\documentclass[11pt]{amsart} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs} \usepackage{array} \usepackage{graphicx} \usepackage{xcolor} \usepackage[margin=1in]{geometry} \usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\re}{\operatorname{Re}} \newcommand{\im}{\operatorname{Im}} \newcommand{\Ei}{\operatorname{Ei}} \newcommand{\Ci}{\operatorname{Ci}} \newcommand{\Ein}{\operatorname{Ein}} \newcommand{\Off}{\mathcal O_{\mathrm{off}}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{doi:#1}} \title[Cross-energy of Dirichlet partial sums, Part 18]{Cross-Energy of Dirichlet Partial Sums.\\ Part 18: The Euler--Caceres bound} \author{Pedro Caceres} \address{Universidad Europea de Valencia, Valencia, Spain} \email{pedrojesus.caceres@universidadeuropea.es} \subjclass[2020]{Primary 11M26; Secondary 11M06, 30H15} \keywords{Riemann hypothesis, simple zeros, Jensen formula, Poisson--Jensen formula, explicit formula, Euler product, mollifier, Davenport--Heilbronn function, pair correlation, Weil positivity, Beurling generalized primes} \date{November 2026; version 2. Part 18 of the book-series \emph{Cross-Energy of Dirichlet Partial Sums} (forty-seven parts, Parts 0--46; block IV: Weil positivity). Part 0 \cite{CE0} is the guide. Renumbered 2 Nov 2026 (formerly Part 11). In book version~1 (2 November 2026) this was Part~17; the present numbering is that of book version~2 (10 November 2026).} \begin{document} \begin{abstract} Gonek, Hughes and Keating factored $\zeta\approx P_XZ_X$, an Euler product times a product over the zeros. We use an exact form $\zeta=E^*_XZ_X$, $E^*_X$ a smoothed Euler factor, and for $\re s_0>\frac12$ study the Euler--Caceres bound $B_X(s_0)=\frac12\log\int|Z_X(\frac12+it)|^2P_{s_0}(t)\,dt-\log|Z_X(s_0)|$, $P_{s_0}$ the Poisson kernel. It is the off-line Blaschke term of the Poisson--Jensen formula at $s_0$, independent of $X$, plus a positive Jensen gap. Under RH, $(\log X)B_X(s_0)\to\sum_\gamma\kappa_{m(\gamma)}P_{s_0}(\gamma)$, with universal constants $\kappa_m$ for multiplicity $m$ ($\kappa_1=1.8279\ldots$). Hence, for any single $s_0$, RH holds if and only if $B_X(s_0)=O(1/\log X)$, and RH with simple zeros if and only if $(\log X)B_X(s_0)$ tends to an explicit number ($0.040186\ldots$ at $s_0=2$). If $\Theta=\sup\re\rho>\frac12$, Landau's theorem in the variable $\log X$ gives $B_X(s_0)=\Omega_+(X^{\Theta-\frac12-\varepsilon})$ for $\re s_0>1$, attained or not, so RH holds iff $B_X(s_0)\ll X^\varepsilon$ at one point. Unconditionally, $B_X(s_0)=O(\sqrt X/\log X)$ for $\re s_0>1$, and the off-line term at $s_0=2$ is below $2.3\cdot10^{-12}$. Under RH we identify the second-order term up to an interaction of the wells (known only to be $o(\log X)$), related heuristically to Montgomery's pair correlation; it is governed by the gaps between zeros, and under a small-gap hypothesis the law holds with error $O(\log\log X/\log X)$. We give a function-field model where the bound holds, and controls (a Beurling system with the zeros of $\zeta$ and a zero at $0.6$, the Davenport--Heilbronn function) suggesting that a proof must couple the Euler product with the functional equation. Numerically the bound follows the law, stays below the floor $0.0386$ of Dirichlet-polynomial mollifiers at the centre $s_0=1$, and grows for the Davenport--Heilbronn function and for a planted off-line zero. We do not prove RH. \end{abstract} \maketitle \section{Introduction} \subsection{The Jensen moment and its mollifiers} Let $F(s)=(s-1)\zeta(s)/s$. Balazard, Saias and Yor [BSY] proved that \begin{equation}\label{eq:bsy} \int_{-\infty}^{\infty}\log|\zeta(\tfrac12+it)|\,\frac{dt}{2\pi(\frac14+t^2)}=\sum_{\re\rho>\frac12}\log\Big|\frac{\rho}{1-\rho}\Big|, \end{equation} the Poisson--Jensen formula for $F$ in the half-plane $\re s>\frac12$ at the point $s=1$. RH holds if and only if the left side vanishes. Jensen's inequality turns any analytic $M$ into an upper bound for the right side, $\frac12\log\int|\zeta M|^2d\mu-\log|M(1)|$. For Dirichlet polynomials $M$ this bound cannot fall below $\frac12\log(1+c_0)=0.0386$, $c_0=\sum_{k\ge1}(1-k(k+1)\log^2(1+\frac1k))$ (Proposition~\ref{prop:floor} in the appendix): the normalization at the pole forces $M$ to approximate a step function. We replace $M$ by the reciprocal of the prime factor of an exact form of the hybrid Euler--Hadamard product of Gonek, Hughes and Keating [GHK], which carries the pole itself; the barrier disappears (Table~\ref{tab:barrier}), and the resulting bound decays like $1/\log X$. This paper is about that bound. We determine its exact asymptotic behaviour under RH, show that it measures the multiplicities of the zeros, and turn it into criteria for RH and for RH together with the simplicity of the zeros. \subsection{Setting} Fix $g_0\in C_c^\infty(\mathbb R)$ with $g_0\ge0$, $\int g_0=1$, $\operatorname{supp}g_0\subset[-a,a]$. For $X>e^{a+1}$ put $L=\log X$, $g_X(u)=g_0(u-L)$, $\psi_X(v)=\int_v^\infty g_X$, and \begin{equation}\label{eq:Estar} \log E^*_X(s)=\sum_{n\ge2}\frac{\Lambda(n)\,\psi_X(\log n)}{n^s\log n}+K_X(s-1)-\sum_{k\ge1}K_X(s+2k),\qquad K_X(w)=\int g_X(u)E_1(wu)\,du . \end{equation} Here $E_1(z)=\int_z^\infty e^{-\tau}\tau^{-1}d\tau$ is the exponential integral and, below, $\Ci(v)=-\int_v^\infty\tau^{-1}\cos\tau\,d\tau$ is the cosine integral. The prime sum is finite (it stops at $e^{a}X$). By the explicit formula (Lemma~\ref{lem:factor}), $\zeta=E^*_XZ_X$ with \[ Z_X(s)=\prod_\rho e^{-K_X(s-\rho)}, \] a product over the nontrivial zeros, each factor entire with a simple zero at $\rho$. $E^*_X$ has a simple pole at $s=1$ and the trivial zeros $-2k$, and no other zeros or poles; in particular it has no zeros in $\re s>-2$. \emph{Relation to the hybrid Euler--Hadamard product.} This factorization is a variant of the hybrid Euler--Hadamard product of Gonek, Hughes and Keating [GHK, Thm.~1] (see also [BGM]). For $\sigma>0$, $|t|\ge2$, they prove $\zeta(s)=P_X(s)Z_X(s)\big(1+O(X^{K+2}(|s|\log X)^{-K})+O(X^{-\sigma}\log X)\big)$ with $P_X(s)=\exp\sum_{n\le X}\Lambda(n)n^{-s}/\log n$ and $Z_X(s)=\exp\big(-\sum_\rho U((s-\rho)\log X)\big)$, $U(z)=\int u(x)E_1(z\log x)\,dx$, $u$ a smooth weight on $[e^{1-1/X},e]$; and they observe that the second error term disappears if $P_X$ is smoothed. Our $Z_X$ is their zero factor with the weight $u$ replaced by $g_X$, which has fixed width in $\log x$ around $\log X$; our $E^*_X$ is the corresponding smoothed prime factor, completed by the pole term $K_X(s-1)$ and the trivial zeros so that the identity is exact for every $s$ (Lemma~\ref{lem:factor}). The difference matters here: we fix $s$ and let $X\to\infty$, a limit in which the error term of [GHK] does not decrease, while the moments of $Z_X$ studied in [GHK] are averages over $t\in[T,2T]$ with $X\ll(\log T)^{2-\varepsilon}$. The profile $\re E_1(ix)=-\Ci(|x|)$ of $Z_X$ on the critical line under RH (Lemma~\ref{lem:profile}) already appears in their moment computation. The name \emph{Euler--Caceres bound} refers to the quantity below, not to the factorization. \emph{The bound.} For $s_0=\sigma_0+it_0$, $\sigma_0>\frac12$, let \[ P_{s_0}(t)=\frac{\sigma_0-\frac12}{\pi\big((\sigma_0-\frac12)^2+(t-t_0)^2\big)} \] be the Poisson kernel of the half-plane $\re s>\frac12$ at $s_0$, a probability density on $\mathbb R$, and let $\phi_X(t)=\log|Z_X(\frac12+it)|$. \begin{center} \emph{The Euler--Caceres bound:}\quad $B_X(s_0)=\tfrac12\log\displaystyle\int e^{2\phi_X(t)}P_{s_0}(t)\,dt-\log|Z_X(s_0)|$. \end{center} \subsection{Results} Let $b_\rho(s)=(s-\rho)/(s-1+\bar\rho)$ and let \[ \Off(s_0)=\sum_{\re\rho>\frac12}\log\frac1{|b_\rho(s_0)|}\ \ge0 \] be the off-line contribution; $\Off(1)$ is the right side of \eqref{eq:bsy}. \begin{theorem}[decomposition]\label{thm:decomp} For every $s_0$ with $\re s_0>\frac12$ and $\zeta(s_0)\ne0$, and every $X>e^{a+1}$, \[ B_X(s_0)=\Off(s_0)+J_X(s_0),\qquad J_X(s_0)=\tfrac12\log\int e^{2\phi_X}P_{s_0}-\int\phi_X\,P_{s_0}\ >0 . \] \end{theorem} The first term does not depend on $X$; the second is the Jensen gap of $\phi_X$ under the probability measure $P_{s_0}(t)\,dt$, and it is strictly positive because $\phi_X\to-\infty$ at the ordinates of the zeros on the line. In particular $B_X(s_0)>\Off(s_0)$ for every $X$. For the multiplicity $m$ of a zero define \[ \kappa_m=\int_0^\infty\big(e^{2m\Ci(v)}-1-2m\Ci(v)\big)\,dv,\qquad \Ci(v)=-\int_v^\infty\frac{\cos\tau}{\tau}\,d\tau . \] The integrand is $O(m^2/v^2)$ at infinity and $O(m\log(1/v))$ at $0$, so $\kappa_m$ is finite; Table~\ref{tab:kappa} gives the first values. Since $\int_0^\infty\Ci=0$, $\kappa_1=\int_0^\infty(e^{2\Ci(v)}-1)\,dv$ as an improper integral. \begin{theorem}[the law of the bound]\label{thm:law} Assume RH. For every $s_0$ with $\re s_0>\frac12$, \[ \lim_{X\to\infty}(\log X)\,B_X(s_0)=\Omega(s_0):=\sum_{\gamma}\kappa_{m(\gamma)}P_{s_0}(\gamma), \] where $\gamma$ runs over the distinct ordinates of the zeros and $m(\gamma)$ is the multiplicity of $\frac12+i\gamma$; the series converges. \end{theorem} \begin{table}[h]\centering\small \begin{tabular}{ccccc} \toprule $m$ & $1$ & $2$ & $3$ & $4$\\ \midrule $\kappa_m$ & $1.82792$ & $7.34579$ & $20.036$ & $48.806$\\ $\kappa_m/m$ & $1.82792$ & $3.67289$ & $6.679$ & $12.202$\\ \bottomrule \end{tabular} \caption{The constants $\kappa_m$.}\label{tab:kappa} \end{table} \begin{corollary}[criteria]\label{cor:criteria} Let $\re s_0>\frac12$ and $\zeta(s_0)\ne0$. Unconditionally: \begin{enumerate} \item[(a)] RH $\iff B_X(s_0)\to0\iff B_X(s_0)=O(1/\log X)$; \item[(b)] RH and all zeros simple $\iff\displaystyle\lim_{X\to\infty}(\log X)\,B_X(s_0)=\frac{\kappa_1}{\pi}\,\re\frac{\xi'}{\xi}(s_0)$; \item[(c)] if RH holds, $\lim(\log X)B_X(s_0)\ge\frac{\kappa_1}{\pi}\re\frac{\xi'}{\xi}(s_0)$, and the excess is $\sum_\gamma\big(\kappa_{m(\gamma)}-m(\gamma)\kappa_1\big)P_{s_0}(\gamma)$, which vanishes only if all zeros are simple. \end{enumerate} \end{corollary} Here $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)$, and $\re(\xi'/\xi)(s_0)=\sum_\rho\re\frac1{s_0-\rho}$ is an explicit number; at $s_0=2$, $(\kappa_1/\pi)\re(\xi'/\xi)(2)=0.040186\ldots$. Both RH and the simple-zeros conjecture are thus encoded in a single limit at a single point, whose value is known in advance. \begin{remark}[a zero detector]\label{rem:detector} Assume RH. As a function of $s_0$, $\Omega$ is the Poisson integral of the positive measure $\sum_\gamma\kappa_{m(\gamma)}\delta_\gamma$ on the critical line; it is harmonic in $\re s>\frac12$, and its boundary measure recovers every ordinate together with its multiplicity (through $\kappa_m$, which is strictly increasing). The Euler--Caceres bound, rescaled by $\log X$, converges to this harmonic function. \end{remark} The rate of convergence in Theorem~\ref{thm:law} is governed by the shape of the weight. Let $W,W'$ be independent random variables with density $g_0$, so that $E|W-W'|$ (the mean difference of $g_0$) measures how sharply $E^*_X$ cuts the primes at $X$, and let $\hat g_0(y)=\int g_0(w)e^{iyw}\,dw$ (we use this convention throughout, except in Theorem~\ref{thm:isometry}, where $\widehat F_X$ is the boundary value of a Laplace transform, with $e^{-itu}$; for the even functions of Sections~\ref{sec:second} and~\ref{sec:pair} the sign is immaterial). For a single ordinate $\gamma$ of multiplicity $m$, let \[ H_{X,\gamma}=\int h\big(m\,c_X(t-\gamma)\big)P_{s_0}(t)\,dt,\qquad h(x)=e^{2x}-1-2x, \] be the cost of its well alone, where $c_X$ is the well profile \eqref{eq:cX}, and let \[ \mathcal I_X(s_0)=\tfrac12(\log X)^2\Big(\int h(\phi_X)P_{s_0}-\sum_\gamma H_{X,\gamma}\Big) \] be the interaction of the wells. \begin{theorem}[the second-order term]\label{thm:second} Assume RH and that $g_0$ is even. For every $s_0$ with $\re s_0>\frac12$, \[ (\log X)\,B_X(s_0)=\Omega(s_0)+\frac{\Omega_1(s_0)-\Omega(s_0)^2+\mathcal I_X(s_0)}{\log X}+o\Big(\frac1{\log X}\Big), \] \[ \Omega_1(s_0)=\frac12\sum_\gamma m(\gamma)^2\Big[-\pi E|W-W'|\,P_{s_0}(\gamma)+\mathrm{PV}\!\!