\documentclass[11pt]{amsart} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{longtable,booktabs,array,calc} \usepackage{newunicodechar} \usepackage{etoolbox} \AtBeginEnvironment{longtable}{\small} \usepackage[margin=1in]{geometry} \usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} \usepackage{xcolor}\usepackage{graphicx} \newunicodechar{−}{\ensuremath{-}} \newunicodechar{σ}{\ensuremath{\sigma}} \newunicodechar{½}{\ensuremath{\tfrac12}} \newunicodechar{✓}{\ensuremath{\checkmark}} \newunicodechar{✗}{\ensuremath{\times}} \newunicodechar{–}{--} \newunicodechar{≤}{\ensuremath{\le}} \newunicodechar{…}{\ldots} \DeclareRobustCommand{\S}{\ensuremath{\mathsection}} \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \setlength{\LTleft}{0pt} \title[Cross-energy of Dirichlet partial sums, Part 9]{Cross-Energy of Dirichlet Partial Sums.\\ Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence} \author{Pedro Caceres} \address{Universidad Europea de Valencia, Valencia, Spain} \email{pedrojesus.caceres@universidadeuropea.es} \subjclass[2020]{Primary 11M41; Secondary 30B50, 11K60, 11Y35} \keywords{prime zeta function, zeros of Dirichlet series, Jessen function, almost periodic functions, Kronecker's theorem} \date{October 2026. Part 9 of the series \emph{Cross-energy of Dirichlet partial sums} (seventeen parts; the plan of the series is in Part~1 \cite{P1}).} \begin{document} \begin{abstract} The prime zeta function $P(s)=\sum_pp^{-s}$ has zeros in its half-plane of absolute convergence. Belovas, \v{C}epaityt\.{e} and Sabaliauskas proved that $P\ne0$ for $\operatorname{Re}s>\sigma_P=1.7795446535\ldots$, the solution of $P(\sigma)=2^{1-\sigma}$. We show that this is sharp: the closure of the real parts of the zeros in $\operatorname{Re}s>1$ is exactly $[1,\sigma_P]$, there are no zeros on $\operatorname{Re}s=\sigma_P$, and zeros occur arbitrarily close to every line $\operatorname{Re}s=\sigma$ with $1<\sigma<\sigma_P$ (Bohr's method). By Jessen's theorem the density of real parts is $J''(\sigma)/2\pi$, where $J(\sigma)=-\sigma\log2+\mathbb E\log^+(2^\sigma|Y_\sigma|)$ and $Y_\sigma=\sum_{p\ge3}p^{-\sigma}e^{i\theta_p}$ has independent uniform phases. $J$ is smooth, $\sigma_P=\inf\{\sigma:J(\sigma)+\sigma\log2=0\}$, and the density vanishes at $\sigma_P$ faster than any power. The argument principle in ball arithmetic certifies exactly $68$ zeros with $1.1\le\operatorname{Re}s\le1.85$ below height $2000$ ($70.0$ predicted) and $12$ with $\operatorname{Re}s\ge1.5$ below height $5000$ ($12.0$ predicted). A heuristic large-deviation extrapolation predicts zeros with real part above $1.7$ about once every $6\cdot10^6$ units of height. In $\frac12<\operatorname{Re}s<1$, $P$ is single-valued on the half-plane minus $(\frac12,1]$ if and only if RH holds (a reformulation of a classical equivalence), its zeros are canonical below height $3\cdot10^{12}$, and they are the solutions of $\zeta(s)=e^{A(s)}$, $A(s)=\sum_{m\ge2}P(ms)/m$, with $|P(s)|<2\pi$. Below height $600$ their counts for $\operatorname{Re}s\ge0.7$ agree numerically with the random model. The zeros are cancellations between $2^{-s}$ and the remaining primes, the prime-sum analogue of the Euler-product interference behind the off-line zeros of the Davenport--Heilbronn function. We do not prove the Riemann hypothesis. \end{abstract} \maketitle \section{Introduction} The prime zeta function \[P(s)=\sum_{p}p^{-s}=\sum_{k\ge1}\frac{\mu(k)}{k}\log\zeta(ks)\] converges absolutely for \(\operatorname{Re}s>1\). The second expression continues it to \(\operatorname{Re}s>0\), where it has logarithmic branch points at \(s=\rho/k\) and \(s=1/k\) (\(k\) squarefree, \(\rho\) a zero of \(\zeta\)) and a natural boundary on \(\operatorname{Re}s=0\) \cite{LW}. Fröberg remarked in 1968 that very little was known about its roots \cite{Fr}. Recently Belovas, \v{C}epaityt\.