_F(1-s)\) for \(\xi_F=\xi H_{\beta,\gamma}H_{1-\beta,\gamma}\), (b) a positive prime measure (the measure \(\mu\ge0\) on \((0,\infty)\) with \(-F'/F(s)=\int_0^\infty e^{-su}\mu(du)\) for \(\operatorname{Re}s>1\)), and (c) a positive level \(\alpha\) density \(\pi^{-1}\operatorname{Re}\,(\xi_F'/\xi_F)(\alpha+iy)\), and it has the zero \(\beta+i\gamma\) with real part in \((\alpha-\varepsilon,\alpha)\). Hence the inputs (a), (b), (c) imply for no \(\varepsilon>0\) that there is no zero with real part above \(\alpha-\varepsilon\), not even with \(\varepsilon=\varepsilon(\alpha)\to0\) as \(\alpha\to1\), and they give no limit level below \(1\).

Source: ch/nc_rigid.tex, label rg:cor:nolimit.

No uniform level step exists over a class containing zeta and functions with zeros at every real part

Volume II, Proposition 24.6.9, page 733 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\mathfrak{F}\) be a class of functions \(F\), each with a statement \(\mathrm{QRH}_F(\alpha)\) that \(F\) has no zero with real part greater than \(\alpha\), such that \(\zeta\in\mathfrak{F}\) and, for every \(\beta\in(\frac12,1)\), some \(F_\beta\in\mathfrak{F}\) has a zero of real part \(\beta\) and satisfies \(\mathrm{QRH}_{F_\beta}(\alpha)\) for every \(\alpha\ge\max(\beta,\Theta_\zeta)\), where \(\Theta_\zeta\) is the supremum of the real parts of the zeros of \(\zeta\) in the critical strip. Let \(\frac12<\alpha_1<1\). Then there is no function \(\varepsilon:(\alpha_1,1]\to(0,\infty)\) such that \(\mathrm{QRH}_F(\alpha)\Rightarrow\mathrm{QRH}_F(\alpha-\varepsilon(\alpha))\) for all \(F\in\mathfrak{F}\) and all \(\alpha\in(\alpha_1,1]\).

Source: ch/nc_rigid.tex, label rg:prop:class.

The level step is a Taylor radius bound for the level alpha density

Volume II, Theorem 24.6.11, page 734 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12<\alpha\le1\), let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function, suppose that \(\xi\) has no zero with real part at least \(\alpha\), and let \(\Theta\le\alpha\) be the supremum of the real parts of its zeros. Let \(v_\alpha(\theta)=\operatorname{Re}\,(\xi'/\xi)(\alpha+i\theta)\) for real \(\theta\), which is positive under the stated hypothesis, and put \(\Lambda(\alpha)=\sup_{\theta\in\mathbb{R},\,k\ge1}\big(|v_\alpha^{(k)}(\theta)|/(k!\,v_\alpha(\theta))\big)^{1/k}\). Then: (a) for \(\varepsilon>0\), no zero of \(\xi\) has real part above \(\alpha-\varepsilon\) if and only if \(v_\alpha\) continues analytically to the strip \(|\operatorname{Im}\theta|<\varepsilon\), if and only if \(|v_\alpha^{(k)}(\theta)|\le k!\,\varepsilon^{-k}v_\alpha(\theta)\) for all \(k\ge1\) and \(\theta\in\mathbb{R}\); (b) \(\Lambda(\alpha)=1/(\alpha-\Theta)\), read as \(+\infty\) when \(\Theta=\alpha\); (c) the Riemann hypothesis holds if and only if \(|v_1^{(k)}(\theta)|\le k!\,2^k\,v_1(\theta)\) for all \(k\ge1\) and \(\theta\in\mathbb{R}\).

Source: ch/nc_rigid.tex, label rg:thm:taylor.

The P'olya law of the modified zeta function with double real poles

Volume II, Proposition 24.6.13, page 735 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: proved.

Statement. Let \(\frac12<\beta<1\), \(\gamma>0\), \(\eta=\beta-\frac12\), \(H_{\beta,\gamma}(s)=1+\gamma^2/(s-\beta)^2\), \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) the completed Riemann zeta function, \(\xi_F=\xi H_{\beta,\gamma}H_{1-\beta,\gamma}\), \(\Xi(t)=\xi(\frac12+it)\) and \(\Xi_F(t)=\xi_F(\frac12+it)\). Then \(\Xi_F=\Xi\,m\) with \(m(t)=|1+\gamma^2/(it-\eta)^2|^2\ge0\) and \(m(t)-1=\gamma^2(\gamma^2+2\eta^2-2t^2)/(t^2+\eta^2)^2\). Let \(\Phi_{\mathrm P}\) be the P'olya kernel of \(\Xi\), the even positive function with \(\Xi(t)=\int\Phi_{\mathrm P}(u)e^{itu}\,du\). Then the kernel of \(\Xi_F\) is \(\Phi_F=\Phi_{\mathrm P}*(\delta_0+r)\) with \(r(u)=\frac{\gamma^2}{4\eta^3}e^{-\eta|u|}\big[\gamma^2(1+\eta|u|)+4\eta^3|u|\big]>0\) and \(\int r=m(0)-1\), so \(\Phi_F\ge\Phi_{\mathrm P}>0\) is even. The P'olya variable of \(\xi_F\) (the variable with density \(\Phi_F/\int\Phi_F\)) is \(X+R\), where \(X\) has density proportional to \(\Phi_{\mathrm P}\) and \(R\) is independent of \(X\) with law \((\delta_0+r(u)\,du)/m(0)\) and characteristic function \(m(t)/m(0)\), which vanishes at \(\pm\gamma\pm i\eta\). The moment generating function of \(X+R\) exists only for \(|\operatorname{Re}h|<\eta\), so \(\Phi_F\) has no Gaussian moments and \(\Xi_F\) is not entire. Finally \(K_F(h)=\log\mathbb{E}e^{h(X+R)}\) has third derivative tending to \(+\infty\) as \(h\uparrow\eta\), so the single spin Griffiths, Hurst and Sherman inequality (third derivative of the cumulant generating function nonpositive) fails near the pole.