\int\frac{P_{s_0}(\gamma+y)-P_{s_0}(\gamma)}{y^2}\,|\hat g_0(y)|^2\,dy\Big], \] the series converging absolutely, and $\mathcal I_X(s_0)=o(\log X)$. \end{theorem} Since we prove only $\mathcal I_X=o(\log X)$, the term $\mathcal I_X/\log X$ is only known to be $o(1)$, and it may be larger than the explicit part $(\Omega_1-\Omega^2)/\log X$. Theorem~\ref{thm:second} is therefore an exact decomposition of the error, in which the single wells are computed to second order (Proposition~\ref{prop:onewell}) and the interaction is isolated; it becomes a second-order asymptotic exactly when $\mathcal I_X=O(1)$. Section~\ref{sec:gaps} shows that $\mathcal I_X$ is governed by the gaps between the zeros: it is $O(\log\log X)$ under a hypothesis on small gaps (Theorem~\ref{thm:rate}), and numerically $O(1)$. The leading part of $\Omega_1$ is $-\frac\pi2E|W-W'|\sum m^2P_{s_0}(\gamma)$: a smooth cut-off of the primes costs a deficit proportional to its width, the price of $E^*_X$ being zero-free. The interaction $\mathcal I_X$ collects the overlaps of neighbouring wells; its phases depend on the zeros only through their differences $\gamma-\gamma'$, and it is damped, roughly, by $\hat g_0(\gamma-\gamma')$ (Section~\ref{sec:second}); at large height its quadratic part is related, by a heuristic derivation, to the pair correlation of the zeros (Section~\ref{sec:pair}). A wider weight reduces the interaction and makes $\Omega_1$ more negative; numerically, at $s_0=2$, $\mathcal I_X$ is a fraction of $|\Omega_1|$ that decreases with the width of the weight, from about $35\%$ to about $4\%$ in Table~\ref{tab:second}. \begin{theorem}[an attained off-line supremum makes the bound explode]\label{thm:explode} Suppose $\Theta=\sup\re\rho>\frac12$ is attained. Then for every $s_0$ with $\re s_0>\Theta$ there are $r\in(0,\Theta-\frac12)$, $C$, $X_0$ with \[ B_X(s_0)\ \ge\ \tfrac12\Ei\big(r(\log X-a)\big)-C\log\log X\ \ge\ \frac{X^r}{3re^{ra}\log X}\qquad(X\ge X_0). \] \end{theorem} Theorem~\ref{thm:explode} covers only the case in which the supremum $\Theta$ is attained. Without attainment, a Laplace transform in the variable $\log X$ and Landau's theorem on Laplace integrals with an eventually one-signed integrand give the growth along a sequence (Section~\ref{sec:landau}). \begin{theorem}[the bound explodes along a sequence]\label{thm:landau} Suppose $\Theta=\sup\re\rho>\frac12$, attained or not. Then for every $s_0$ with $\re s_0>1$ and every $\varepsilon>0$, \[ \limsup_{X\to\infty}X^{-(\Theta-\frac12-\varepsilon)}B_X(s_0)=+\infty;\qquad\text{in particular}\quad\limsup_{X\to\infty}\frac{\log B_X(s_0)}{\log X}\ \ge\ \Theta-\tfrac12 . \] \end{theorem} \begin{corollary}[the bound at one point]\label{cor:landau} Let $\re s_0>1$. Then RH $\iff B_X(s_0)\ll_\varepsilon X^\varepsilon$ for every $\varepsilon>0$ $\iff\limsup_{X\to\infty}\log B_X(s_0)/\log X\le0$. \end{corollary} \begin{corollary}[one mean square]\label{cor:energy} Let $\mathcal E(X)=\int|\zeta/E^*_X(\frac12+it)|^2\,dt/(1+t^2)$, finite for every $X$. If RH holds, $\mathcal E(X)\to\pi$. Unconditionally, RH $\iff\limsup_X\mathcal E(X)<\infty$. If the supremum $\Theta$ of the real parts of the zeros is attained (this is automatic under RH, where $\Theta=\frac12$), then also RH $\iff\liminf_X\mathcal E(X)<\infty$. \end{corollary} \subsection{What is new, and what is not} The factorization is the hybrid Euler--Hadamard product of [GHK] in an exact form, and the profile of $Z_X$ on the line under RH is in [GHK]. Theorem~\ref{thm:decomp} and Corollary~\ref{cor:criteria}(a) are Jensen-type reformulations: the off-line term does not depend on $X$. The new content is Theorem~\ref{thm:law}: under RH the gap $J_X$ is asymptotically $\Omega(s_0)/\log X$ with the universal constants $\kappa_m$, which come from the profile $e^{m\Ci(|v|)}$, $v=(t-\gamma)\log X$, of $|Z_X|$ at a zero of multiplicity $m$ (Lemma~\ref{lem:profile}). This gives the rate in (a) and the simple-zero criterion (b), and Theorems~\ref{thm:explode} and~\ref{thm:landau} give the growth on the other side. Theorem~\ref{thm:landau} uses no attainment hypothesis: it is Landau's argument for $\psi(x)-x=\Omega_\pm(x^{\Theta-\varepsilon})$ [I, Ch.~V], applied to $\log|Z_X(s)|$ as a function of $\log X$, whose Laplace transform has a logarithmic singularity at $\rho-s$ for every zero $\rho$ (Lemma~\ref{lem:laplace}); it makes the bound $B_X(s_0)\ll X^\varepsilon$ at one point an equivalent of RH (Corollary~\ref{cor:landau}). Theorem~\ref{thm:second} identifies the next term up to the interaction of the wells, which we bound only by $o(\log X)$; Section~\ref{sec:gaps} shows that the interaction is a sum over pairs of zeros of a two-well function, whose excess at coincidence is $\kappa_2-2\kappa_1$, and that the rate in Theorem~\ref{thm:law} is controlled by the gaps between the zeros (Theorem~\ref{thm:rate}). The closest known analogue of Theorem~\ref{thm:law} is the Nyman--Beurling approximation problem: Burnol [Bu] proved $\liminf_N(\log N)d_N^2\ge\sum m(\rho)^2/|\rho|^2$, the sum over the distinct zeros $\rho$ on the critical line, for the Nyman--Beurling distance $d_N$, and Bettin, Conrey and Farmer [BCF] showed, under RH and a bound for $\sum_{|\gamma|1$ not a prime power, the truncated explicit formula with the term $\frac12\log(1-x^{-2})$ of $\psi(x)$ kept exactly gives $\sum_{n\le x}\Lambda(n)n^{-s}/\log n+E_1((s-1)\log x)=\log\zeta(s)+\sum_\rho E_1((s-\rho)\log x)+\sum_kE_1((s+2k)\log x)$ for $t\ne0$ and $s$ off the horizontal cuts (this is the formula for $\sum_{n\le x}\Lambda(n)n^{-s}$ \cite[(14.20)]{T} integrated in $s$, as in [GHK, Thm.~1]); multiply by $g_X(\log x)$ and integrate in $x$. Equivalently, and without any conditionally convergent sum, the $s$-derivative of the identity follows by Mellin inversion of $\sum_n\Lambda(n)\psi_X(\log n)n^{-s}$ against $-\zeta'/\zeta$ with the smooth weight $\psi_X$, and one integrates in $s$ from $+\infty$: the transform of $\psi_X$ decays faster than any power on vertical lines, the contour is moved to the left across the pole, the zeros and the trivial zeros, and the sum over the zeros converges absolutely. The symmetric partial sums over $|\gamma|\frac12$, and $Z_X=F/P$ lies in the Smirnov class $N^+$. \end{lemma} \begin{proof} $\log|P(s)|=-\log|s|+\big(\log|s-1|+\re K_X(s-1)\big)+\re D(s)-\sum_k\re K_X(s+2k)$, $D$ the finite prime sum, $|\re D|\le\sum\Lambda(n)n^{-1/2}/\log n$. Since $\log|z|+\re E_1(z)=-\gamma_E+\re\Ein(z)$, the bracket is continuous, and bounded on $|s-1|\le R$. For $|s-1|\ge R$, $z=(s-1)u$ has either $\re z\ge0$ and $d(z)=|z|$, or $|\im z|\ge u\sqrt{R^2-1}$, so $|E_1(z)|\le e^{u/2}/d(z)$ by \eqref{eq:phi} and $K_X(s-1)$ is bounded, by $e^{(L+a)/2}/((L-a)\sqrt{R^2-1})$, as is $\log|(s-1)/s|$; and $|E_1(z)|\le e^{-\re z}/|z|$ for $\re z>0$ bounds the trivial-zero terms. A zero-free analytic function whose $\log|\cdot|$ is bounded, harmonic, and continuous up to the line is the Poisson integral of its boundary values, i.e.\ outer. By [BSY] $F$ is a Blaschke product times an outer function (no singular inner factor, also at $\infty$, since $F(\sigma)\to1$); hence $Z_X\in N^+$. The bounds depend on $X$: $|\re D|\le\mathcal M(e^aX)\sim2\sqrt{e^aX}/\log X$ in the notation of Theorem~\ref{thm:uncond}, and $K_X(s-1)$ is of the same order on the line; this is the size used in Theorem~\ref{thm:uncond}(b). \end{proof} \begin{lemma}[the kernel]\label{lem:kernel} Let $w=-x+iy$, $|x|\le x_{\max}$. For every integer $A\ge0$ there is $C$ depending on $A$, $x_{\max}$, $g_0$ only, such that: \begin{enumerate} \item[(L1)] if $\re w\ge\eta>0$, then $|K_X(w)|\le C_\eta X^{-\re w}(1+|y|)^{-A-1}/L$; \item[(L2)] if $\re w\ge0$, $w\ne0$, then $|K_X(w)|\le C(1+|\log(|w|L)|)$, and if moreover $|y|\ge1$, $|K_X(w)|\le C|y|^{-A-1}/L$; \item[(L3)] if $x>0$, $y\ne0$, then $|K_X(w)|\le e^{x(L+a)}/(|y|(L-a))$, and if moreover $|y|\ge1$, $|K_X(w)|\le Ce^{xa}X^{x}|y|^{-A-1}/L$; \item[(L4)] for $x>0$, $-\re K_X(-x)=\int g_X(u)\Ei(xu)\,du\ge\Ei\big(x(L-a)\big)$. \end{enumerate} \end{lemma} \begin{proof} $K_X(w)=e^{-wL}\int g_0(v)e^{-wv}\varphi(w(v+L))\,dv$; put $u=v+L\in[L-a,L+a]$, and note $d(wu)=u\,d(w)$ with $d(w)=|w|$ if $x\le0$, $d(w)=|y|$ if $x>0$. Without integration by parts, \eqref{eq:phi} gives $|K_X(w)|\le e^{x(L+a)}/(d(w)(L-a))$, the first part of (L3). Integrating by parts $A$ times against $e^{-wv}$, $K_X(w)=w^{-A}e^{-wL}\int e^{-wv}\partial_v^A[g_0(v)\varphi(w(v+L))]\,dv$, and the $m$-th derivative of $\varphi(w(v+L))$ is at most $m!(|w|/d(w))^m/(u\,d(w))$, with $|w|/d(w)\le1+x_{\max}$ in all cases used. Hence $|K_X(w)|\le Ce^{x(L+a)}|w|^{-A}/(d(w)(L-a))$, which gives the second parts of (L2), (L3) and (L1). For the first part of (L2), $|E_1(wu)|\le|\varphi(wu)|\le1$ if $|wu|\ge1$, and $|E_1(wu)|\le\gamma_E+|\log|wu||+\pi+e$ if $|wu|<1$. For (L4), $E_1(-\xi\pm i0)=-\Ei(\xi)\mp i\pi$ and $\Ei$ increases. \end{proof} By \eqref{eq:cX} and (L2), for all real $y\ne0$, \begin{equation}\label{eq:cbounds} \begin{gathered} c_X(y)\le\Ci(\tfrac\pi2)=0.4720,\qquad |c_X(y)|\le C\big(1+|\log(|y|L)|\big),\\ |c_X(y)|\le\frac{2}{|y|(L-a)},\qquad |c_X(y)|\le\frac{C}{|y|^3L}\quad(|y|\ge1), \end{gathered} \end{equation} the first because $g_X\ge0$, $\int g_X=1$, the third from $|\Ci(x)|\le2/x$ (indeed $\le1/x$). \section{The decomposition}\label{sec:decomp} \begin{proof}[Proof of Theorem~\ref{thm:decomp}] The hypotheses of the Poisson--Jensen formula on $\re s>\frac12$ are as follows. $F=(s-1)\zeta(s)/s$ is analytic on $\re s\ge\frac12$ and of bounded type there: $F(s)/(s+1)$ is bounded on $\re s>\frac12$ by the convexity bound. Its zeros satisfy the Blaschke condition, $\log|F(\frac12+it)|$ is integrable against $P_{s_0}$, and $F$ has no singular inner factor, neither on the line nor at infinity [BSY]. The Poisson--Jensen formula at any centre $s_0$ then follows from this canonical factorization. $P=(s-1)E^*_X/s$ is outer, with $\log|P|$ bounded and continuous up to the line (Lemma~\ref{lem:outer}). Finally $F(s_0)\ne0$ because $\zeta(s_0)\ne0$ (and $F(1)=1$). So $\log|F|$ and $\log|P|$ satisfy the Poisson--Jensen formula at $s_0$ without singular part, $\log|P|$ without zeros; subtracting, \[ \int\phi_X\,P_{s_0}=\log|Z_X(s_0)|+\Off(s_0). \] Substituting $\log|Z_X(s_0)|=\int\phi_XP_{s_0}-\Off(s_0)$ in the definition of $B_X$ gives $B_X=\Off+J_X$, and $J_X\ge0$ is Jensen's inequality, with equality only if $\phi_X$ is constant almost everywhere. Since $\phi_X(t)\to-\infty$ as $t\to\gamma_1=14.1347\ldots$ (the first zero lies on the line), $J_X>0$. \end{proof} Since $|F|=|\zeta|$ on the critical line, the Poisson--Jensen formula for $F$ used in the proof reads, at a general centre, \begin{equation}\label{eq:pj} \int\log|\zeta(\tfrac12+it)|\,P_{s_0}(t)\,dt=\log\Big|\frac{(s_0-1)\zeta(s_0)}{s_0}\Big|+\Off(s_0), \end{equation} which is \eqref{eq:bsy} for $s_0=1$. Moving the centre to $s_0=\sigma_0+it_0$ concentrates the sensitivity near the height $t_0$: an off-line zero at height close to $t_0$ contributes of order $(\beta-\frac12)(\sigma_0-\frac12)/|s_0-\rho|^2$ instead of $(\beta-\frac12)/|\rho|^2$. Table~\ref{tab:identity} checks \eqref{eq:pj} numerically (script \texttt{mollifier\_tables.py}), together with its analogue for the Davenport--Heilbronn function $f$ (Section~\ref{sec:numerics}), which has no pole and whose off-line term we compute from its eight off-line zeros below height $400$; we do not prove the Poisson--Jensen formula without singular part for $f$, and the agreement in the last row is numerical. \begin{table}[h]\centering\small \begin{tabular}{llccc} \toprule function & $s_0$ & line mean & interior (and pole) & excess\\ \midrule $\zeta$ & $2+20i$ & $-0.03463$ & $-0.03408$ & $-0.0006$\\ $\zeta$ & $1.5+100i$ & $0.27150$ & $0.27144$ & $+0.0001$\\ $\zeta$ & $1+85.7i$ & $0.13177$ & $0.13158$ & $+0.0002$\\ $f$ & $1+85.7i$ & $-0.05385$ & $-1.49465$ & $+1.4408$ (off-line zeros: $+1.4407$)\\ \bottomrule \end{tabular} \caption{The identity \eqref{eq:pj} at three centres, with the window $|t-t_0|<4000$ and step $0.002$; the remaining kernel mass is below $3\cdot10^{-4}$, and for $\zeta$ the excess, which should be $\Off(s_0)<10^{-11}$ (Theorem~\ref{thm:uncond}), is of the order of the truncation error. For $f$ the interior term is $\log|f(s_0)|$, and the excess agrees with the off-line Blaschke term of its zeros.