{e} and Sabaliauskas \cite{BCS} proved that \(P(s)\ne0\) for \(\operatorname{Re}s>\sigma_P\), with \(\sigma_P\) as below, computed more than \(10^4\) zeros in \((0.1,1.65)\times(0,10^4)\), found the largest real part \(1.6826\ldots\) for \(|\operatorname{Im}s|<2\cdot10^5\), and tested conjectures on the distribution of the zeros statistically. In 2018 the author computed zeros of \(P\) numerically and plotted them on a logarithmic scale \cite{C0}. That computation found a cluster of zeros with \(\operatorname{Re}s<1.6\) and \(\operatorname{Im}s<800\). It also reported three further clusters with \(\operatorname{Re}s>10^9\) or \(\operatorname{Im}s>10^{13}\). This note explains the first cluster and shows that the other three are numerical artifacts. We restrict mainly to the half-plane \(\operatorname{Re}s>1\), where \(P\) is single-valued and its Dirichlet series converges absolutely. There the zeros can be described completely. \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{The right edge (Theorem 2.1).} Let \(\sigma_P\) be the unique real solution of \(P(\sigma)=2^{1-\sigma}\), that is, \(2^{-\sigma}=\sum_{p\ge3}p^{-\sigma}\). Then \(\sigma_P=1.7795446535\ldots\) and \(P\) has no zeros with \(\operatorname{Re}s\ge\sigma_P\). Conversely, for every \(1<\sigma<\sigma_P\) there are zeros arbitrarily close to the line \(\operatorname{Re}s=\sigma\); we do not claim that every such line contains a zero. So the closure of the real parts of the zeros in \(\operatorname{Re}s>1\) is exactly \([1,\sigma_P]\). \item \textbf{The density (Section 3).} By Jessen's theorem on the zeros of almost periodic functions \cite{Je,JT}, the zeros with \(\operatorname{Re}s\) in an interval have a frequency given by the second derivative of the mean \(J(\sigma)\) of \(\log|P|\) on vertical lines. We show that \(J(\sigma)=-\sigma\log2+\mathbb E\log^+\big(2^{\sigma}|Y_\sigma|\big)\), where \(Y_\sigma=\sum_{p\ge3}p^{-\sigma}e^{i\theta_p}\) has independent uniform phases, that \(J\) is \(C^\infty\), and that \(\sigma_P=\inf\{\sigma>1:J(\sigma)+\sigma\log2=0\}\); we compute \(J\) through a Bessel-product formula. \item \textbf{Certified counts (Section 4).} By the argument principle in ball arithmetic there are exactly \(68\) zeros (with multiplicity) with \(1.1\le\operatorname{Re}s\le1.85\) and \(0<\operatorname{Im}s<2000\), against \(70.0\) predicted, and exactly \(12\) with \(1.5\le\operatorname{Re}s\le1.85\), \(0<\operatorname{Im}s<5000\), against \(12.0\) predicted. The distribution of real parts matches the prediction bin by bin (numerically). The edge is sharp but practically invisible: by a heuristic large-deviation extrapolation (not certified), zeros with \(\operatorname{Re}s>1.7\) are expected about once every \(6\cdot10^6\) units of height. \end{enumerate} \textbf{What is new.} The zero-free half-plane \(\operatorname{Re}s>\sigma_P\) is due to \cite{BCS}. Theorem 2.1 adds the boundary line and shows that the half-plane is sharp; it is an instance of the classical theory of Bohr, Jessen and Tornehave for Dirichlet series with linearly independent frequencies \cite{Bo,JT}, in which a line \(\operatorname{Re}s=\sigma\) carries zeros nearby exactly when no single term dominates the sum of the others, and we include a self-contained proof. New here are also the Jessen density of the real parts (Section 3), including its edge characterization and its flatness at \(\sigma_P\); the certified counts (Section 4); the heuristic edge asymptotics; and the analysis inside the critical strip (Section 5). The value \(1.6826\) found in \cite{BCS} below height \(2\cdot10^5\) is consistent with Section 3, which predicts about \(4\) zeros with real part above \(1.65\) and about \(0.4\) above \(1.68\) in that range. The mechanism is the one studied in \cite{P17} for the Davenport--Heilbronn function: \(P\) is a sum, not a product, and its zeros are cancellations between the term \(2^{-s}\) and the other primes. An Euler product cannot vanish where it converges absolutely, but a sum over primes can. \section{The right edge} \textbf{Theorem 2.1.