Source: ch/nc_rigid.tex, label rg:prop:FPolya.

Is zeta the only admissible positive minorant of zeta

Volume II, Problem 24.1, page 736 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is there a meromorphic \(G\not\equiv1\) with \(G(s)=G(1-s)\), \(-G'/G=\sum\rho_nn^{-s}\) with \(0\le\rho_n\le\Lambda(n)\) (\(\Lambda\) the von Mangoldt function), regular and nonzero at \(s=1\)? Equivalently, is there a positive minorant of \(\zeta\) other than \(\zeta\) itself? Such a \(G\) would have \(\frac12\le\sigma_\rho<1\), coefficients in \([0,1]\), infinitely many poles or infinite order, and infinitely many primes in the support of \(\rho\). The answer does not bear on the Riemann hypothesis.

Source: ch/nc_rigid.tex, label rg:prob:minorant.

Beurling zeta function of conductor above one with a positive theta kernel and a zero off the line

Volume II, Problem 24.2, page 736 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Is there a Beurling zeta function with the functional equation of \(\zeta\) of conductor \(Q>1\) (that is, \(Q^{s/2}\pi^{-s/2}\Gamma(\frac s2)X(s)\) invariant under \(s\mapsto1-s\)), a positive theta kernel, a simple pole at \(1\) and a zero off the critical line? By the rigidity theorems for positive integer measures and for conductors it would have to miss zeros of \(\zeta\) or have atoms involving infinitely many primes.

Source: ch/nc_rigid.tex, label rg:prob:conductor.

Is there a rational modification of zeta with positive prime and level measures, a zero left of the level and no low zero

Volume II, Problem 24.3, page 737 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\zeta\) be the Riemann zeta function, \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(\frac12<\alpha\le1\) and \(\varepsilon>0\). Is there a rational function \(R\) with \(R(s)=R(1-s)\) and \(R(s)\to1\) as \(|s|\to\infty\), with finitely many quadruples of zeros and pairs of real poles, such that \(F=\zeta R\) has a positive prime measure (a measure \(\mu\ge0\) on \((0,\infty)\) with \(-F'/F(s)=\int_0^\infty e^{-su}\mu(du)\) for \(\operatorname{Re}s>1\)), a positive level \(\alpha\) density \(\pi^{-1}\operatorname{Re}\,((\xi R)'/(\xi R))(\alpha+iy)\), a zero with real part in \((\alpha-\varepsilon,\alpha)\) and no zero with \(|\operatorname{Im}s|<14\)? The modification \(\zeta H_{\beta,\gamma}H_{1-\beta,\gamma}\) with \(H_{\beta,\gamma}(s)=1+\gamma^2/(s-\beta)^2\), which has the zeros \(\beta\pm i\gamma\), needs \(\gamma<0.03\).

Source: ch/nc_rigid.tex, label rg:prob:modif.

Is there a Lee and Yang strip for the Polya kernel divided by cosh, or a Williams type kernel with a zero off the line

Volume II, Problem 24.4, page 737 of Polson and Sokolov (2026), Computing the Riemann Hypothesis, edition 2. New in this book.

Status: open.

Statement. Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) be the completed Riemann zeta function and \(\Phi_{\mathrm P}\) its P'olya kernel, the even positive function on \(\mathbb{R}\) with \(\xi(\frac12+it)=\int\Phi_{\mathrm P}(x)e^{itx}\,dx\). Does the Fourier transform of \(x\mapsto\Phi_{\mathrm P}(x)/\cosh(ax)\) have all its zeros in \(|\operatorname{Im}t|\le\frac12\) for small \(a>0\)? Is there an entire function \(\Lambda\) with \(\Lambda(s)=\Lambda(1-s)\) whose kernel \(\Phi\), defined by \(\Lambda(\frac12+it)=\int\Phi(x)e^{itx}\,dx\), is positive and even, whose Williams variable \(W\), defined by \(\mathbb{E}W^{s/2}=\Lambda(s)/\Lambda(0)\), is a generalised gamma convolution (a law on \([0,\infty)\) with Laplace transform \(\exp\{-bz-\int_{(0,\infty)}\log(1+z/x)\,U(dx)\}\) for some \(b\ge0\) and some nonnegative measure \(U\)), and which has a zero off the critical line \(\operatorname{Re}s=\frac12\)?

Source: ch/nc_rigid.tex, label rg:prob:cosh.

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Footnotes

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