}\label{tab:identity} \end{table} Under RH, $\Off=0$, $Z_X$ is outer, and \begin{equation}\label{eq:ell} \ell_X:=\int\phi_X\,P_{s_0}=\log|Z_X(s_0)|,\qquad |\ell_X|\le C_{s_0}\frac{X^{-(\sigma_0-\frac12)}}{L} \end{equation} by (L1), since $\re(s_0-\rho)=\sigma_0-\frac12$ for every zero and $\sum_\rho(1+|t_0-\gamma|)^{-3}<\infty$. Writing $h(x)=e^{2x}-1-2x\ge0$ and $H_X=\int h(\phi_X)P_{s_0}$, we get $\int e^{2\phi_X}P_{s_0}=1+H_X+2\ell_X$ and \begin{equation}\label{eq:BH} B_X(s_0)=\tfrac12\log(1+H_X+2\ell_X)-\ell_X . \end{equation} So under RH everything is in $H_X$, an integral of a nonnegative function. \section{Proof of the law}\label{sec:law} \begin{lemma}[profile at a zero; cf.\ {[GHK, Sect.~3]}]\label{lem:profile} Assume RH and let $\frac12+i\gamma$ be a zero of multiplicity $m$. Then for every $v\ne0$, $\phi_X(\gamma+v/\log X)\to m\Ci(|v|)$ as $X\to\infty$. \end{lemma} \begin{proof} By Lemma~\ref{lem:factor} and \eqref{eq:cX}, under RH $\phi_X(t)=\sum_\rho c_X(t-\gamma_\rho)$ for $t$ not an ordinate. The $m$ terms of $\gamma$ give $m\,c_X(v/L)=m\int g_0(w)\Ci(|v|(1+w/L))\,dw\to m\Ci(|v|)$. The other ordinates are at distance $\ge\delta>0$ from $\gamma$; by \eqref{eq:cbounds} those within distance $1$ contribute $O(1/L)$ and the rest $O(\log(|\gamma|+3)/L)$. \end{proof} \begin{lemma}[uniform bounds under RH]\label{lem:upper} Assume RH. (i) For every $\varepsilon>0$ there are $X_\varepsilon$, $C_\varepsilon$ with $\phi_X(t)\le\varepsilon\log(|t|+3)+C_\varepsilon$ for all $t$ and $X\ge X_\varepsilon$. (ii) For every interval $I$ of length $1$ at distance $\le n$ from $0$, $\int_I\phi_X^2\le C\log^2(n+3)/L$. \end{lemma} \begin{proof} (i) Under RH, $S(T)\ll\log T/\log\log T$ [T, Thm.~14.13], so an interval of length $h\le1$ centred at $t$ contains at most $\frac h{2\pi}\log(|t|+3)+C\log(|t|+3)/\log\log(|t|+3)$ ordinates. By \eqref{eq:cbounds}, the ordinates with $|t-\gamma|<1/L$ contribute at most $0.472\big(\frac1{\pi L}+\frac C{\log\log(|t|+3)}\big)\log(|t|+3)$, those on the dyadic shells $2^k/L\le|t-\gamma|<\min(1,2^{k+1}/L)$ contribute $\ll\big(\frac{\log L}L+\frac1{\log\log(|t|+3)}\big)\log(|t|+3)$, and those with $|t-\gamma|\ge1$ contribute $\ll\log(|t|+3)/L$. Treating $|t|\le T_\varepsilon$ separately gives (i). (ii) For $t\in I$ split $\phi_X=\phi'+\phi''$, with $\phi'$ the sum over the $N\ll\log(n+3)$ ordinates (with multiplicity) within distance $1$ of $I$. By \eqref{eq:cbounds}, $|\phi''|\le C\log(n+3)/L$, and $\int_{\mathbb R}c_X^2\le C/L$ (split at $|y|=1/L$). By Cauchy--Schwarz, $\int_I\phi'^2\le N\sum\int_{\mathbb R}c_X^2\le CN^2/L$. \end{proof} Numerically, with the Gaussian weight of Table~\ref{tab:barrier} and step $0.01$, the maximum of $\phi_X(t)/\log(t+2)$ over $0T}h(\phi_X)P_{s_0}\le C''\sum_{n\ge T}\frac{\log^2(n+3)}{(n+1)^{3/2}}=:\eta(T)\xrightarrow[T\to\infty]{}0, \] uniformly in $X\ge X_\varepsilon$. \emph{Bounded part.} Fix $T$ such that $\pm T$ are not ordinates. Let $\gamma_1,\dots,\gamma_k$ be the distinct ordinates in $(-T,T)$, with multiplicities $m_j$, and let $\delta\in(0,1]$ be at most the distance between any two distinct ordinates in $[-T-1,T+1]$ and the distance from $\pm T$ to the ordinates. Let $U_j=(\gamma_j-\delta/2,\gamma_j+\delta/2)$. For $t\in U_j$, write $\phi_X(t)=m_jc_X(t-\gamma_j)+r_X(t)$: every other ordinate is at distance $\ge\delta/2$ from $t$, so by \eqref{eq:cbounds} $|r_X(t)|\le C_T/L$. For $t\in(-T,T)\setminus\bigcup U_j$ the same bounds give $|\phi_X(t)|\le C_T/L$, hence $L\int h(\phi_X)P_{s_0}\le C_T^2/L\to0$ there. On $U_j$ substitute $t=\gamma_j+v/L$: \[ L\int_{U_j}h(\phi_X)P_{s_0}\,dt=\int_{|v|<\delta L/2}h\big(m_jc_X(v/L)+r_X\big)P_{s_0}(\gamma_j+v/L)\,dv . \] The integrand tends to $h(m_j\Ci(|v|))P_{s_0}(\gamma_j)$ for every $v\ne0$ (Lemma~\ref{lem:profile}). For domination we use $h(x)\le C_M(1+|x|)$ and $h(x)\le C_Mx^2$ for $x\le M$ (note $0\le h(x)<2|x|$ for $x<0$). If $|v|\le1$, $m_jc_X(v/L)+r_X\le M$ and $|m_jc_X(v/L)|\le C(1+|\log|v||)$, so the integrand is at most $C(1+|\log|v||)$. If $|v|\ge1$, $|m_jc_X(v/L)|\le3m_j/|v|$, and $h(x)\le C_Mx^2\le2C_M(9m_j^2/v^2)+2C_Mr_X^2$; the second piece is at most $2C_MC_T^2/L^2$, whose integral over $|v|<\delta L/2$ is $O(1/L)$, and we split it off. Dominated convergence applies to the rest, with the integrable majorant $C(1+|\log|v||)\mathbf 1_{|v|\le1}+18C_Mm_j^2v^{-2}\mathbf 1_{|v|>1}$ times $\max P_{s_0}$. Hence \[ \lim_{X\to\infty}L\int_{U_j}h(\phi_X)P_{s_0}=P_{s_0}(\gamma_j)\int_{\mathbb R}h(m_j\Ci(|v|))\,dv=2\kappa_{m_j}P_{s_0}(\gamma_j). \] \emph{Conclusion.} Let $\Omega_T=\sum_{|\gamma|T$ no local analysis is used: the bound $\eta(T)$ comes from Lemma~\ref{lem:upper}, which holds uniformly in $X\ge X_\varepsilon$ and uses only the counting function of the zeros under RH ($S(T)\ll\log T/\log\log T$); it bounds the total, wells and interactions together. (2) For $|t|\frac12$; in particular the harmonic function $\Omega$ of Remark~\ref{rem:detector} is the locally uniform limit of $(\log X)B_X$. The constant $C_T$ depends on the smallest gap $\delta$ between the ordinates below $T$; Section~\ref{sec:gaps} shows that this dependence is genuine. The same interactions are what Theorem~\ref{thm:second} isolates at the next order, where we can only show that they are $o(\log X)$. \section{The constants \texorpdfstring{$\kappa_m$}{kappa\_m} and the criteria}\label{sec:kappa} \begin{lemma}\label{lem:kappa} $\kappa_m/m$ is strictly increasing in $m$; in particular $\kappa_m>m\kappa_1$ for $m\ge2$. \end{lemma} \begin{proof} For fixed $x\ne0$ let $f(\mu)=(e^{2\mu x}-1-2\mu x)/\mu$, $\mu>0$. Then $\mu^2f'(\mu)=(2\mu x-1)e^{2\mu x}+1>0$, since $(u-1)e^u+1>0$ for $u\ne0$. Integrate against $dv$ with $x=\Ci(v)$, which vanishes only on a null set. \end{proof} \begin{lemma}\label{lem:W} $\sum_\rho\re\frac{1}{s_0-\rho}=\re\frac{\xi'}{\xi}(s_0)$ for $\re s_0>\frac12$ with $\zeta(s_0)\ne0$, the sum absolutely convergent and counting multiplicity. Under RH, $\re\frac1{s_0-\rho}=\pi P_{s_0}(\gamma)$, so $\sum_\rho P_{s_0}(\gamma)=\varpi(s_0)$, where $\varpi(s_0):=\frac1\pi\re\frac{\xi'}{\xi}(s_0)$. \end{lemma} \begin{proof} Hadamard's product gives $\xi'/\xi(s)=B+\sum_\rho\big(\frac1{s-\rho}+\frac1\rho\big)$ with $\re B=-\sum_\rho\re\frac1\rho$ [D, Ch.~12]. Under RH, $s_0-\rho=(\sigma_0-\frac12)+i(t_0-\gamma)$. \end{proof} \begin{proof}[Proof of Corollary~\ref{cor:criteria}] (a) If RH fails, $B_X(s_0)\ge\Off(s_0)>0$ for all $X$ by Theorem~\ref{thm:decomp}. If RH holds, $B_X(s_0)\sim\Omega(s_0)/\log X$ by Theorem~\ref{thm:law}. (c) Under RH, by Lemma~\ref{lem:W}, $\frac{\kappa_1}\pi\re\frac{\xi'}\xi(s_0)=\kappa_1\sum_\gamma m(\gamma)P_{s_0}(\gamma)$, so $\Omega(s_0)-\frac{\kappa_1}\pi\re\frac{\xi'}\xi(s_0)=\sum_\gamma(\kappa_{m(\gamma)}-m(\gamma)\kappa_1)P_{s_0}(\gamma)$, a sum of nonnegative terms, each positive when $m(\gamma)\ge2$ (Lemma~\ref{lem:kappa}; $P_{s_0}>0$). (b) If RH holds and all zeros are simple, $\Omega=\frac{\kappa_1}\pi\re\frac{\xi'}\xi(s_0)$ by (c). Conversely, if $(\log X)B_X(s_0)$ has a finite limit, then $B_X\to0$ and RH holds by (a), and equality of the limit with $\frac{\kappa_1}\pi\re\frac{\xi'}\xi(s_0)$ forces all zeros simple by (c). \end{proof} \begin{remark} Corollary~\ref{cor:criteria}(b) needs one point $s_0$, because $P_{s_0}(\gamma)>0$ for every $\gamma$: a single multiple zero anywhere raises the limit by $(\kappa_m-m\kappa_1)P_{s_0}(\gamma)$, for instance by $3.69\,P_{s_0}(\gamma)$ for a double zero. The test is quantitative, but the effect of a multiple zero at height $\gamma$ is of size $\gamma^{-2}$ at a fixed centre; moving $s_0$ to height $\gamma$ makes it of order one. \end{remark} \section{The second-order term}\label{sec:second} \begin{proposition}[one well to second order]\label{prop:onewell} Let $g_0$ be even, $s_0=\sigma_0+it_0$ with $\sigma_0>\frac12$, $\gamma\in\mathbb R$ and $m\ge1$. With $L=\log X$, \[ L\,H_{X,\gamma}=2\kappa_mP_{s_0}(\gamma)+\frac{m^2}{L}\Big[-\pi E|W-W'|P_{s_0}(\gamma)+\mathrm{PV}\!\!\int\frac{P_{s_0}(\gamma+y)-P_{s_0}(\gamma)}{y^2}|\hat g_0(y)|^2dy\Big]+R_{X,\gamma}, \] with $|R_{X,\gamma}|\le C\,m^3e^{2m}(1+|\gamma-t_0|)^{-2}\,L^{-2}\log L$, where $C$ depends on $s_0$ and $g_0$ only. \end{proposition} \begin{proof} Write $P_\gamma(y)=P_{s_0}(\gamma+y)$, $c=c_X$, and $h(x)=2x^2+h_3(x)$. Constants $C$ depend on $s_0,g_0$ only, and we use $\sup_{|y|\le1}(|P_\gamma|+|P_\gamma'|+|P_\gamma''|+|P_\gamma'''|)\le C(1+|\gamma-t_0|)^{-2}=:C\,p_\gamma$ and $\int_{|y|\ge1}P_\gamma(y)y^{-2}dy\le Cp_\gamma$. \emph{(i) The quadratic part at the centre.} Let $G_X(\xi)=\int_{u\le\xi}g_X(u)\,du$, which vanishes for $\xiL+a$, \begin{equation}\label{eq:cfourier} c(y)=-\int_0^\infty\frac{G_X(\xi)}{\xi}\cos(y\xi)\,d\xi, \end{equation} a cosine transform of the $L^2$ function $G_X(\xi)/\xi$. By Plancherel, $\int_{\mathbb R}c^2=\pi\int_0^\infty G_X(\xi)^2\xi^{-2}d\xi$. Now $G_X^2$ is the distribution function of $L+M$ with $M=\max(W,W')$, so $\int_0^\infty G_X^2\xi^{-2}d\xi=E[(L+M)^{-1}]=L^{-1}-E[M]L^{-2}+O(L^{-3})$, and $E[M]=\frac12E|W-W'|$ because $g_0$ is even. Hence \[ 2m^2L\,P_\gamma(0)\int c^2=2\pi m^2P_\gamma(0)-\frac{\pi m^2E|W-W'|}{L}P_\gamma(0)+O\Big(\frac{m^2p_\gamma}{L^2}\Big). \] \emph{(ii) The quadratic part off the centre.} Let $\vartheta(y)=P_\gamma(y)-P_\gamma(0)-P_\gamma'(0)y\mathbf 1_{|y|\le1}$; since $c$ is even, $2m^2L\int c^2(P_\gamma-P_\gamma(0))=2m^2L\int c^2\vartheta$, and $|\vartheta(y)|\le Cp_\gamma\min(y^2,1)+P_\gamma(y)$. Integrating \eqref{eq:cfourier} by parts twice, \[ yL\,c(y)=\im\big(e^{iyL}\hat g_0(y)\big)+\varrho(y),\qquad|\varrho(y)|\le\frac CL+\frac C{|y|L}\quad(|y|\ge1/L), \] the first term coming from $L\int g_0(w)\sin(y(L+w))(L+w)^{-1}dw$ and $\varrho$ from the factor $L/(L+w)$ and from $L\int G_X\xi^{-2}\sin(y\xi)\,d\xi$. Thus $(yLc)^2=\frac12|\hat g_0(y)|^2-\frac12\re\big(e^{2iyL}\hat g_0(y)^2\big)+O(L^{-1}+(|y|L)^{-1})$ for $|y|\ge1/L$, and \[ 2m^2L\int c^2\vartheta=\frac{2m^2}L\int\frac{(yLc)^2\vartheta(y)}{y^2}\,dy . \] On $|y|<1/L$, $|c|\le C(1+|\log(|y|L)|)$ by \eqref{eq:cbounds} and $|\vartheta|\le Cp_\gamma y^2$, a contribution $O(m^2p_\gamma L^{-2})$. On $|y|\ge1/L$ the term $\frac12|\hat g_0|^2$ gives $\frac{m^2}L\,\mathrm{PV}\!\int|\hat g_0|^2(P_\gamma-P_\gamma(0))y^{-2}dy+O(m^2p_\gamma L^{-2})$ (the term $P'_\gamma(0)y$ is odd); the oscillating term is $O(m^2p_\gamma L^{-2})$ after one integration by parts in $y$; and the remainder is at most $\frac{Cm^2}L\int_{|y|\ge1/L}(L^{-1}+(|y|L)^{-1})|\vartheta|y^{-2}dy\le Cm^2p_\gamma L^{-2}\log L$. \emph{(iii) The cubic part.} Since $\int wg_0(w)\,dw=0$ and $\partial_\varepsilon\Ci(x(1+\varepsilon))=\cos(x(1+\varepsilon))/(1+\varepsilon)$, Taylor's formula gives $|c(v/L)-\Ci(|v|)|\le Ca^2(1+|v|)L^{-2}$. For $x\le M_m:=0.472m+1$, $|h_3(x)|\le C_m|x|^3$, $|h_3'(x)|\le C_mx^2$ if $|x|\le M_m$ and $|h_3'(x)|\le4|x|+2$ if $x<0$, with $C_m\le Ce^{2m}$. Hence, substituting $y=v/L$, \[ L\int h_3(mc)P_\gamma=\int_{|v|\le L}h_3(m\Ci(|v|))P_\gamma(v/L)\,dv+O\Big(\frac{m^3e^{2m}p_\gamma\log L}{L^2}\Big), \] the region $|v|>L$ contributing $O(m^3L^{-2}\int_{|y|>1}|y|^{-3}P_\gamma(y)\,dy)$, and the difference of the profiles at most $Cm^3e^{2m}p_\gamma L^{-2}$ times $\int_{|v|\le1}(1+|\log|v||)^2dv+\int_{1\le|v|\le L}(1+|v|)v^{-2}dv=O(\log L)$, since $|\Ci(|v|)|\le\min(C(1+|\log|v||),1/|v|)$. Expanding $P_\gamma(v/L)$ to second order (the first-order term is odd), and using $|h_3(m\Ci(|v|))|\le Ce^{2m}m^3|v|^{-3}$ for $|v|\ge1$, the integral is $P_\gamma(0)\int_{\mathbb R}h_3(m\Ci(|v|))\,dv+O(m^3e^{2m}p_\gamma L^{-2}\log L)$. \emph{(iv) The constant.} $\int_{\mathbb R}2m^2\Ci(|v|)^2dv=4m^2\int_0^\infty\Ci^2=2\pi m^2$, so $2\pi m^2+\int h_3(m\Ci(|v|))dv=\int h(m\Ci(|v|))\,dv=2\kappa_m$. Adding (i)--(iii) proves the proposition. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:second}] Under RH, $m(\gamma)\le N(\gamma+\eta)-N(\gamma-\eta)$ for every $\eta>0$, and Littlewood's bound for $S(T)$ gives $m(\gamma)\ll\log(|\gamma|+3)/\log\log(|\gamma|+3)$, so $m^3e^{2m}\ll_\varepsilon(|\gamma|+3)^\varepsilon$ and $\sum_\gamma m^3e^{2m}(1+|\gamma-t_0|)^{-2}<\infty$. Summing Proposition~\ref{prop:onewell} over the distinct ordinates, \[ \tfrac12L\sum_\gamma H_{X,\gamma}=\Omega(s_0)+\frac{\Omega_1(s_0)}L+O\Big(\frac{\log L}{L^2}\Big), \] and by definition $\frac12L\int h(\phi_X)P_{s_0}=\frac12L\sum_\gamma H_{X,\gamma}+\mathcal I_X/L$. By Theorem~\ref{thm:law} both $\frac12LH_X$ and $\frac12L\sum H_{X,\gamma}$ tend to $\Omega$, so $\mathcal I_X=o(L)$ and $LH_X=2\Omega+o(1)$. Finally \eqref{eq:BH} and \eqref{eq:ell} give $LB_X=\frac12LH_X-\frac14LH_X^2+O(LH_X^3+L|\ell_X|)=\frac12LH_X-\Omega^2/L+o(1/L)$. \end{proof} \begin{remark} The proof uses only $\int wg_0(w)\,dw=0$, so ``even'' can be replaced by ``of mean zero''. For a weight with mean $\mu_1\ne0$, $E[M]=\mu_1+\frac12E|W-W'|$, and the first moment also enters through the profile in (iii); to first order it amounts to replacing $\log X$ by $\log X+\mu_1$. The interaction $\mathcal I_X$ is the only part of the second-order term that depends on more than one zero. By the identity $h(x+y)=h(x)+h(y)+(e^{2x}-1)(e^{2y}-1)$ its leading part is $\frac12L^2\sum_{\gamma<\gamma'}\int f_\gamma f_{\gamma'}P_{s_0}$ with $f_\gamma=e^{2m(\gamma)c_X(\cdot-\gamma)}-1$; the tails of $f_\gamma$ oscillate like $e^{\pm i|t-\gamma|\log X}\hat g_0(t-\gamma)$, so the phases of $\mathcal I_X$ are $(\gamma-\gamma')\log X$, with a damping that is roughly $\hat g_0(\gamma-\gamma')$. Proving $\mathcal I_X=O(1)$, or $O(\log\log X)$, would need control of small gaps between zeros. \end{remark} \section{The interaction at large height: pair correlation}\label{sec:pair} At a fixed centre the wells are far apart and their interaction is small. With the centre at height $T$ and $X=(T/2\pi)^\alpha$ (which agrees with $X=T^\alpha$ up to a factor $1+O(1/\log T)$ in $\alpha$) the wells have width of order $1/(\alpha\log T)$ and the zeros have mean spacing $2\pi/\log(T/2\pi)$, so the interaction becomes a statistic of the spacings. Its quadratic part, \[ \mathcal Q_X(s_0)=\int2\phi_X^2P_{s_0}-\sum_\gamma m(\gamma)^2\!\int2c_X(t-\gamma)^2P_{s_0}(t)\,dt=2\sum_{\rho\ne\rho'}\int c_X(t-\gamma)c_X(t-\gamma')P_{s_0}(t)\,dt, \] the sum running over ordered pairs of distinct zeros counted with multiplicity, is a pair sum, and we compare it with Montgomery's form factor \[ F(\beta)=\frac{2\pi}{T\log T}\sum_{0<\gamma,\gamma'\le T}T^{i\beta(\gamma-\gamma')}w(\gamma-\gamma'),\qquad w(u)=\frac4{4+u^2}. \] Under RH, Montgomery proved $F(\beta)=\beta+o(1)$ for $0<\beta<1$ [M], and the pair-correlation conjecture is $F(\beta)=1+o(1)$ for $\beta\ge1$. (Gonek, Hughes and Keating model the moments of $Z_X$ by random matrix theory for $X\ll(\log T)^{2-\varepsilon}$ [GHK]; here $X=T^\alpha$ is much larger, and the wells are separated.) \emph{Derivation.} By \eqref{eq:cfourier} and Plancherel, $\int c_X(y)c_X(y-d)\,dy=\frac1{2\pi}\int|\hat c(\xi)|^2e^{i\xi d}d\xi$ with $|\hat c(\xi)|^2=\pi^2G_X(|\xi|)^2/\xi^2$, a function supported on $|\xi|\ge L-a$. Averaging the centre $t_0$ over a long interval near $T$ removes the kernel $P_{s_0}$ from each pair, so the average of $\mathcal Q_X$ is $\frac1\pi\int|\hat c(\xi)|^2\,\frac1{H}\sum_{\rho\ne\rho'}e^{i\xi(\gamma-\gamma')}d\xi$, the pair sum being over a window of length $H$ near $T$. In the normalization of $F$, the pair sum at frequency $\xi=\beta\ell$, $\ell=\log\frac T{2\pi}$, is $\frac\ell{2\pi}(F(\beta)-1)$ per unit length, the diagonal $\rho=\rho'$ being excluded. The single wells cost on average $\frac\ell{2\pi}\int h(c_X)=\frac{\ell}{2\pi}\cdot\frac{2\kappa_1}{L}(1+O(L^{-1}))$ per unit length. Since $L\int_0^\infty G_X^2\xi^{-2}d\xi=1+O(L^{-1})$, this gives \begin{equation}\label{eq:pair} \frac{\langle\mathcal Q_X\rangle}{\langle\sum_\gamma H_{X,\gamma}\rangle}=\frac{\pi}{\kappa_1}\,\alpha\int_\alpha^\infty\frac{F(\beta)-1}{\beta^2}\,d\beta+O\Big(\frac1{\log T}\Big),\qquad X=\Big(\frac T{2\pi}\Big)^\alpha, \end{equation} where $\langle\cdot\rangle$ is the average over the centre. By Montgomery's theorem $\alpha$ times the part of the integral over $\alpha<\beta<1$ equals $-(1-\alpha+\alpha\log\alpha)$, so under RH \[ \frac{\langle\mathcal Q_X\rangle}{\langle\sum H_{X,\gamma}\rangle}=-\frac\pi{\kappa_1}\big(1-\alpha+\alpha\log\alpha\big)+\frac\pi{\kappa_1}\,\alpha\int_1^\infty\frac{F(\beta)-1}{\beta^2}d\beta+o(1)\qquad(0<\alpha<1), \] and the pair-correlation conjecture implies that the last integral vanishes, so that the quadratic interaction is $0$ for $\alpha\ge1$. We have not written out the error analysis (the smooth truncation of the pair sums and of the window in $t_0$), which follows the lines of Goldston and Montgomery [GM]; \eqref{eq:pair} should be read as a derivation, not a theorem. Conversely, \eqref{eq:pair} can be inverted: $F(\alpha)-1=-\alpha^2\frac{d}{d\alpha}\big(\alpha^{-1}R(\alpha)\big)\cdot\frac{\kappa_1}\pi$, $R$ the left side, so the Euler--Caceres energy determines $F$. \emph{Numerical test.} We computed the zeros in $[T-70,T+470]$, $T=10^6$, with the Riemann--Siegel formula ($1030$ zeros), built $\phi_X=\sum_\gamma c_X(\cdot-\gamma)$ for $X=(T/2\pi)^\alpha$, $0.3\le\alpha\le3$, the normalization of \eqref{eq:pair}, and averaged over $200$ centres $s_0=\sigma_0+it_0$, $t_0\in[T,T+400]$, with $\sigma_0-\frac12=1$ and with Montgomery's weight, $\sigma_0-\frac12=2$. Figure~\ref{fig:pair}a shows the quadratic interaction: it follows Montgomery's curve for $\alpha\le0.7$ ($-0.587\pm0.011$ against $-0.582$ at $\alpha=0.3$, $-0.257\pm0.009$ against $-0.264$ at $\alpha=0.5$, $-0.088\pm0.010$ against $-0.087$ at $\alpha=0.7$) and is close to $0$ for $\alpha\ge1.2$, as the pair-correlation conjecture predicts; uncorrelated zeros would give $0$ for every $\alpha$. Around $\alpha=1$ the data lie above both, by up to $0.04$ ($4\sigma$); lower-order terms in the pair correlation at finite height (Bogomolny and Keating [BK]) and the width of the cut-off, which smooths the kink of $F$ at $\beta=1$, are possible causes that we have not separated. The full interaction (Figure~\ref{fig:pair}b) is positive, with a maximum of $0.15$ near $\alpha=0.9$: the nonlinear terms more than compensate the repulsion seen in the quadratic part, and we do not have a formula for them. \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Montgomery_pair_correlation.png} \caption{The interaction of the wells at height $T=10^6$, $X=(T/2\pi)^\alpha$, averaged over $200$ centres, relative to the sum of the single wells. (a) The quadratic part against $-\frac\pi{\kappa_1}(1-\alpha+\alpha\log\alpha)$ for $\alpha<1$ (Montgomery's theorem under RH, with the contribution of $\beta\ge1$ set to $0$) and $0$ for $\alpha>1$ (pair-correlation conjecture). (b) The full interaction.}\label{fig:pair}\end{figure} \begin{remark} The total $\int2\phi_X^2P_{s_0}=\int2\log^2|\zeta/E^*_X(\frac12+it)|P_{s_0}(t)\,dt$ is computed from $\zeta$ and the primes up to $e^aX$ alone, and by Proposition~\ref{prop:onewell} the single-well part is $2\pi E[(L+M)^{-1}]\sum m^2P_{s_0}(\gamma)$ up to $O(L^{-2})$, which under RH and simple zeros is $2E[(L+M)^{-1}]\re\frac{\xi'}\xi(s_0)$. So \eqref{eq:pair} expresses the pair correlation through the mean square of the logarithm of the tail of the Euler product beyond $X$, in the spirit of the equivalences of Goldston, Gonek and Montgomery between pair correlation and mean values of $\zeta'/\zeta$ [GGM]. \end{remark} \section{The interaction of two wells and the gaps between the zeros}\label{sec:gaps} The interaction $\mathcal I_X$ of Theorem~\ref{thm:second} is the only part of the error in Theorem~\ref{thm:law} that depends on more than one zero. Its leading part is a sum over pairs of zeros (the remark after the proof of Theorem~\ref{thm:second}); here we compute the contribution of one pair exactly, show that it is controlled by the distance between the two zeros in the scale $1/\log X$, and deduce a rate in Theorem~\ref{thm:law} under a hypothesis on small gaps. \subsection{The two-well function} For two simple zeros at distance $\delta$, put $d=\delta\log X$, the distance in the variable $v=(t-\gamma)\log X$ of Lemma~\ref{lem:profile}, and \[ \kappa_{\rm pair}(d)=\tfrac12\int_{\mathbb R}h\big(\Ci|v-\tfrac d2|+\Ci|v+\tfrac d2|\big)\,dv,\qquad E(d)=\kappa_{\rm pair}(d)-2\kappa_1 . \] By dominated convergence, $\kappa_{\rm pair}(d)\to\kappa_2$ as $d\to0$ and $\kappa_{\rm pair}(d)\to2\kappa_1$ as $d\to\infty$; by the identity $h(x+y)-h(x)-h(y)=(e^{2x}-1)(e^{2y}-1)$, $E(d)=\frac12\int(e^{2\Ci|v-d/2|}-1)(e^{2\Ci|v+d/2|}-1)\,dv$. The proof of Theorem~\ref{thm:law}, applied to a configuration in which two simple zeros near height $\gamma$ are at distance $d/\log X$, shows that the pair contributes $P_{s_0}(\gamma)\kappa_{\rm pair}(d)$ to the limit of $(\log X)B_X$, in place of $2\kappa_1P_{s_0}(\gamma)$ for two separated zeros; $P_{s_0}(\gamma)E(d)$ is its interaction. Table~\ref{tab:kpair} gives the function; the script \texttt{ec\_phase2\_controls.py} computes it. \begin{table}[h]\centering\small \begin{tabular}{c|ccccccc} \toprule $d$ & $0$ & $0.1$ & $0.5$ & $1$ & $2$ & $3$ & $5$\\ $\kappa_{\rm pair}(d)$ & $7.3458$ & $7.3092$ & $6.4935$ & $4.6008$ & $2.4720$ & $3.5843$ & $4.3443$\\ $E(d)$ & $3.690$ & $3.653$ & $2.838$ & $0.945$ & $-1.184$ & $-0.072$ & $0.688$\\ \midrule $d$ & $8$ & $12$ & $20$ & $40$ & $80$ & $160$ & $\infty$\\ $\kappa_{\rm pair}(d)$ & $3.2330$ & $3.8722$ & $3.5021$ & $3.5875$ & $3.7013$ & $3.6509$ & $3.6558$\\ $E(d)$ & $-0.423$ & $0.216$ & $-0.154$ & $-0.068$ & $0.045$ & $-0.005$ & $0$\\ \bottomrule \end{tabular} \caption{The two-well function. $E(0)=\kappa_2-2\kappa_1$: two coalescing simple zeros cost what a double zero costs. $E<0$ for $1.5\lesssim d\lesssim3.1$.}\label{tab:kpair} \end{table} \emph{Decay.} On $10\le d\le400$ (781 points) the maximum of $|dE(d)|$ over windows of length $6\pi$ is $3.64$ to $3.66$, constant in $d$, and the least-squares fit $dE(d)=A\cos d+B\sin d+(C\cos d+D\sin d)\log d$ gives $A=0.451$, $B=-3.564$, $C=-0.078$, $D=-0.017$, with residual $0.04$. So, numerically, \[ E(d)=\frac{0.45\cos d-3.56\sin d}{d}+O\Big(\frac{\log d}{d^2}\Big). \] Heuristically, away from the wells $e^{2x}-1\approx2x$ and $\Ci(v)\approx\sin v/v$; the region between the wells contributes $-4\cos d\,\log d/d$ and the two outer regions $+4\cos d\,\log d/d$, while the terms of order $\log d/d$ from the neighbourhoods of the wells cancel in pairs. Status: the limits at $0$ and $\infty$ are proved (elementary); the decay is numerical and heuristic. \emph{A planted pair.} We replaced the first zero $\pm\gamma_1$ by the pair $\pm(\gamma_1\pm\delta/2)$, keeping the other zeros below height $1000$, and compared $L\Delta B$, the change of $(\log X)B_X(2)$ against the configuration without $\pm\gamma_1$, with $2P_2(\gamma_1)\kappa_{\rm pair}(\delta L)$ (Table~\ref{tab:pairtest}). The ratio tends to $1$ at the rate $O(1/\log X)$ of the single-well deficit of Proposition~\ref{prop:onewell}; for the single zero alone the ratios are $0.899$, $0.948$, $0.964$ at the same $X$, and the relative deficit $m^2/\kappa_m$ is the same for $m=1$ and for a merged pair ($0.547$ and $0.545$). \begin{table}[h]\centering\small \begin{tabular}{cccccc} \toprule $X$ & $\delta$ & $d=\delta\log X$ & $L\Delta B$ & $2P_2(\gamma_1)\kappa_{\rm pair}(d)$ & ratio\\ \midrule $10^2$ & $0.004$ & $0.018$ & $0.03064$ & $0.03471$ & $0.883$\\ $10^2$ & $0.5$ & $2.30$ & $0.01189$ & $0.01250$ & $0.951$\\ $10^4$ & $0.02$ & $0.184$ & $0.03213$ & $0.03414$ & $0.941$\\ $10^4$ & $0.5$ & $4.61$ & $0.01975$ & $0.02076$ & $0.951$\\ $10^6$ & $0.004$ & $0.055$ & $0.03331$ & $0.03467$ & $0.961$\\ $10^6$ & $0.1$ & $1.38$ & $0.01485$ & $0.01555$ & $0.955$\\ $10^6$ & $2$ & $27.6$ & $0.01695$ & $0.01690$ & $1.003$\\ \bottomrule \end{tabular} \caption{A planted close pair at $\gamma_1$, $s_0=2$, bump weight $a=1.2$, zeros below height $1000$ (script \texttt{ec\_phase2\_controls.py pair}).