} Let \(\sigma_P>1\) be defined by \(2^{-\sigma_P}=\sum_{p\ge3}p^{-\sigma_P}\). \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item \(P(s)\ne0\) for \(\operatorname{Re}s\ge\sigma_P\). \item For every \(1<\sigma<\sigma_P\) and every \(\varepsilon>0\), the set of \(t\) for which \(P\) has a zero in the disc \(|s-(\sigma+it)|<\varepsilon\) has positive lower density. \item Consequently \(\overline{\{\operatorname{Re}\rho:P(\rho)=0,\ \operatorname{Re}\rho>1\}}=[1,\sigma_P]\). \end{enumerate} The number \(\sigma_P\) is well defined. The function \(\sigma\mapsto\sum_{p\ge3}(2/p)^{\sigma}\) is strictly decreasing, tends to \(+\infty\) as \(\sigma\to1^+\), and tends to \(0\) as \(\sigma\to\infty\). We computed \(\sigma_P\) from \(P(\sigma)=\sum_k\mu(k)k^{-1}\log\zeta(k\sigma)\) (it agrees with \cite{BCS}); a direct sum over the primes below \(2\cdot10^6\), without tail correction, confirms it to about six digits. \emph{Proof of (a).} For \(\operatorname{Re}s=\sigma\ge\sigma_P\), \[|P(s)|\ge2^{-\sigma}-\sum_{p\ge3}p^{-\sigma}\ge0.\] The second inequality is strict for \(\sigma>\sigma_P\). Let \(\sigma=\sigma_P\) and \(P(s)=0\), \(s=\sigma_P+it\). Equality in the triangle inequality forces every \(p^{-s}\), \(p\ge3\), to point opposite to \(2^{-s}\), that is, \(t\log(p/2)\equiv\pi\pmod{2\pi}\) for all \(p\ge3\). If \(t=0\) this is impossible, since \(P(\sigma_P)>0\). If \(t\ne0\), write \(t\log\frac32=b\pi\) and \(t\log\frac52=a\pi\) with \(a,b\) odd integers. Then \(a,b\ne0\) have the same sign and \(b\log\frac52=a\log\frac32\), so \((5/2)^{|b|}=(3/2)^{|a|}\), that is, \(5^{|b|}2^{|a|}=3^{|a|}2^{|b|}\). The left side is divisible by \(5\) and the right side is not, which contradicts unique factorization. \(\square\) \textbf{Lemma 2.2 (polygon lemma).} Let \(\ell_1\ge\ell_2\ge\dots\ge\ell_n>0\), \(n\ge2\), with \(\ell_1\le\ell_2+\dots+\ell_n\). Then \(\{\sum_j\ell_je^{i\theta_j}:\theta_j\in\mathbb R\}\) is the closed disc of radius \(\sum_j\ell_j\) centred at \(0\). The hypothesis is necessary: for lengths \(2\) and \(1\) the set of sums is the annulus \(1\le|z|\le3\). \emph{Proof.} The set \(K\) of sums is the image of a torus, so it is compact and connected, and it is invariant under rotations; so \(\{|z|:z\in K\}\) is an interval containing \(\sum_j\ell_j\), and it suffices to show \(0\in K\). Choose signs \(\varepsilon_2,\dots,\varepsilon_n\) greedily (\(\varepsilon_j=-1\) if the partial sum \(\sum_{2\le iN}a_pN\). The tail \(R=\sum_{p>N}a_pe^{i\theta_p}\) has modulus at most \(\sum_{p>N}a_p1\) and not identically zero (\(2^sF(s)\to e^{i\theta_2}\) as \(\operatorname{Re}s\to\infty\)), so there is \(\delta<\min(\varepsilon,\sigma-1)\) with \(m:=\min_{|s-\sigma|=\delta}|F(s)|>0\). The numbers \(\log p\) are linearly independent over \(\mathbb Q\). By the Kronecker--Weyl theorem, for every \(Y\) and \(\eta>0\) the set of \(t\) with \(|p^{-it}-e^{i\theta_p}|<\eta\) for all \(p\le Y\) has positive density (the measure of the corresponding box in the torus \(\mathbb T^{\pi(Y)}\)). For such \(t\) and \(|s-\sigma|=\delta\), \[|P(s+it)-F(s)|\le\eta\sum_{p\le Y}p^{-\sigma+\delta}+2\sum_{p>Y}p^{-\sigma+\delta}.\] Take \(Y\) large and \(\eta\) small to make the right side smaller than \(m\). By Rouché's theorem \(P(\cdot+it)\) has a zero in \(|s-\sigma|<\delta\). \(\square\) \emph{Proof of (c).} By (a) and (b), the closure contains \((1,\sigma_P)\), hence \([1,\sigma_P]\), and is contained in \([1,\sigma_P]\). \(\square\) This is Bohr's classical argument \cite{Bo}, \cite[Ch. XI]{T}. Mantovanelli recently used the same scheme to find the exact right edge of the zeros of the Fibonacci zeta function \cite{Ma}. For \(P\) no Diophantine obstruction arises, because all the frequencies \(\log p\) are independent. The arithmetic enters only through unique factorization, in (a) and in the Kronecker--Weyl theorem. \section{The Jessen function and the density of zeros} For \(\sigma>1\) put \[J(\sigma)=\lim_{T\to\infty}\frac1{2T}\int_{-T}^{T}\log|P(\sigma+it)|\,dt.\] \begin{figure}[h]\centering\includegraphics[width=\textwidth]{PrimeZeta_edge.png} \caption{The Jessen excess near the edge (numerical, not certified). (a), (b): $\log(J(\sigma)+\sigma\log2)$ from the Bessel formula, for $1.2\le\sigma\le1.68$ (above the error floor), against $1/(\sigma_P-\sigma)$ and $1/\sqrt{\sigma_P-\sigma}$, with $-I(\sigma)$ shifted by a constant, $I$ the Cram\'er rate of Section 3. (c): $I(\sigma)$ and $-\log(J+\sigma\log2)$ against $\sigma_P-\sigma$ on logarithmic axes.}\label{fig:edge}\end{figure} \textbf{The theorem used.} On every closed strip \(\sigma_1\le\operatorname{Re}s\le\sigma_2\) with \(\sigma_1>1\), \(P\) is the sum of a uniformly and absolutely convergent Dirichlet series, hence an analytic almost periodic function in the sense of Bohr, and it is not identically zero. For such functions Jessen \cite{Je} proved: the limit \(J(\sigma)\) exists for every \(\sigma\), uniformly on compact sets; \(J\) is convex; and the zeros, \emph{counted with multiplicity}, have a frequency in the following sense. If \(\nu\) is the measure \(J''/2\pi\) (the distributional second derivative, a positive measure because \(J\) is convex), then for all \(1<\sigma_1<\sigma_2\) at which \(J\) is differentiable, \begin{align*}N(\sigma_1,\sigma_2;T)&:=\#\{\rho:\ \sigma_1<\operatorname{Re}\rho<\sigma_2,\ 0<\operatorname{Im}\rho-\sigma\log2\) for \(1<\sigma<\sigma_P\). \emph{Proof.} Condition on \(Y_\sigma\) and average over \(\theta_2\) alone. Jensen's formula for the linear function \(z\mapsto2^{-\sigma}z+Y\) on the unit circle gives \(\frac1{2\pi}\int_0^{2\pi}\log|2^{-\sigma}e^{i\theta}+Y|\,d\theta=\max(\log2^{-\sigma},\log|Y|)\). The event \(|Y_\sigma|>2^{-\sigma}\) has positive probability exactly when \(\sup|Y_\sigma|=\sum_{p\ge3}p^{-\sigma}\) exceeds \(2^{-\sigma}\), that is, when \(\sigma<\sigma_P\): \(Y_\sigma\) is a continuous function on the torus \(\mathbb T^\infty\), so \(\{|Y_\sigma|>2^{-\sigma}\}\) is open, and Haar measure charges every nonempty open set. \(\square\) \textbf{Corollary 3.2 (edge characterization).} \(J(\sigma)+\sigma\log2\ge0\) for all \(\sigma>1\), with equality exactly for \(\sigma\ge\sigma_P\); hence \[\sigma_P=\inf\{\sigma>1:\ J(\sigma)+\sigma\log2=0\}.\] Moreover \(J\) is strictly convex on \((1,\sigma_P]\) and affine on \([\sigma_P,\infty)\). \emph{Proof.} The first statements are Proposition 3.1. By Theorem 2.1(b) every strip \(\sigma-\varepsilon<\operatorname{Re}s<\sigma+\varepsilon\) with \(1<\sigma<\sigma_P\) contains zeros with positive lower frequency (a set of \(t\) of lower density \(c>0\) gives at least \(cT/(2\varepsilon)+o(T)\) zeros, since each zero serves a \(t\)-interval of length at most \(2\varepsilon\)), so \(\nu\) charges every open subinterval of \((1,\sigma_P)\) and \(J'\) is strictly increasing there; on \([\sigma_P,\infty)\), \(J=-\sigma\log2\). \(\square\) \textbf{Probability and density are different quantities.} The excess \(J(\sigma)+\sigma\log2=\mathbb E\log^+(2^\sigma|Y_\sigma|)\) is positive exactly when the event \(|Y_\sigma|>2^{-\sigma}\) has positive probability, so this event decides \emph{where} zeros can occur. The density of zeros, however, is \(d_P(\sigma)=J''(\sigma)/2\pi\), the second derivative of the excess; it is not the probability \(\Pr(|Y_\sigma|>2^{-\sigma})\). Both are determined by the distribution of \(Y_\sigma\), and both collapse as \(\sigma\to\sigma_P\), because the event requires almost all the phases to align. \textbf{An exact formula.} For \(r>0\), \(\int_0^\infty(e^{-x}-J_0(rx))\,\frac{dx}x=\log\frac r2\). Since \(Z=\sum_pp^{-\sigma}e^{i\theta_p}\) is isotropic with characteristic function \(\mathbb EJ_0(x|Z|)=\prod_pJ_0(p^{-\sigma}x)\), taking expectations (Fubini on \([0,X]\), then \(X\to\infty\)) gives \[J(\sigma)=\log2+\int_0^\infty\Big(e^{-x}-\prod_pJ_0\big(p^{-\sigma}x\big)\Big)\frac{dx}{x}.