}\label{tab:pairtest} \end{table} \subsection{A rate under a gap hypothesis} For a distinct ordinate $\gamma$ let $\delta_\gamma\in(0,1]$ be the smaller of $1$ and the distance from $\gamma$ to the nearest other distinct ordinate, and for $\varepsilon>0$ put \[ \mathcal G_\varepsilon(s_0)=\sum_\gamma P_{s_0}(\gamma)\,\frac{(|\gamma|+3)^\varepsilon}{\delta_\gamma}\ \le\infty . \] $\mathcal G_\varepsilon(s_0)<\infty$ follows from any bound $\delta_\gamma\gg|\gamma|^{-1+\eta}$ with $\eta>\varepsilon$, since $P_{s_0}(\gamma)\asymp\gamma^{-2}$ and the zeros have density $\frac1{2\pi}\log\gamma$; under the GUE model the smallest gap below $T$ is of order $T^{-1/3}$ and the mean of $1/\delta_\gamma$ is of order $\log\gamma$, so $\mathcal G_\varepsilon(s_0)<\infty$ is expected for every $\varepsilon>0$. It is not known, even under RH. \begin{theorem}[a rate for the law]\label{thm:rate} Assume RH and $\mathcal G_\varepsilon(s_0)<\infty$ for some $\varepsilon\in(0,\frac14)$. Then there are $C$ and $X_0$, depending on $s_0$, $g_0$ and $\varepsilon$, with \[ \big|(\log X)B_X(s_0)-\Omega(s_0)\big|\le C\big(1+\mathcal G_\varepsilon(s_0)\big)\,\frac{\log\log X}{\log X}\qquad(X\ge X_0). \] \end{theorem} \begin{proof} By \eqref{eq:BH} and \eqref{eq:ell} it suffices to bound $\frac12LH_X-\Omega$. Under RH, $m(\gamma)\ll\log|\gamma|/\log\log|\gamma|$, so every factor depending on the multiplicity, such as $m^3e^{2m}$, is $\ll_\varepsilon(|\gamma|+3)^{\varepsilon/2}$; and by Lemma~\ref{lem:upper}(i) with $\varepsilon/4$, $\phi_X\le\frac\varepsilon4\log(|t|+3)+C$ for $X\ge X_\varepsilon$. Call an ordinate \emph{separated} if $\delta_\gamma L\ge4$ and \emph{crowded} otherwise. \emph{Crowded ordinates.} Let $\mathcal B$ be the union of the intervals $[\gamma-\frac32,\gamma+\frac32]$ over the crowded $\gamma$. On a unit interval $I$ near height $n$, Lemma~\ref{lem:upper} gives $L\int_Ih(\phi_X)\le C_\varepsilon(n+3)^{\varepsilon/2}\log^2(n+3)$, and the part of $\Omega$ carried by the ordinates in $I$ is bounded in the same way. Since a crowded ordinate satisfies $1<4/(\delta_\gamma L)$, the contribution of $\mathcal B$ to $|\frac12LH_X-\Omega|$ is at most $\frac{C}L\sum_\gamma P_{s_0}(\gamma)(|\gamma|+3)^\varepsilon/\delta_\gamma=C\mathcal G_\varepsilon/L$. \emph{Separated ordinates outside $\mathcal B$.} On $U_\gamma=(\gamma-\frac{\delta_\gamma}2,\gamma+\frac{\delta_\gamma}2)$ write $\phi_X=m\,c_X(\cdot-\gamma)+r_X$. The $\ll\log(|\gamma|+3)$ ordinates within distance $1$ are at distance $\ge\delta_\gamma/2$ from $U_\gamma$ and contribute at most $4/(\delta_\gamma L)$ each by \eqref{eq:cbounds}, the others $O(\log(|\gamma|+3)/L)$; so $|r_X|\le C\log(|\gamma|+3)/(\delta_\gamma L)$. By the mean value theorem $|h(x+r)-h(x)|\le2|r|\,|e^{2(x+\vartheta r)}-1|$, and, since $|r_X|$ may be as large as $\frac C4\log(|\gamma|+3)$, Lemma~\ref{lem:upper}(i) caps the exponential by $C(|\gamma|+3)^{\varepsilon/2}$ and gives $\int_{|v|<\delta_\gamma L/2}|e^{2mc_X(v/L)+2\vartheta r_X}-1|\,dv\ll(|\gamma|+3)^{\varepsilon/2}\big(m\log(\delta_\gamma L)+\delta_\gamma L\,|r_X|\big)$: the integrand is at most $C(|\gamma|+3)^{\varepsilon/2}$ for $|v|\le1$ and at most $C(|\gamma|+3)^{\varepsilon/2}(6m/|v|+2|r_X|)$ beyond. Hence \[ L\Big|\int_{U_\gamma}\big(h(\phi_X)-h(mc_X(\cdot-\gamma))\big)P_{s_0}\Big|\ll P_{s_0}(\gamma)\,(|\gamma|+3)^{\varepsilon/2}\,\log(|\gamma|+3)\,\frac{m\log L+\log(|\gamma|+3)}{\delta_\gamma L}\ \ll\ P_{s_0}(\gamma)\,(|\gamma|+3)^{\varepsilon}\,\frac{\log L}{\delta_\gamma L}. \] By Proposition~\ref{prop:onewell}, $L\int_{U_\gamma}h(mc_X(\cdot-\gamma))P_{s_0}=2\kappa_mP_{s_0}(\gamma)+O(m^3e^{2m}P_{s_0}(\gamma)/L)$, up to the truncation of the well at $|v|=\delta_\gamma L/2$, which costs $O(m^2P_{s_0}(\gamma)/(\delta_\gamma L))$ because $h(m\Ci(|v|))\ll m^2/v^2$ for $|v|\ge1$. \emph{Between the wells.} Outside $\mathcal B$ and the $U_\gamma$, $\phi_X$ is a sum of $c_X(t-\gamma')$ with $|t-\gamma'|\ge\delta_{\gamma'}/2$. By Cauchy--Schwarz over the $\ll\log(|t|+3)$ ordinates within distance $1$ and $h(x)\le2x^2e^{2\max(x,0)}$, $L\int h(\phi_X)P_{s_0}$ over the gap next to $\gamma$ is $\ll P_{s_0}(\gamma)(|\gamma|+3)^{\varepsilon/2}\log^2(|\gamma|+3)\,L\int_{|y|\ge\delta_\gamma/2}(yL)^{-2}dy\ll P_{s_0}(\gamma)(|\gamma|+3)^{\varepsilon}/(\delta_\gamma L)$. Summing over all ordinates, with $m\log^2(|\gamma|+3)\ll(|\gamma|+3)^{\varepsilon/2}$, the three parts give $|\frac12LH_X-\Omega|\le C(1+\mathcal G_\varepsilon)\log L/L$, with no cut-off in height. \end{proof} \noindent Status: proved; the constants are not made explicit. The factor $\log\log X$ comes only from the absolute value in the first display of the second step, which discards the oscillation of the pair interaction. If $E(d)=O(1/d)$, as Table~\ref{tab:kpair} indicates, and the interactions of three or more wells are of lower order, the same proof gives \[ (\log X)B_X(s_0)=\Omega(s_0)+\frac{\Omega_1(s_0)-\Omega(s_0)^2+\mathcal I_X(s_0)}{\log X}+o\Big(\frac1{\log X}\Big),\qquad \mathcal I_X(s_0)=O\big(\mathcal G_\varepsilon(s_0)\big), \] that is, Theorem~\ref{thm:second} would be a genuine second-order asymptotic under the gap hypothesis. This is consistent with Table~\ref{tab:second}, where $\mathcal I_X$ is small and does not grow. \begin{remark}[the gaps are the obstruction]\label{rem:gapsobstruct} Our proof gives no rate in Theorem~\ref{thm:law} without information on small gaps. By Table~\ref{tab:pairtest}, a pair of simple zeros at height $\gamma$ with gap $\delta$ adds about $(\kappa_2-2\kappa_1)P_{s_0}(\gamma)=3.69P_{s_0}(\gamma)$ to $(\log X)B_X(s_0)$ for all $X$ with $\delta\log X\lesssim\frac12$: in that range it is indistinguishable from a double zero. The speed of convergence is thus controlled, from both sides, by $\sum_{\delta_\gamma\lesssim1/\log X}P_{s_0}(\gamma)$, which the hypothesis $\mathcal G_\varepsilon<\infty$ makes $O(\mathcal G_\varepsilon/\log X)$ by Markov's inequality. Corollary~\ref{cor:criteria}(b) is the limit of this picture, and Open problem~3 becomes quantitative: a finite $X$ resolves multiplicities only at the scale $\delta\approx1/\log X$. \end{remark} \section{Off the line: explosion}\label{sec:explode} \begin{proof}[Proof of Theorem~\ref{thm:explode}] Write $Z=Z_X$ and fix $s_0$ with $\re s_0>\Theta$. (a) \emph{Reduction to a point.} By Lemma~\ref{lem:outer} and Jensen, $\log|Z(s_1)|\le\int\log|Z|P_{s_1}\le\frac12\log\int|Z|^2P_{s_1}$ for $\re s_1>\frac12$, and $P_{s_1}\le HP_{s_0}$ with $H$ independent of $X$, so $B_X(s_0)\ge\log|Z(s_1)|-\frac12\log H-\log|Z(s_0)|$. (b) \emph{Choice of $s_1$.} Take a zero $\rho_0=\Theta+i\tau_0$; let $d\in(0,1]$ be at most the distance from $\tau_0$ to the other ordinates of zeros of real part $\Theta$, and $S=\sum'\min(1/|y|,C|y|^{-3})$ over those zeros, $y=\tau_0-\gamma$, with $C$ from (L3) ($A=2$, $x_{\max}=\frac12$); $S$ does not depend on $r$. Choose $r\in(0,\Theta-\frac12)$ with $2re^{2ra}S\le\frac14$ and $s_1=\rho_0-r$ not a zero; let $\delta>0$ be the distance from $s_1$ to the zeros. (c) \emph{Lower bound at $s_1$.} $\log|Z(s_1)|=-\sum_\rho\re K_X(s_1-\rho)$, $w=s_1-\rho$, $x=r-(\Theta-\beta)$. The term of $\rho_0$ is $\ge\Ei(r(L-a))$ by (L4). The other zeros with $\beta=\Theta$ contribute at most $X^re^{ra}S/(L-a)\le\frac18X^re^{-ra}/(r(L-a))$ by (L3). The zeros with $\Theta-r<\beta<\Theta$ contribute $\ge0$ if $y=0$ (finitely many, $X\ge X_0$), $o(X^r/L)$ if $0<|y|<1$ (finitely many), and $o(X^r/L)$ in total if $|y|\ge1$, by (L3) and dominated convergence. The zeros with $\beta\le\Theta-r$ contribute $O(\log L)$ by (L2). Since $\Ei(\xi)\ge e^\xi/\xi$ for $\xi\ge1.35$ and $X^re^{-ra}/(r(L-a))\ge X^re^{-ra}/(rL)$, the bound $2re^{2ra}S\le\frac14$ makes the other zeros of real part $\Theta$ cost at most $\frac18\Ei(r(L-a))$, and for $X\ge X_0$ the $o(X^r/L)$ terms cost at most another $\frac38\Ei(r(L-a))$; hence $\log|Z(s_1)|\ge\frac12\Ei(r(L-a))-C\log L$. (d) \emph{The centre.} $\re(s_0-\rho)\ge\re s_0-\Theta>0$, so $\log|Z(s_0)|\to0$ by (L1). \end{proof} \begin{proof}[Proof of Corollary~\ref{cor:energy}] $\log|E^*_X|$ is bounded below on the line (proof of Lemma~\ref{lem:outer}) and $|\zeta(\frac12+it)|\ll(|t|+2)^{1/4}$, so $\mathcal E(X)<\infty$. Under RH, $\phi_X\to0$ almost everywhere (Lemma~\ref{lem:profile} and its proof) and $e^{2\phi_X}\le C(|t|+3)^{1/2}$ by Lemma~\ref{lem:upper}(i), so $\mathcal E(X)\to\pi$ by dominated convergence. If RH fails, take $\re s_0>1$; then $P_{s_0}\le C/(1+t^2)$ gives $B_X(s_0)\le\frac12\log(C\mathcal E(X))-\log|Z_X(s_0)|$, and $\log|Z_X(s_0)|\to0$ by (L1), since $\Theta\le1<\re s_0$. Theorem~\ref{thm:landau} makes $B_X(s_0)$ unbounded, so $\limsup\mathcal E(X)=\infty$; if $\Theta$ is attained, Theorem~\ref{thm:explode} forces $\mathcal E(X)\to\infty$. \end{proof} \subsection{\texorpdfstring{$\Theta$}{Theta} not attained: Landau's theorem in the variable \texorpdfstring{$\log X$}{log X}}\label{sec:landau} The proof of Theorem~\ref{thm:explode} needs a zero of maximal real part, because the other zeros near the line $\re s=\Theta$ could cancel its volcano. Landau's argument for $\psi(x)-x=\Omega_\pm(x^{\Theta-\varepsilon})$ [I, Ch.~V] avoids this: it treats $\log X$ as a variable, takes a Laplace transform, and uses that every zero is a singularity of it. Write $L=\log X$, $L_0=a+1$, $\hat g_0(w)=\int g_0(v)e^{wv}\,dv$ (an entire function, real on the real axis, not identically $0$), and, for $s=\sigma+it$ with $\sigma>\frac12$ and $\zeta(s)\ne0$, \[ f_s(L)=\log\big|Z_{e^L}(s)\big|=-\sum_\rho\re K_{e^L}(s-\rho)\qquad(L\ge L_0). \] By Lemma~\ref{lem:kernel}, $f_s$ is continuous, and $|f_s(L)|\le C_se^{L/2}$, uniformly for $s$ in compact subsets of $\{\sigma>\frac12\}$ that contain no zero: every zero has $\re(\rho-s)<\frac12$, the finitely many zeros within distance $1$ of $s$ are bounded by (L2) and the first part of (L3), and the others by the second parts of (L2) and (L3). \begin{lemma}[the Laplace transform in $\log X$]\label{lem:laplace} For $\re w>2$, \[ \int_{L_0}^\infty f_s(L)\,e^{-wL}\,dL=\tfrac12\big(\Phi_s(w)+\Phi_{\bar s}(w)\big),\qquad \Phi_s(w)=-\frac{\hat g_0(w)}{w}\,\log\big((s+w-1)\zeta(s+w)\big)+R_s(w), \] where $R_s$ is holomorphic in $\{\re w>-\sigma\}\setminus\{0\}$. Consequently the left side continues to $\{\re w>-\sigma\}$ with singular points only at $w=0$, at $w=\rho-s$ and at $w=\rho-\bar s$, $\rho$ running over the zeros; near $w_0=\rho-s\ne0$ the function $\Phi_s(w)+m(\rho)\frac{\hat g_0(w)}{w}\log(w-w_0)$ is holomorphic. \end{lemma} \begin{proof} Let first $\re s>1$. By~\eqref{eq:Estar} and $\log\zeta(s)=\sum\Lambda(n)n^{-s}/\log n$, with $\int_1^\infty G_X(\log x)x^{-s}dx/\log x=K_X(s-1)$ (integrate by parts against $dE_1((s-1)u)=-e^{-(s-1)u}du/u$), we have, with $\theta(u)=\psi(e^u)-e^u$, \[ \log Z_{e^L}(s)=\int_0^\infty G_0(u-L)\,\frac{e^{-su}}u\,d\theta(u)+\sum_{k\ge1}K_{e^L}(s+2k), \] as in the proof of Theorem~\ref{thm:isometry}. For $u\ge U:=L_0+a$, the substitution $v=u-L$ and $\int_{-a}^aG_0e^{wv}dv=(e^{wa}-\hat g_0(w))/w$ give \[ \int_{L_0}^\infty G_0(u-L)e^{-wL}dL=-\frac{\hat g_0(w)}w\,e^{-wu}+\frac{e^{-wL_0}}w , \] and for $u1$), \begin{multline*} \Phi_s(w):=\int_{L_0}^\infty\log Z_{e^L}(s)e^{-wL}dL\\ =-\frac{\hat g_0(w)}w\int_U^\infty\frac{e^{-(s+w)u}}u\,d\theta(u)+\frac{e^{-wL_0}}w\int_U^\infty\frac{e^{-su}}u\,d\theta(u)+E_s(w)+T_s(w), \end{multline*} with $E_s$ entire and $T_s$ the transform of the trivial-zero terms. With $z=s+w$, $\int_U^\infty e^{-zu}u^{-1}d\theta=\log\zeta(z)-\sum_{n\le e^U}\Lambda(n)n^{-z}/\log n-E_1((z-1)U)=\log((z-1)\zeta(z))+(\text{entire in }z)$, because $E_1(\xi)=-\gamma_E-\log\xi+\Ein(\xi)$. For the trivial-zero terms, $\int_{L_0}^\infty g_0(u-L)e^{-wL}dL=\hat g_0(w)e^{-wu}$ for $u\ge U$, and $\int_U^\infty E_1(zu)e^{-wu}du$, $z=s+2k$, equals $w^{-1}\log(1+w/z)$ minus an entire function; its singular point $w=-s-2k$ has real part $<-\sigma$, and the sum over $k$ converges locally uniformly in $\re w>-\sigma$, with a removable singularity at $w=0$. This gives the formula for $\Phi_s$ when $\re s>1$, with $R_s$ holomorphic in $\{\re w>-\sigma\}\setminus\{0\}$: the trivial zeros of $\zeta(s+w)$ have $\re w<-\sigma$, and $(z-1)\zeta(z)$ is holomorphic and nonzero at $z=1$. Taking real parts, $\int f_se^{-wL}dL=\frac12(\Phi_s(w)+\overline{\Phi_s(\bar w)})$, and $\overline{\Phi_s(\bar w)}=\Phi_{\bar s}(w)$ because $Z_X(\bar