\] \textbf{Lemma 3.3.} \(J\in C^\infty(1,\infty)\). \emph{Proof.} Let \(\Phi(x,\sigma)=\prod_pJ_0(p^{-\sigma}x)\) and \(1<\sigma_0\le\sigma\le\sigma_1\). Since \(|J_0|\le1\), \(|J_0(y)|\le(2/(\pi y))^{1/2}\) for \(y>0\) \cite[\S13.74]{W}, so \(|J_0(y)|<0.4\) for \(y\ge4\), and \(|J_0^{(j)}|\le1\), every derivative \(\partial_\sigma^k\Phi\) is a sum, absolutely convergent because \(\sum_p\log^kp\,p^{-\sigma_0}<\infty\), of terms bounded by \(C_kx^k\) times a product of all but at most \(k\) of the factors. At least \(\pi((x/4)^{1/\sigma_1})-k\) of the remaining factors are below \(0.4\) in modulus, so \(\partial_\sigma^k\Phi(x,\sigma)\) decays faster than any power of \(x\), uniformly in \(\sigma\). Near \(x=0\), \(\partial_\sigma^k\Phi=O(x^2)\) uniformly, since \(\Phi=1-\frac{x^2}4\sum_pp^{-2\sigma}+O(x^4)\). So all derivatives may be taken under the integral sign. \(\square\) We evaluate the product exactly for \(p\le10^4\). For \(10^42\cdot10^6\). For \(\sigma\ge1.8\), where the excess is exactly \(0\), the computed \(J(\sigma)+\sigma\log2\) is below \(10^{-8}\) in modulus; this is the error floor of the computation (double precision, not certified). The values also agree with the time average of \(\log|P(\sigma+it)|\) over \(0\) & 1.1 & 1.3 & 1.5 & 1.6 & 1.65 & 1.7 & 1.72 \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot zeros per unit height & 0.0350 & 0.0149 & 0.0024 & \(2.4\cdot10^{-4}\) & \(2.3\cdot10^{-5}\) & \(\approx1.6\cdot10^{-7}\) & \(\approx3\cdot10^{-9}\) \\ \end{longtable} The entries for \(\sigma\ge1.7\) are below the error floor of the Bessel computation and are extrapolations by the large-deviation rate described next (heuristic). With them, a zero with real part above \(1.7\) is expected about once every \(6\cdot10^6\) units of height, and one above \(1.72\) about once every \(3\cdot10^8\). \textbf{Corollary 3.4 (flatness at the edge).} For every \(N\), \(J(\sigma)+\sigma\log2\), \(J'(\sigma)+\log2\) and \(J''(\sigma)\) are \(O((\sigma_P-\sigma)^N)\) as \(\sigma\to\sigma_P^-\). In particular the density of the real parts of the zeros and the number of zeros per unit height with real part above \(\sigma\) tend to \(0\) faster than any power of \(\sigma_P-\sigma\), and \(J\) is not real-analytic at \(\sigma_P\). \emph{Proof.} By Lemma 3.3 and Corollary 3.2, \(g(\sigma)=J(\sigma)+\sigma\log2\) is \(C^\infty\) on \((1,\infty)\) and vanishes identically on \([\sigma_P,\infty)\); so all its derivatives vanish at \(\sigma_P\), and Taylor's theorem gives the bounds. If \(J\) were real-analytic at \(\sigma_P\), \(g\) would vanish near \(\sigma_P\), contradicting Corollary 3.2. \(\square\) \textbf{Behaviour at the edge (heuristic).} The event \(|Y_\sigma|>2^{-\sigma}\) is a large deviation. Its Cramér rate in a fixed direction, \[I(\sigma)=\sup_{\lambda>0}\Big(\lambda2^{-\sigma}-\sum_{p\ge3}\log I_0(\lambda p^{-\sigma})\Big),\] can be computed accurately for every \(\sigma<\sigma_P\), and \(I(\sigma)\to\infty\) as \(\sigma\to\sigma_P\). On the range where the Bessel values are reliable (\(1.2\le\sigma\le1.68\)), \(\log(J(\sigma)+\sigma\log2)+I(\sigma)\) varies only from \(-1.6\) to \(-3.4\) while \(I\) goes from \(2.2\) to \(12.6\) (Figure~\ref{fig:edge}(a),(b)), consistent with \(\log(J(\sigma)+\sigma\log2)=-I(\sigma)(1+o(1))\). A saddle-point analysis of \(I\) (the primes \(p\lesssim\lambda^{1/\sigma}\) are nearly aligned and the remaining tail is of the size of the deficit \(\sum_{p\ge3}p^{-\sigma}-2^{-\sigma}\asymp\sigma_P-\sigma\)) suggests \[I(\sigma)\asymp x^{-1/(\sigma_P-1)}\big(\log\tfrac1x\big)^{-\sigma_P/(\sigma_P-1)},\qquad x=\sigma_P-\sigma,\] with \(1/(\sigma_P-1)=1.283\). The local exponent \(d\log I/d\log\frac1x\) computed from \(I\) is \(0.88\), \(0.95\), \(1.02\) at \(x=10^{-2},10^{-3},10^{-4}\), against \(0.79\), \(0.95\), \(1.03\) for \(1.283-2.283/\log\frac1x\). Decay faster than any power is proved (Corollary 3.4); if the heuristic holds, the excess decays eventually faster than \(e^{-c/(\sigma_P-\sigma)}\) for every \(c\), which is not proved. Figure~\ref{fig:edge} shows \(\log(J+\sigma\log2)\) against \(1/(\sigma_P-\sigma)\) and \(1/\sqrt{\sigma_P-\sigma}\). \section{Numerical verification} \textbf{Evaluation of \(P\).