s)=\overline{Z_X(s)}$. For fixed $w$ with $\re w>2$, the left side is harmonic in $s$ on the connected set $\{\sigma>\frac12\}\setminus\{\rho\}$, by the locally uniform bound $|f_s(L)|\le C_se^{L/2}$. The right side is the sum of a holomorphic and an antiholomorphic function of $s$, and it is single-valued: the term of $\Phi_s(w)$ that contains $\log((s+w-1)\zeta(s+w))$ has no monodromy in $s$, since $\re(s+w)>\frac52$; the multivalued term is $c(w)\log((s-1)\zeta(s))$ with $c(w)=e^{-wL_0}/w$, so a change of branch adds $2\pi in\,c(w)$ to $\Phi_s(w)$ and $-2\pi in\,c(w)$ to $\Phi_{\bar s}(w)$, because $\overline{c(\bar w)}=c(w)$. The two sides agree for $\sigma>1$, hence on the whole set. The statement about singular points follows; a logarithmic branch point multiplied by the entire function $\hat g_0(w)/w$, not identically zero, is not removable. \end{proof} \begin{theorem}[$\log|Z_X|$ at a fixed point]\label{thm:omega} Let $s=\sigma+it$ with $\sigma>\frac12$ and $\zeta(s)\ne0$, and suppose that $t\notin\mathcal T$, the countable set of $t$ for which $t=\pm\gamma$ for some zero, or $\gamma-\gamma'=2t$ for two zeros with $\beta=\beta'$. Then for every zero $\rho=\beta+i\gamma$ with $\beta>\sigma$ and every $\delta\in[0,\beta-\sigma)$, \[ \limsup_{X\to\infty}X^{-\delta}\log|Z_X(s)|=+\infty,\qquad \liminf_{X\to\infty}X^{-\delta}\log|Z_X(s)|=-\infty . \] In particular $\log|Z_X(s)|=\Omega_\pm(X^{\Theta-\sigma-\varepsilon})$ for every $\varepsilon>0$, whether or not $\Theta$ is attained. \end{theorem} \begin{proof} Suppose $f_s(L)\le Ce^{\delta L}$ for $L\ge L_1$, and let $A(L)=Ce^{\delta L}-f_s(L)\ge0$ on $[L_1,\infty)$. Its Laplace transform $\hat A(w)=\int_{L_1}^\infty A(L)e^{-wL}dL$ has a finite abscissa of convergence $w_c$, since $|f_s|\le C_se^{L/2}$, and by Lemma~\ref{lem:laplace} \[ \hat A(w)=\frac{Ce^{(\delta-w)L_1}}{w-\delta}-\tfrac12\big(\Phi_s(w)+\Phi_{\bar s}(w)\big)+(\text{entire}). \] Because $t\notin\mathcal T$, no point $\rho-s$ or $\rho'-\bar s$ is real, so the right side continues holomorphically to a neighbourhood of every real $w>\max(\delta,0)=\delta$. By Landau's theorem (a Laplace integral with an eventually nonnegative integrand is singular at the real point of its abscissa of convergence; [Wi, Ch.~II, Thm.~5b]), $w_c\le\delta$, and $\hat A$, hence $\frac12(\Phi_s+\Phi_{\bar s})$, is holomorphic in $\re w>\delta$. But $w_0=\rho-s$ has real part $\beta-\sigma>\delta$, it is a singular point of $\Phi_s$, and, because $t\notin\mathcal T$, it is not a singular point of $\Phi_{\bar s}$, whose singular points are the $\rho'-\bar s$. This is a contradiction. The statement for $\liminf$ follows in the same way from $A(L)=Ce^{\delta L}+f_s(L)$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:landau} and Corollary~\ref{cor:landau}] Let $\varepsilon>0$, choose a zero with $\beta>\Theta-\varepsilon/2$, and let $s_1=\sigma_1+it_1$ with $\sigma_1=\frac12+\varepsilon/4$, $\zeta(s_1)\ne0$ and $t_1\notin\mathcal T$. Step (a) of the proof of Theorem~\ref{thm:explode} gives $B_X(s_0)\ge\log|Z_X(s_1)|-\frac12\log H-\log|Z_X(s_0)|$ with $H$ independent of $X$, and $\log|Z_X(s_0)|\to0$ by (L1), since $\re s_0>1\ge\Theta$. Theorem~\ref{thm:omega} at $s_1$ with $\delta=\beta-\sigma_1-\varepsilon/4>\Theta-\frac12-\varepsilon$ gives Theorem~\ref{thm:landau}. For Corollary~\ref{cor:landau}: under RH, $B_X(s_0)\to0$ (Theorem~\ref{thm:law}); if RH fails, Theorem~\ref{thm:landau} contradicts $B_X(s_0)\ll X^{\varepsilon}$ for $\varepsilon<\frac12(\Theta-\frac12)$. \end{proof} Theorem~\ref{thm:landau} gives growth along a sequence of $X$; Theorem~\ref{thm:explode}, under attainment, gives it for all large $X$, which is what the $\liminf$ form of Corollary~\ref{cor:energy} uses. For an isolated off-line zero the exponent $\Theta-\frac12$ of Theorem~\ref{thm:landau} is the true one (Table~\ref{tab:planted}); the restriction to a small $r$ in Theorem~\ref{thm:explode} comes only from the possible cancellation of the volcano of $\rho_0$ by other zeros of real part $\Theta$. \begin{remark}[$\Theta$ not attained: all large $X$]\label{rem:cluster} If $\Theta>\frac12$ is not attained, Theorem~\ref{thm:landau} gives growth along a sequence, but the proof of Theorem~\ref{thm:explode}, which gives growth for all large $X$, has no fixed zero $\rho_0$ to work with. What follows is a heuristic outline only. For each $X$ one can choose a zero $\rho^*=\beta^*+i\gamma^*$ maximizing $\beta\log X-A\log(2+|\gamma|)$, $A>0$ fixed; the zeros at distance at least $1$ from $\rho^*$ then contribute at most a small multiple of the main term, and a vertical shift of $s_1$ aligns the common phase $e^{-i\tau\log X}$. Since $K_X(w)\approx e^{-w\log X}\hat g_0(iw)/(w\log X)$ for $|w|\log X$ large, what remains is the cluster amplitude \[ \sum_{|\gamma-\gamma^*|<1}X^{\beta-\beta^*}e^{i(\gamma-\gamma^*)\log X}\,\frac{\hat g_0(iw)}{w},\qquad w=s_1-\rho, \] a sum of Cauchy kernels, which can be small only if off-line zeros of nearly maximal real part close to $\rho^*$ cancel one another. For two such zeros at distance $\Delta$ we expect a remainder of order $X^r\Delta/(r^2\log X)$, so a lower bound $\Delta\ge X^{-o(1)}$ for their separation would plausibly close this case; no such bound is known. \end{remark} \section{What holds without RH}\label{sec:uncond} The decomposition, the law and the explosion theorem leave open how large the bound can be, unconditionally, for a given $X$. The answer has three parts: the term $\Off(s_0)$ that does not depend on $X$ is tiny because RH has been verified to a large height; the bound is at most of the order of the length of the prime sum on the critical line; and between these two the position of the bound is decided by the real parts of the zeros above that height. Let $T_0=3\cdot10^{12}$; all zeros with $0<\gamma\le T_0$ lie on the critical line [PT]. (The numbers below are reproduced by the ancillary script \texttt{uncond\_numbers.py}.) \begin{theorem}[unconditional bounds]\label{thm:uncond} \begin{enumerate} \item[(a)] For every $s_0=\sigma_0+it_0$ with $\sigma_0>\frac12$, $\zeta(s_0)\ne0$ and $|t_0|\le T_0/2$, \[ 0\le\Off(s_0)\le(2\sigma_0-1)\,\frac{\log(T_0/2\pi)+2}{\pi T_0}<3.1\cdot10^{-12}\,(2\sigma_0-1), \] and for $t_0=0$ the bound improves by the factor $\frac14$; in particular $\Off(2)<2.3\cdot10^{-12}$. \item[(b)] For every $s_0$ with $\re s_0>1$ there is $C$, depending on $s_0$ and $g_0$ only, such that for every $X>e^{a+1}$ \[ 0\le B_X(s_0)\le \mathcal M(e^aX)+C\,\frac{e^{(L+a)/2}}{L-a}+C,\qquad \mathcal M(Y)=\sum_{n\le Y}\frac{\Lambda(n)}{\sqrt n\,\log n}\sim\frac{2\sqrt Y}{\log Y}; \] hence $B_X(s_0)\ll\sqrt X/\log X$. \end{enumerate} \end{theorem} \begin{proof} (a) For a zero $\rho=\beta+i\gamma$ with $\beta>\frac12$ and $y=t_0-\gamma$, \[ \log\frac1{|b_\rho(s_0)|}=\frac12\log\Big(1+\frac{(2\sigma_0-1)(2\beta-1)}{(\sigma_0-\beta)^2+y^2}\Big)\le\frac{(2\sigma_0-1)(2\beta-1)}{2y^2}\le\frac{2\sigma_0-1}{2y^2}. \] Every such zero has $|\gamma|>T_0$, so $|y|\ge|\gamma|/2$ (and $|y|=|\gamma|$ if $t_0=0$). The zeros with $\beta>\frac12$ and $\gamma>0$ are at most half of all the zeros with $\gamma>T_0$, since $\rho\mapsto1-\bar\rho$ pairs each of them with a zero of real part $<\frac12$ at the same height, and the same holds for $\gamma<0$. Hence $\Off(s_0)\le2(2\sigma_0-1)\Sigma$, and $\Off(s_0)\le\frac12(2\sigma_0-1)\Sigma$ if $t_0=0$, with $\Sigma=\sum_{\gamma>T_0}\gamma^{-2}$. By partial summation, $\Sigma=-N(T_0)T_0^{-2}+2\int_{T_0}^\infty N(t)t^{-3}dt$, and the Riemann--von Mangoldt formula with Trudgian's explicit remainder, $|N(t)-\frac t{2\pi}\log\frac t{2\pi e}-\frac78|\le0.112\log t+0.278\log\log t+2.51$ for $t\ge e$ [Tr], gives $\Sigma=\frac{\log(T_0/2\pi)+1}{2\pi T_0}+O(T_0^{-2}\log T_0)\le\frac{\log(T_0/2\pi)+2}{2\pi T_0}$; numerically $\Sigma<1.48\cdot10^{-12}$. (b) $B_X(s_0)\ge0$ by Theorem~\ref{thm:decomp}. By \eqref{eq:Estar}, on the critical line $-\log|E^*_X(s)|=-\re D(s)-\re K_X(s-1)+\sum_k\re K_X(s+2k)$, with $D$ the finite prime sum. First, $|\re D(\frac12+it)|\le \mathcal M(e^aX)$, because $0\le\psi_X\le1$ and $\psi_X(\log n)=0$ for $n>e^aX$. Second, $s-1=-\frac12+it$ and $K_X(s-1)=\int g_X(u)E_1(z)\,du$ with $z=(-\frac12+it)u$, $u\in[L-a,L+a]$; by the asymptotic expansion of $E_1$, valid uniformly in the closed sector $|\operatorname{ph}z|\le\pi$ with the boundary values $E_1(-x\pm i0)=-\Ei(x)\mp i\pi$ [DLMF, 6.12.1], $|E_1(z)|\le C|e^{-z}|/|z|+\pi\le2Ce^{u/2}/u+\pi$ for $|z|\ge\frac12$ (on the closed sector $E_1$ is bounded on $\frac12\le|z|\le1$), so $|K_X(s-1)|\le2Ce^{(L+a)/2}/(L-a)+\pi$ for every $t$. Third, the trivial-zero terms are bounded (proof of Lemma~\ref{lem:outer}). Hence $e^{2\phi_X(t)}\le|\zeta(\frac12+it)|^2\exp\big(2\mathcal M(e^aX)+C'e^{(L+a)/2}/(L-a)+C'\big)$, and $\int|\zeta(\frac12+it)|^2P_{s_0}(t)\,dt<\infty$ since $\zeta(\frac12+it)\ll(|t|+2)^{1/4}$ [T]. Finally $\re(s_0-\rho)\ge\re s_0-1>0$ for every zero, so $|\log|Z_X(s_0)||\le CX^{-(\re s_0-1)}/L$ by (L1). The asymptotic of $\mathcal M$ follows from the prime number theorem by partial summation (Chebyshev's bound already gives $\mathcal M(Y)\ll\sqrt Y/\log Y$). \end{proof} Together with Theorem~\ref{thm:explode}, Theorem~\ref{thm:uncond} brackets the bound. Unconditionally, for $\re s_0>1$ and $|t_0|\le T_0/2$, $B_X(s_0)$ lies between $\Off(s_0)$, which is below $3.1\cdot10^{-12}(2\sigma_0-1)$, and $O(\sqrt X/\log X)$, the length of the prime sum on the critical line; for $\frac12<\re s_0\le1$ we prove no upper bound. Under RH it decays like $\Omega(s_0)/\log X$ (Theorem~\ref{thm:law}). If $\Theta>\frac12$ it exceeds $X^{\Theta-\frac12-\varepsilon}$ for arbitrarily large $X$ (Theorem~\ref{thm:landau}), and if $\Theta$ is attained it grows at least like $X^r/\log X$ for all large $X$, for some $r\in(0,\Theta-\frac12)$; since an attained $\Theta$ is $<1$, the exponent $r$ of Theorem~\ref{thm:explode} is always below $\frac12$, the exponent of the upper bound. Without information on the zeros above $T_0$ nothing in between can be excluded, and no finite computation of $B_X$ can confirm RH. \begin{remark}[detection length] In the proof of Theorem~\ref{thm:explode} the height of the zero $\rho_0=\Theta+i\tau_0$ enters through the comparison $P_{s_1}\le HP_{s_0}$, with $\frac12\log H=\log|\tau_0-t_0|+O(1)$, and through the zeros near $\rho_0$, about $\log\tau_0$ per unit length, which affect the constant of the term $C\log\log X$ and $X_0$; the other constants depend on the configuration of the zeros of real part $\Theta$ near $\rho_0$. The bound at $s_0$ therefore exceeds a fixed level once $\frac12\Ei(r(\log X-a))$ exceeds $\log|\tau_0|+O(\log\log X)$, that is for $\log X$ of order $r^{-1}\log\log|\tau_0|$. For a zero just above $T_0$, ignoring the constants and with $a=1.2$, this means $\log X\approx56$ if the theorem applies with $r=0.1$ (which needs $\Theta>0.6$) and $\log X\approx550$ if it applies with $r=0.01$. The detection length grows like the logarithm of the logarithm of the height, the same dependence on the height as for Weil's form with two windows in {\cite{CE19}}, which detects an isolated zero with $\beta=\frac12+\delta$ with the prime powers up to $(C\log\gamma)^{1/\delta}$; the two mechanisms differ and we have not compared constants. Our computations, with $X\le10^6$, give $B_X(2)\approx3\cdot10^{-3}$, nine orders of magnitude above the largest possible value of $\Off(2)$ and far below the detection length of any zero above $T_0$ that is not very far from the line: they agree with the law of Theorem~\ref{thm:law}, and they would agree with it whether or not RH holds above $T_0$. \end{remark} \section{Numerical results}\label{sec:numerics} Unless stated otherwise (Table~\ref{tab:barrier} and Figure~\ref{fig:BX}), all values are computed in double precision with $\zeta$ and Hurwitz zeta functions by Euler--Maclaurin summation, on $|t-t_0|\le800$ with step $0.01$ (the missing kernel mass is added with $\phi_X=0$). The weight is the bump $g_0(v)\propto\exp(-1/(1-(v/1.2)^2))$ on $[-1.2,1.2]$, for which the prime sum is exactly finite. The numbers illustrate the theorems; the proofs do not use them. \textbf{The law at $s_0=2$.} Here $\varpi(2)=\frac1\pi\re\frac{\xi'}\xi(2)=0.0219844$ (Lemma~\ref{lem:W}) and $\kappa_1\varpi(2)=0.040186$. A computation on a finite window only sees the zeros in it: with $|t|\le T$ and $\phi_X=0$ outside, the limit of $(\log X)B_X(2)$ is $\kappa_1\varpi_T(2)$, $\varpi_T(2)=\sum_{|\gamma|1$ [DH], so $\Theta_f>1$ and its attainment is unknown, the zero $\rho_f$ is not of maximal real part, and we do not prove the factorization and the Smirnov property for $f$. Numerically the mechanism of its proof is nevertheless visible for moderate $X$: at $s_1=\rho_f-r$, $\log|Z_f(s_1)|$ is $1.0332$ and $1.6702$ for $X=10^3,10^4$ with $r=0.1$, against the main term $\int g_X(u)\Ei(ru)\,du=1.0373$ and $1.6800$ of (L4), and $2.9489$, $4.3721$ with $r=0.2$, against $2.9564$, $4.3976$ (script \texttt{explosion.py}). \textbf{The decomposition for $f$.} Lemma~\ref{lem:factor} and the Smirnov property are not proved for $f$, but the decomposition $B_f=\Off^{f}+J_f$ of Theorem~\ref{thm:decomp} can be tested. With the bump weight $a=1.2$, the window $|t-t_0|\le200$, and the eight off-line zeros of $f$ below height $400$, which a grid search with Newton refinement returns as $0.808517+85.699348i$, $0.650830+114.163343i$, $0.574356+166.479306i$, $0.724258+176.702461i$, $0.869531+240.404672i$, $0.819550+320.876490i$, $0.768223+331.050259i$ and $0.628508+366.640908i$, the identity $\int\phi_XP_{s_0}-\log|Z_f(s_0)|=\Off^{f}(s_0)$ holds to $10^{-3}$ at $1+85.7i$ and $2+85.7i$, and to $10^{-2}$ at $1+100i$, where the window error dominates the small floor (Table~\ref{tab:dhdecomp}). The floors do not depend on the weight and agree with Table~\ref{tab:barrier}; the bound itself, with this weight, agrees with the Gaussian values of Table~\ref{tab:barrier} to within $0.04$. The growth is entirely in the Jensen gap. \begin{table}[h]\centering\small \begin{tabular}{cccccc} \toprule $s_0$ & $X$ & $B_f(s_0)$ & $\Off^{f}(s_0)$ & $J_f(s_0)$ & $\int\phi_XP_{s_0}-\log|Z_f(s_0)|$\\ \midrule $1+85.7i$ & $10$ & $1.6335$ & $1.4407$ & $0.1933$ & $1.4403$\\ & $10^2$ & $2.5790$ & & $1.1375$ & $1.4415$\\ & $10^3$ & $4.4904$ & & $3.0506$ & $1.4398$\\ & $10^4$ & $7.6043$ & & $6.1637$ & $1.4406$\\ $2+85.7i$ & $10$ & $0.7555$ & $0.4181$ & $0.3382$ & $0.4173$\\ & $10^4$ & $7.1070$ & & $6.6890$ & $0.4180$\\ $1+100i$ & $10$ & $0.3349$ & $0.0024$ & $0.3231$ & $0.0119$\\ & $10^4$ & $5.9504$ & & $5.9462$ & $0.0042$\\ \bottomrule \end{tabular} \caption{The decomposition of Theorem~\ref{thm:decomp} for the Davenport--Heilbronn function, bump weight $a=1.2$ (script \texttt{ec\_phase2\_controls.py dh}).}\label{tab:dhdecomp} \end{table} \textbf{A planted off-line zero.} To see Theorems~\ref{thm:decomp}, \ref{thm:explode} and~\ref{thm:landau} at work on a configuration in which $\Theta$ is known, we kept the zeros of $\zeta$ below height $1000$ and moved the first one to $\beta+i\gamma_1$, adding $1-\beta+i\gamma_1$ and the conjugates, and computed $B_X(2)$ from $\phi_X=-\sum_\rho\re K_X(\frac12+it-\rho)$ (Table~\ref{tab:planted}). In every case $\int\phi_XP_2-\log|Z_X(2)|$ equals the floor $\Off(2)$ to within $4\cdot10^{-4}$, the window error, and $B_X(2)>\Off(2)$. The maximum of $\phi_X$ on the line agrees with the main term $\int g_X(u)\Ei((\beta-\frac12)u)\,du$ of (L4) to two or three digits once it dominates, and the logarithmic slope of $B_X$ tends to $(\beta-\frac12)-1/\log X$, as $B_X\approx X^{\beta-\frac12}/((\beta-\frac12)\log X)$ predicts: $0.329$ against $0.40-0.072=0.328$ at $\beta=0.9$, $X=10^6$. For $\beta=0.55$ the bound first decreases with the on-line wells and turns up only once the main term is positive, near $(\beta-\frac12)\log X\approx0.37$ where $\Ei$ changes sign, with a minimum $0.0081$ near $X=10^5$, three times the value $0.0027$ of the unperturbed zeros. \begin{table}[h]\centering\small \begin{tabular}{ccccccc} \toprule $\beta$ & $X$ & $B_X(2)$ & $\Off(2)$ & $\max\phi_X$ & $\int g_X\Ei((\beta-\frac12)u)du$ & $d\log B_X/d\log X$\\ \midrule $0.55$ & $10^2$ & $0.0129$ & $0.00149$ & $0.98$ & $-0.65$ & $-0.24$\\ $0.55$ & $10^5$ & $0.0081$ & $0.00149$ & $1.17$ & $0.70$ & $-0.02$\\ $0.55$ & $10^6$ & $0.0085$ & $0.00149$ & $1.29$ & $1.04$ & $0.02$\\ $0.60$ & $10^6$ & $0.0998$ & $0.00297$ & $2.84$ & $2.96$ & $0.354$\\ $0.75$ & $10^6$ & $9.363$ & $0.00742$ & $13.54$ & $13.54$ & $0.255$\\ $0.90$ & $10^4$ & $11.904$ & $0.01187$ & $15.93$ & $15.93$ & $0.392$\\ $0.90$ & $10^6$ & $55.710$ & $0.01187$ & $60.32$ & $60.33$ & $0.329$\\ \bottomrule \end{tabular} \caption{The first zero of $\zeta$ moved off the line to $\beta+i\gamma_1$, $s_0=2$, bump weight $a=1.2$; the slope is taken over one decade (script \texttt{ec\_phase2\_controls.py planted}).}\label{tab:planted} \end{table} \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{B_X_plot.png} \caption{The Euler--Caceres bound (data of Table~\ref{tab:barrier} and of the window-$600$ computation above). (a) $B_X(s_0)$ for $\zeta$ at four centres, with $\kappa_1\varpi(s_0)/\log X$ (dashed) and the floor $0.0386$ of Dirichlet mollifiers (dotted; proved for $s_0=1$ only). (b) $(\log X)B_X(2)$, window $|t|\le600$, against $\kappa_1\varpi(2)=0.0402$ and $\kappa_1\varpi_{600}(2)$. (c) $\zeta$ (solid) against the Davenport--Heilbronn function (dashed). Panels (a) and (c) use a Gaussian weight $g_0$ of standard deviation $0.4$, with the prime sum truncated at $4.5$ standard deviations, on $|t-t_0|\le300$ ($|t|\le2000$ at $s_0=1$).}\label{fig:BX}\end{figure} \section{The geometry behind the law}\label{sec:geometry} Figure~\ref{fig:geom} shows the factorization. On the critical line the truncated Euler product does not converge and oscillates around $|\zeta|$ (the Euler product inside the critical strip is studied in {\cite{CE8}}), while $E^*_X$ follows $|\zeta(\frac12+it)|$ as a zero-free envelope built from the primes, which passes over each zero without touching it: with the Gaussian weight of Table~\ref{tab:barrier}, $|E^*_X(\rho_1)|=0.097$ for $X=10^2$ and $0.048$ for $X=10^4$ (script \texttt{mollifier\_tables.py}), and the zero is carried by $Z_X$ alone; $|Z_X|$ is close to $1$ except for a well of width of order $1/\log X$ at each zero, with the profile $e^{m\Ci(|v|)}$ of Lemma~\ref{lem:profile} and an overshoot of height $e^{0.472m}$. Theorem~\ref{thm:law} is the statement that each well costs exactly $\kappa_mP_{s_0}(\gamma)/\log X$ in the Jensen gap: the wells narrow as $X$ grows, and their cost in the mean square does not depend on anything else. Off the line the picture changes: the factor of a zero $\rho$ grows like $\exp\big(X^{\beta-\sigma}/((\beta-\sigma)\log X)\big)$ where $\sigma<\beta$, a \emph{volcano} to the left of the zero, which for an off-line zero crosses the critical line; this is the mechanism of Theorem~\ref{thm:explode}. From the right, in $\re s>\frac12$, a zero on the line is seen only as a \emph{crater}. In this language, Corollary~\ref{cor:criteria}(a) and Theorem~\ref{thm:explode} say that the bound decays when only craters reach the critical line and, if $\Theta$ is attained, explodes when a volcano crosses it. \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Geometry_ZX.png} \caption{(a) $|\zeta(\frac12+it)|$, the truncated Euler product $\prod_{p\le X}|1-p^{-s}|^{-1}$ with $X=100$, and $|E^*_X|$ for $X=10^2,10^4$. (b) $|Z_X|$ on the line: wells with the profile $e^{\Ci(|v|)}$. (c) $\log|Z_X(\sigma+it)|$ for $\zeta$, $X=10^3$: craters on the line, volcanoes to the left. (d) The same for the Davenport--Heilbronn function near its off-line zero $0.8085+85.699i$: its volcano crosses the line.}\label{fig:geom}\end{figure} \subsection{The zero factor seen through the cross-energy}\label{sec:zerofactor} The factorization separates $|\zeta|$ into a zero-free factor built from the primes and a factor carried by the zeros near the point. Here we look at both through the cross-energy $C_2$ of consecutive partial sums of $\zeta$ {\cite{CE1}}: the radius of the helix that $C_2$ describes on the critical line is $|E^*_X||Z_X|$, and $Z_X$ can be computed from the partial sums alone, as the field of the charges of $\log|\zeta|$, without dividing by $E^*_X$ and without a list of zeros. The computations of this subsection use for $g_0$ the Gaussian of standard deviation $0.4$ of Table~\ref{tab:barrier}, truncated at nine standard deviations (so $a=3.6$, and the prime sum stops at $e^{3.6}X$; the jumps of the truncated Gaussian, below $10^{-17}$, are invisible in double precision), and $X=1000.5$; the scripts are in \texttt{anc/zero\_factor/}. \begin{remark}[the zeros act locally]\label{rem:locality} Let $s=\sigma+it$ with $0\le\sigma\le1$. For a zero $\rho$ with $|\gamma-t|\ge1$, (L2) and (L3) of Lemma~\ref{lem:kernel}, with $x_{\max}=1$, give $|K_X(s-\rho)|\le C_Ae^{a}X^{(\beta-\sigma)_+}|t-\gamma|^{-A-1}/L$ for every $A$, and since there are $O(\log(|t|+k+2))$ zeros with $k\le|\gamma-t|1-2K$, and $n\ge2$. Then \[ \zeta(s)=\tfrac12n^{\bar s}\,\mathcal W(n,s)-\frac{n^{1-s}}{1-s}+\tfrac12n^{-s}+\sum_{k=1}^K\frac{B_{2k}}{(2k)!}(s)_{2k-1}n^{-s-2k+1}+O_K\big(|(s)_{2K+1}|\,n^{-\sigma-2K}\big). \] In particular, for $0<\sigma<1$, \[ \mathcal W(n,s)=2\zeta(s)\,n^{-\bar s}+\frac{2n^{1-2\sigma}}{1-s}-n^{-2\sigma}-\frac s6\,n^{-2\sigma-1}+O\big(|(s)_3|\,n^{-2\sigma-2}\big), \] and on the critical line, for $|t|\ge1$, \[ C_2(n,\tfrac12+it)=\frac1{\frac14+t^2}+2n^{-1/2}\re\big(n^{it}\zeta(\tfrac12+it)\big)-\frac1n+O\big(|s|^3n^{-2}\big). \] \end{lemma} \begin{proof} The Euler--Maclaurin formula for $\sum_{k1-2K$, $\widetilde B_j$ the periodic Bernoulli functions; and $D_{n-1}(s)=\frac12n^{\bar s}\mathcal W(n,s)$. Multiply by $2n^{-\bar s}$ for the second statement, and take real parts with $\sigma=\frac12$ for the third. \end{proof} \begin{remark}[the radius of the helix]\label{rem:helix} By Lemma~\ref{lem:c2zero}, on the critical line $\frac12\mathcal W(n,s)$ winds around the centre $\frac1{1-s}$ with radius $|\zeta(\frac12+it)|\,n^{-1/2}$ and angle $t\log n$, so that $\sqrt n\,\big(\frac12\mathcal W(n,s)-\frac1{1-s}\big)$ is, up to $O(|s|^3n^{-1/2})$, a helix of radius $|\zeta(\frac12+it)|$ {\cite[Thm.~3.2]{CE1}}, and the oscillating part of $C_2$, rescaled by $\sqrt n$, is $2\re(n^{it}\zeta(\frac12+it))$, with envelope $\pm2|\zeta(\frac12+it)|$. By Lemma~\ref{lem:factor} the radius factorizes as $|E^*_X(s)|\,|Z_X(s)|$: the first factor comes from the primes, the pole and the trivial zeros, and by Remark~\ref{rem:locality} the second is in effect a finite product over the zeros near $t$. Figure~\ref{fig:localzeros}(a) shows the oscillating part of $C_2$ at $n=2000$ inside the envelope $\pm2|E^*_X||Z_X|$ computed with the zeros within distance $20$; the prime factor $2|E^*_X|$ alone follows the envelope away from the zeros and passes over each zero without touching it. At a simple zero the collapse of the helix is carried by the single factor \[ e^{-K_X(s-\rho)}=e^{\gamma_E+\int g_X(u)\log u\,du}\,(s-\rho)\big(1+O_X(|s-\rho|)\big), \] since $E_1(z)=-\gamma_E-\log z+\Ein(z)$, and the remaining factors give \[ \zeta'(\rho)=E^*_X(\rho)\,e^{\gamma_E+\int g_X(u)\log u\,du}\prod_{\rho'\ne\rho}e^{-K_X(\rho-\rho')} \] (see {\cite[Rem.~4]{CE8}}, where this form is checked numerically), the constant that governs the behaviour of $C_2$ near a zero {\cite[Sect.~2]{CE2}}. Status: the factorization of the radius and the expansion at a zero are proved (Lemmas~\ref{lem:factor} and~\ref{lem:c2zero}); the figure is numerical (\texttt{fig\_local\_zeros.py}). \end{remark} \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Euler_product_local_zeros.png} \caption{The factorization $\zeta=E^*_XZ_X$ seen by $C_2$, Gaussian weight of standard deviation $0.4$, $X=1000.5$. (a) $C_2(n,\frac12+it)$ minus its centre $1/(\frac14+t^2)$ and its Bernoulli terms, multiplied by $\sqrt n$, for $n=2000$, against $\pm2|E^*_X||Z_X|$ with the zeros $|\gamma-t|\le20$ and against the prime factor $2|E^*_X|$ alone; dots: zeros. (b) The relative error of $E^*_XZ_X$ against the window $W$ of zeros used, at four points (Table~\ref{tab:locality}); dotted: the sharp cut-off at $x=1000.5$ with the $1517$ zeros below height $2000$.