} For \(\operatorname{Re}s\ge1.1\) we use \[P(s)=\operatorname{Log}\zeta(s)+\sum_{p\le M}\big(\log(1-p^{-s})+p^{-s}\big)+E,\qquad |E|\le\frac{M^{1-2\sigma}}{2(2\sigma-1)(1-M^{-\sigma})},\] which follows from \(\log\zeta(s)=-\sum_p\log(1-p^{-s})\) by bounding \(\sum_{p>M}\sum_{m\ge2}p^{-m\sigma}/m\). Here \(\operatorname{Log}\) is the principal branch: for \(\operatorname{Re}s\ge1.1\), \(|\operatorname{Im}\log\zeta(s)|\le\log\zeta(1.1)<\pi\). We take \(M=10^4\) for \(\operatorname{Re}s<1.3\) and \(M=2000\) otherwise, and evaluate \(\zeta\), the logarithms and the exponentials in ball arithmetic (Arb). A second, independent floating-point implementation (\(P(s)=\log\zeta(s)-\sum_{k=2}^{39}P(ks)/k\), Euler--Maclaurin for \(\zeta\)) reproduces the zeros to six digits. \textbf{Certified counting.} On a side of a rectangle with \(\operatorname{Re}s\ge s_0\) we use \(|P'(s)|\le\sum_p\log p\,p^{-s_0}\le-\zeta'/\zeta(s_0)=:B\). From a point \(s_j\) where the ball for \(P(s_j)\) has modulus at least \(l_j\), we step by \(h_j\le0.45\,l_j/B\). On \([s_j,s_{j+1}]\) the values of \(P\) stay in a disc of radius \(10^9\) are artifacts: there \(P(s)\approx2^{-s}\), which is smaller than any floating-point tolerance, so a root-finder accepts every point. Those with \(\operatorname{Im}s>10^{13}\) are also artifacts, because the phases \(t\log p\) exceed double precision; genuine zeros with \(\operatorname{Re}s<\sigma_P\) do exist at such heights by Theorem 2.1(b), but not at the computed locations. Theorem 2.1(a) excludes all of these with \(\operatorname{Re}s\ge\sigma_P\) rigorously. \begin{figure}[h]\centering\includegraphics[width=\textwidth]{PrimeZeta_zeros.png} \caption{Left: zeros of $P$ with $\operatorname{Re}s\ge1.1$ (all below height $2000$; those with $\operatorname{Re}s>1.5$ up to height $5000$; counts certified, locations floating point) and the edge $\sigma_P$. Centre: distribution of real parts below height $2000$ against the Jessen prediction $TJ''(\sigma)/2\pi$. Right: the excess $J(\sigma)+\sigma\log2$.}\label{fig:zeros}\end{figure} \section{Inside the critical strip} For \(0<\operatorname{Re}s<1\) the continuation \(\sum_k\mu(k)k^{-1}\log\zeta(ks)\) has no poles. It has logarithmic branch points: \begin{itemize} \tightlist \item at \(s=1/k\) for squarefree \(k\), where \(P(s)\approx-\frac{\mu(k)}k\log(s-1/k)\); \item at \(s=\rho/k\) for every zero \(\rho\) of \(\zeta\), where \(P(s)\approx\frac{\mu(k)}k\,m_\rho\log(s-\rho/k)\) and \(m_\rho\) is the multiplicity of \(\rho\). \end{itemize} A zero of \(P\) in this region is only defined once a single-valued branch on a specified domain is fixed. We separate three statements: which domains carry a single-valued branch, the relation with RH, and the branch we compute. \textbf{Proposition 5.1.} Let \(\Omega=\{\operatorname{Re}s>\tfrac12\}\setminus(\tfrac12,1]\), a simply connected domain, and \(A(s)=\sum_{m\ge2}P(ms)/m\), which is analytic in \(\operatorname{Re}s>\tfrac12\). \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item RH holds if and only if \(P\) extends from \(\operatorname{Re}s>1\) to a single-valued analytic function on \(\Omega\). \item Let \(T_0=3\cdot10^{12}\). On the convex domain \(U=\{\operatorname{Re}s>\tfrac12,\ 0<\operatorname{Im}s1\), and \(P=\log\zeta-A\) is single-valued on \(U\). Every continuation of \(P\) along a path inside \(U\) gives this branch, so the zeros of \(P\) in \(U\) are canonical. \end{enumerate} \emph{Proof.} For \(m\ge2\) and \(\operatorname{Re}s>\tfrac12\), \(\operatorname{Re}ms>1\), so \(A\) is analytic there, and \(P=\log\zeta-A\) on \(\operatorname{Re}s>1\). (a) If RH holds, \(\zeta\) has no zeros on \(\Omega\) (it has none on \((\tfrac12,1)\) in any case, and its pole \(s=1\) is not in \(\Omega\)); since \(\Omega\) is simply connected, \(\log\zeta\) has an analytic branch on \(\Omega\) extending the series, and \(\log\zeta-A\) extends \(P\). Conversely, if \(F\) is analytic on \(\Omega\) and equals \(P\) on \(\operatorname{Re}s>1\), then \(\exp(F+A)\) is analytic on \(\Omega\) and equals \(\zeta\) on \(\operatorname{Re}s>1\), hence on \(\Omega\) by the identity theorem; so \(\zeta\) has no zeros in \(\Omega\), and therefore none with \(\operatorname{Re}s>\frac12\), which is RH. (b) \(U\) is convex, hence simply connected, and \(\zeta\ne0,\infty\) on \(U\). \(\square\) Statement (a) is a reformulation of the classical fact that RH is equivalent to the existence of an analytic branch of \(\log\zeta\) in \(\operatorname{Re}s>\tfrac12\) minus the segment \((\frac12,1]\); it is not progress towards RH. If RH failed, a zero \(\rho\) with \(\operatorname{Re}\rho>\frac12\) would be a logarithmic branch point of \(P\), and continuation around it would change \(P\) by \(2\pi i\,m_\rho\). \textbf{Zeros of \(P\) and values of \(\zeta\).} On \(U\), \(P(s)=L(s)-A(s)\), where \(L\) is the branch of \(\log\zeta\) of Proposition 5.1(b). So \[P(s)=0\iff L(s)=A(s)\implies\zeta(s)=e^{A(s)},\] and the converse of the last implication holds only up to the branch: \(\zeta(s)=e^{A(s)}\) if and only if \(P(s)\in2\pi i\,\mathbb Z\). So the zeros of \(P\) are the solutions of \(\zeta(s)=e^{A(s)}\) on which, in addition, \(|P(s)|<2\pi\). For \(\operatorname{Re}s\ge1.1\) the extra condition is automatic, since \(|P(s)|\le P(1.1)<2\pi\). This links the zeros of \(P\) with the value distribution of \(\zeta\): they are points where \(\zeta\) takes the prescribed value \(e^{A(s)}\), whose modulus \(e^{\operatorname{Re}A(s)}\) lies between \(e^{-A(\sigma)}\) and \(e^{A(\sigma)}\) for \(\operatorname{Re}s=\sigma\). \textbf{Numerics.} On rectangles \([0.55,1.1]\times[a,a+5]\) we continue \(\log\zeta\) along the boundary, starting from the principal value at \(\operatorname{Re}s=1.1\). The total change of \(\arg\zeta\) around every rectangle was \(0\), which confirms that no zero of \(\zeta\) lies inside, so the computed values are those of the branch of Proposition 5.1(b). The terms \(m\ge2\) are computed as in Section 4. These computations are in floating point and are not certified. Up to height \(1000\) there are \(240\) zeros of \(P\) with \(0.55\le\operatorname{Re}s\le1.1\). We located the \(128\) below height \(600\) by Newton's method. The Bohr--Jessen random model still describes the values of \(P\) in this range. The mean of \(\log|P(\sigma+it)|\) over \(500\tfrac12\). The counts agree with the model only away from \(\tfrac12\): \begin{longtable}[]{@{}llllll@{}} \toprule\noalign{} band of \(\operatorname{Re}s\) & \([0.55,0.6)\) & \([0.6,0.7)\) & \([0.7,0.8)\) & \([0.8,0.9)\) & \([0.9,1.1)\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot observed, \(t<600\) & 16 & 46 & 27 & 17 & 22 \\ \(\frac{600}{2\pi}\Delta J'\) & 208 & 91 & 32 & 17 & 19 \\ \end{longtable} We separate what is proved, what is observed and what is only a proposed explanation. \emph{Proved:} the asymptotic theory of Section 3 applies to any closed strip inside \(\operatorname{Re}s>1\). Inside the critical strip Jessen's theorem does not apply directly, since \(P\) is not almost periodic there; Borchsenius and Jessen \cite{BoJ} extended the theory to the values of \(\zeta\) in \(\frac12<\operatorname{Re}s<1\) by approximation in mean square, and their method should extend to \(P=\log\zeta-A\), but we do not prove this. The existence of a limiting value distribution of \(P(\sigma+it)\) for \(\sigma>\frac12\) should likewise follow from the Bohr--Jessen method \cite{BJ}; the agreement of the mean of \(\log|P|\) with \(J\) and the zero counts are numerical. \emph{Observed:} for \(\operatorname{Re}s\ge0.7\) the counts below height \(600\) agree with \(\frac T{2\pi}\Delta J'\); nearer to \(\tfrac12\) they fall far below it, and the density in the band \([0.6,0.7)\) doubles between \(t<500\) and \(500\sigma_P\)), Lemma 2.2, Proposition 3.1, Corollary 3.2, Lemma 3.3, Corollary 3.4, Proposition 5.1, the identity \(P=0\iff L=A\) \\ certified computation & the four counts of Section 4 (ball arithmetic) \\ numerical (floating point) & the Jessen values and predictions, the bin counts and locations of the zeros, everything in Section 5 \\ heuristic & the edge rate \(\log(J+\sigma\log2)\sim-I(\sigma)\) and the growth of \(I\); the finite-height explanation near \(\operatorname{Re}s=\frac12\) \\ \end{longtable} Three questions remain open. \begin{itemize} \tightlist \item \textbf{The rate at the edge.} Is \(\log(J(\sigma)+\sigma\log2)\sim-I(\sigma)\) as \(\sigma\to\sigma_P^-\), and is \(I(\sigma)\asymp(\sigma_P-\sigma)^{-1/(\sigma_P-1)}\) up to logarithmic factors? (Decay faster than any power is Corollary 3.4.) \item \textbf{Near the critical line.} Does the Borchsenius--Jessen method give the density of zeros of \(P\) in \(\frac12<\operatorname{Re}s<1\), and is there a finite-height model that predicts the counts for \(\tfrac12<\operatorname{Re}s<0.7\)? What is the precise law of the attraction between zeros of \(P\) and zeros of \(\zeta\)? \item \textbf{Values of \(\zeta\).} The zeros of \(P\) are solutions of \(\zeta(s)=e^{A(s)}\) (Section 5). How are they distributed relative to the \(1\)-points and \(a\)-points of \(\zeta\)? This question is independent of RH. \end{itemize} Nothing in this part bears on the truth of RH beyond the reformulation in Proposition 5.1(a), which is classical. We do not prove the Riemann hypothesis. \appendix \section{Computations} All computations are reproduced by the ancillary scripts in \texttt{anc/} (see \texttt{anc/README.txt}). \textbf{Certified (ball arithmetic, Arb via python-flint).} \texttt{certified/certify\_counts.py} evaluates \(P\) by the formula of Section 4 with the stated truncation bound, bounds \(|P'|\) by \(-\zeta'/\zeta(s_0)\) on each side, walks along the sides with steps \(h_j\le0.45\,l_j/B\), and checks at each step that the arguments are resolved. To prevent the radius from growing with the number of steps (about \(10^5\) per side), the increments are grouped into chunks of length about \(25\): for each chunk the true change of argument lies both in the sum \(S\) of the increment balls and in \(R+2\pi\mathbb Z\), where \(R=\operatorname{Arg}(P(\text{end})/P(\text{start}))\); when \(\operatorname{rad}S+\operatorname{rad}R<1\) the integer is unique and the chunk contributes \(R+2\pi k\). If \(R\) is not resolved (the ratio is near the negative real axis), or if \(\operatorname{rad}S\) exceeds \(0.5\), the chunk is closed and contributes \(S\) itself, which is also rigorous. The script records the minimum of \(|P|\) on each side and the number of steps (\texttt{run\_left.log}, \texttt{run\_right.log}); the windings are in \texttt{certify\_counts.out}. Run time: a few minutes per side. \textbf{Floating point, not certified.} \(\sigma_P\) (Brent's method on the Möbius-inverted real values, checked with a direct sum over the primes below \(2\cdot10^6\)); the Jessen function and its derivatives (\texttt{jessen.py}, \texttt{density.py}); the edge rate and the extrapolations (\texttt{edge.py}, \texttt{fig\_edge.py}); the location of the zeros (\texttt{locate.py}, \texttt{locate\_edge.py}); and all of Section 5 (\texttt{strip.py}, \texttt{count\_strip.py}, \texttt{locate\_strip.py}, \texttt{vavg.py}, \texttt{trunc.py}). \begin{thebibliography}{BCS} \bibitem[BCS]{BCS} I. 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