}\label{fig:localzeros}\end{figure} \paragraph{The prime factor predicts the gaps.} The prime factor also sees how far apart the zeros are. For each of the $1396$ gaps between consecutive ordinates $\gamma_k<\gamma_{k+1}$ with $100<\gamma_k<1900$, let $M_k$ be the maximum of $\log|E^*_X(\frac12+it)|$ over the gap, with the weight and the value of $X$ of this subsection, and let $d_k=(\gamma_{k+1}-\gamma_k)\frac1{2\pi}\log\frac{\gamma_k}{2\pi}$ be the normalized gap. $M_k$ is computed from the prime powers up to $3.7\cdot10^4$ alone; the zeros enter only to delimit the gaps. The Spearman rank correlation of $M_k$ with $d_k$ is $0.951$, as high as that of the maximum of $\log|\zeta|$ itself over the gap ($0.947$). The extreme quartiles do not overlap: every gap in the top quartile of $M_k$ has $d_k\ge1.00$, and every gap in the bottom quartile has $d_k\le0.89$ (the median of $d_k$ is $0.96$, its minimum $0.14$). This is the explicit formula seen through the factorization, not a constraint on the zeros: $\log|E^*_X|=\log|\zeta|-\log|Z_X|$, and $\log|Z_X|$ is small between the wells, so $E^*_X$ inherits the classical correlation between the size of $\zeta$ in a gap and the length of the gap. It does not predict the backward collision times of the zeros under the heat flow, which are fixed by the closest pair in a neighbourhood {\cite[Sect.~6.2]{CE8}}. Status: numerical (\texttt{gap\_predictor.py}). Since $E^*_X$ has no zeros in $\re s>-2$, $Z_X=\zeta/E^*_X$ is determined there by $\zeta$. It can also be computed from the partial sums alone. Let $\Phi=\log|\zeta|$ on $\mathbb C\setminus\{1\}$, and let $\mu_\Phi=\frac1{2\pi}\Delta\Phi$, in the sense of distributions, be its Riesz measure (its charges); write $D_X(s)=\sum_{n\ge2}\Lambda(n)\psi_X(\log n)n^{-s}/\log n$ for the finite prime sum of \eqref{eq:Estar}. \begin{proposition}[the zero factor from the cross-energy]\label{prop:charges} \begin{enumerate} \item[(i)] $\mu_\Phi=\sum_\rho\delta_\rho+\sum_{k\ge1}\delta_{-2k}-\delta_1$, the zeros counted with multiplicity, and for $\sigma>0$ \[ \Phi(s)=\lim_{n\to\infty}\log\Big|\tfrac12n^{\bar s}\,\mathcal W(n,s)-\frac{n^{1-s}}{1-s}\Big|, \] locally uniformly on $\{\sigma>0\}$ minus the zeros and the point $1$. \item[(ii)] For every $s$ that is not a zero, a trivial zero or the pole, \begin{gather*} \log|\zeta(s)|=\re D_X(s)-\int\re K_X(s-w)\,d\mu_\Phi(w),\\ \log|Z_X(s)|=-\int_{0<\re w<1}\re K_X(s-w)\,d\mu_\Phi(w),\\ Z_X(s)=\exp\Big(-\int_{0<\re w<1}K_X(s-w)\,d\mu_\Phi(w)\Big), \end{gather*} the integrals being absolutely convergent sums, the last one defined modulo $2\pi i$. \end{enumerate} \end{proposition} \begin{proof} (i) $\Phi$ is harmonic off the zeros and the pole; near a zero of multiplicity $m$ it is $m\log|s-\rho|$ plus a harmonic function, near $1$ it is $-\log|s-1|$ plus a harmonic function, and $\Delta\log|s-w|=2\pi\delta_w$. By Lemma~\ref{lem:c2zero} with $K=1$, $\frac12n^{\bar s}\mathcal W(n,s)-\frac{n^{1-s}}{1-s}=\zeta(s)+O(n^{-\sigma})$, uniformly on compact subsets of $\sigma>0$, and $\log|\cdot|$ is continuous where $\zeta\ne0$. (ii) By Lemma~\ref{lem:factor} and \eqref{eq:Estar}, $\log|\zeta(s)|=\re D_X(s)+\re K_X(s-1)-\sum_k\re K_X(s+2k)-\sum_\rho\re K_X(s-\rho)$, and $\re K_X$ is continuous across the cut; the last three terms are $-\int\re K_X(s-w)\,d\mu_\Phi(w)$ over the charge $-1$ at the pole, the charges $+1$ at the trivial zeros and the charges in the strip, which are the nontrivial zeros. The sums converge absolutely by Lemma~\ref{lem:kernel}. The last statement is the definition of $Z_X$. \end{proof} So the prime factor is the finite prime sum together with the field of the charges outside the critical strip, and $Z_X$ is the field of the charges in the strip. In this form $Z_X$ is computed from $C_2$ and $S_2$ alone: $\Phi$ from $\mathcal W(n,\cdot)$ by (i), its charges by a discrete Laplacian, and $Z_X$ from them by (ii), with the truncation in height controlled by Remark~\ref{rem:locality}. The construction locates the zeros as charges; it does not say where they must be. RH is the statement that the charges of $\mu_\Phi$ in the strip lie on $\re w=\frac12$. \paragraph{From far away an off-line pair is a quadrupole.} Let $c=\frac12+i\gamma_0$ and $0<\delta<\frac12$. A pair of charges at $c\pm\delta$ and a double charge at $c$ differ by \[ \log|w-c-\delta|+\log|w-c+\delta|-2\log|w-c|=\log\Big|1-\frac{\delta^2}{(w-c)^2}\Big|=-\re\frac{\delta^2}{(w-c)^2}+O\Big(\frac{\delta^4}{|w-c|^4}\Big) \] for $|w-c|>2\delta$: they have the same total charge, so the counting function $N(T)$ does not distinguish them, and the same dipole moment (zero, by symmetry), and the first difference is a quadrupole of strength $\delta^2$, which decays like $|w-c|^{-2}$. In the zero factor the same replacement changes $\log Z_X(s)$ by $-[K_X(s-c-\delta)+K_X(s-c+\delta)-2K_X(s-c)]=-\delta^2K_X''(s-c)+O(\delta^4\sup|K_X^{(4)}|)$, a second difference of the kernel, which by Lemma~\ref{lem:kernel} is concentrated within $O(1)$ of the height $\gamma_0$. So a symmetric off-line pair enters every quantity built linearly from the charges only at order $\delta^2$; in the same way the product of the cross-energies at a reflected pair studied in {\cite{CE3}} dips below zero only by the fraction $4\delta^2/(1-4\delta^2)$ of its limit, and the local reading of the real parts in {\cite[Rem.~9]{CE8}} sees symmetric pairs only at second order. Status: proved (elementary). \begin{remark}[the charges, computed]\label{rem:chargesnum} We computed $\Phi$ from $\mathcal W(N,\cdot)$, $N=200$, by Lemma~\ref{lem:c2zero} with $K=5$, on a grid of step $h=0.01$ in $0.2<\sigma<0.8$, $5\frac12$ adds $\log|\rho/(1-\rho)|>0$ to the left side. The weight $P_1(t)=1/(2\pi(\frac14+t^2))$ is $\frac1{2\pi}$ times the limit of $C_2(n,\frac12+it)$ (Lemma~\ref{lem:c2zero}). For the Gaussian weight, $\log|P(1)|=-6.857\cdot10^{-11}$, and $\log|Z_X(1)|=6.855\cdot10^{-11}$ from the $1517$ zeros below height $2000$, which sum to $-1.1\cdot10^{-14}$. This is Theorem~\ref{thm:decomp} at $s_0=1$, not a new criterion: it separates the part of the Jensen moment fixed by the primes from the part carried by the charges. Status: exact given Lemmas~\ref{lem:factor}, \ref{lem:outer} and [BSY]; the numbers are numerical (\texttt{jensen\_split.py}). \end{remark} \section{The explicit formula in cross-energy coordinates}\label{sec:c2} The wells of $Z_X$ are the zeros seen through a smoothed Euler factor; the explicit formula for $\psi(x)=\sum_{n\le x}\Lambda(n)$ sees the same zeros through the primes. Part~5 writes that formula in the coordinates of the cross-energy {\cite[Sect.~2]{CE5}}, and we recall what we use. By Lemma~\ref{lem:c2zero}, $\mathcal W(n,s)\to2/\bar s$ at every point $s=\frac12+it$ of the critical line, so the limits $C_2(\infty,s)$, $S_2(\infty,s)$ of $C_2(n,s)$, $S_2(n,s)$ exist there, and \[ \frac1s=\frac{C_2(\infty,s)-iS_2(\infty,s)}2,\qquad C_2(\infty,\tfrac12+it)=\frac1{\frac14+t^2}=2\pi P_1(t),\qquad S_2(\infty,s)=2t\,C_2(\infty,s) \] {\cite[Prop.~2.6]{CE5}}. These coordinates hold at every point of the line, zero or not; the cross-energy sees a zero on the line only through the rate of convergence, $1/n$ instead of $n^{-1/2}$, and the Jensen moment \eqref{eq:bsy} integrates $\log|\zeta|$ against $C_2(\infty,\cdot)/2\pi$. In them, von Mangoldt's explicit formula becomes, under RH, a sum of cosines in $\log x$, one for each distinct ordinate $\gamma>0$, with amplitude $2m(\gamma)\sqrt{C_2(\infty,\rho)}=2m(\gamma)/|\rho|$, frequency $\gamma$ and phase $\arctan2\gamma=\arg\rho$ {\cite[Thm.~2.7]{CE5}}; the triangle inequality over these amplitudes gives Schoenfeld's constant $\frac1{8\pi}$ {\cite[Rem.~2.9]{CE5}}, and the phases, almost constant, leave the size of $\psi(x)-x$ to the alignment of the numbers $\gamma\log x$ modulo $2\pi$ {\cite[Rem.~2.10]{CE5}}. Squaring the cosines in mean is Cram\'er's theorem [Cr] (see [MV2, Ch.~13]) in these coordinates {\cite[Rem.~2.8]{CE5}}: under RH, with $\Delta(u)=e^{-u/2}\big(\psi(e^u)-e^u\big)$, \begin{equation}\label{eq:cramer} \lim_{U\to\infty}\frac1U\int_0^U\Delta(u)^2\,du=\sum_{\gamma}m(\gamma)^2\,C_2(\infty,\tfrac12+i\gamma)\ \ge\ \sum_\rho C_2(\infty,\rho)=2+\gamma_E-\log4\pi, \end{equation} where $2+\gamma_E-\log4\pi=0.046191\ldots$, the first sum is over the distinct ordinates of both signs, and the second over the zeros with multiplicity, with equality if and only if all zeros are simple. Since $\sum_\rho1/\rho=(\xi'/\xi)(1)$ is Li's first coefficient $\lambda_1$ [Li], the constant is $2\lambda_1$ {\cite[Prop.~2.4]{CE5}}. Replacing the partial sums of $\zeta$ by $\mathcal P_N(s)=\sum_{n\le N}\Lambda(n)n^{-s}$ gives the arithmetic cross-energy, which, unlike $C_2(\infty,\cdot)$, sees a zero $\rho_0$ of multiplicity $m$: under RH, $\mathcal P_x(\rho_0)=\frac{x^{1-\rho_0}}{1-\rho_0}-m\log x+x^{-\rho_0}\big(\psi(x)-x\big)+O_{\rho_0}(1)$, the resonance of Conrad [Con, (6.12)] {\cite[Rem.~2.22]{CE5}}. What follows is specific to the bound: the law of Theorem~\ref{thm:law} at $s_0=1$ and \eqref{eq:cramer} are sums over the zeros of the same kernel. \begin{corollary}[the bound and the energy of the primes]\label{cor:bound_energy} Assume RH. Then \[ \lim_{X\to\infty}(\log X)\,B_X(1)=\frac1{2\pi}\sum_\gamma\kappa_{m(\gamma)}\,C_2(\infty,\tfrac12+i\gamma). \] If all zeros are simple, this equals $\dfrac{\kappa_1}{2\pi}\displaystyle\lim_{U\to\infty}\frac1U\int_0^U\Delta(u)^2\,du=\frac{\kappa_1}\pi\lambda_1=0.013438\ldots$. \end{corollary} \begin{proof} Theorem~\ref{thm:law} at $s_0=1$, where $P_1(t)=\frac{1/2}{\pi(\frac14+t^2)}=\frac1{2\pi}C_2(\infty,\frac12+it)$ (the Poisson kernel of $\re s>\frac12$ at $s_0=1$, with $\sigma_0-\frac12=\frac12$), and \eqref{eq:cramer}. \end{proof} So, under RH and for simple zeros, the Euler--Caceres bound at $s_0=1$ is $\kappa_1/(2\pi)=0.29092$ times the mean square of the normalized error term of the prime number theorem, divided by $\log X$: both are sums over the zeros of the same kernel $C_2(\infty,\rho)=1/|\rho|^2$. They weigh a multiple zero differently, by $\kappa_m$ and by $m^2$ per ordinate, and both exceed their simple-zero values when a zero is multiple (Corollary~\ref{cor:criteria}(c) and \eqref{eq:cramer}). \subsection*{An isometry at \texorpdfstring{$s_0=1$}{s0=1}} The coincidence of kernels in Corollary~\ref{cor:bound_energy} is not an accident of the limit. At $s_0=1$ the Poisson kernel is $P_1(t)=1/(2\pi|\frac12+it|^2)$, so it cancels the factor $s$ that an integration by parts produces in the prime representation of $\log Z_X$, and the Mellin transform in the prime variable becomes an isometry. Recall $G_X(u)=\int_{-\infty}^ug_X$ (proof of Proposition~\ref{prop:onewell}), which vanishes for $uL+a$, and define the \emph{energy of the primes beyond $X$} \[ \mathcal E_\Delta(X)=\int_0^\infty G_X(u)^2\,\frac{\Delta(u)^2}{u^2}\,du, \] so that \[ \int_{Xe^{a}}^\infty\frac{(\psi(x)-x)^2}{(x\log x)^2}\,dx\le\mathcal E_\Delta(X)\le\int_{Xe^{-a}}^\infty\frac{(\psi(x)-x)^2}{(x\log x)^2}\,dx. \] \begin{proposition}[the energy criterion; standard]\label{prop:energy} The following are equivalent: \begin{enumerate} \item[(a)] RH; \item[(b)] $\displaystyle\int_2^\infty\frac{(\psi(x)-x)^2}{(x\log x)^2}\,dx<\infty$; \item[(c)] $\mathcal E_\Delta(X)<\infty$ for some $X$; \item[(d)] $\mathcal E_\Delta(X)\to0$ as $X\to\infty$. \end{enumerate} Under RH, $\mathcal E_\Delta(X)=c/\log X+o(1/\log X)$ with $c=\sum_\gamma m(\gamma)^2C_2(\infty,\frac12+i\gamma)$ as in \eqref{eq:cramer}. \end{proposition} \begin{proof} With $x=e^u$ the integral in (b) is $\int\Delta(u)^2u^{-2}du$. Since $0\le G_X\le1$, $G_X=1$ for $u>L+a$, and $\Delta$ is bounded on compact sets, (b)$\Leftrightarrow$(c), and (d)$\Rightarrow$(c). (c)$\Rightarrow$(a): for $\sigma>\frac12$, by Cauchy--Schwarz, $\int_{L+a}^\infty|\Delta(u)|e^{-(\sigma-\frac12)u}du\le\big(\int\Delta^2u^{-2}\big)^{1/2}\big(\int u^2e^{-(2\sigma-1)u}du\big)^{1/2}<\infty$, so $\int_1^\infty(\psi(x)-x)x^{-s-1}dx=\int_0^\infty\Delta(u)e^{-(s-\frac12)u}du$ converges absolutely and is analytic in $\re s>\frac12$. For $\re s>1$ it equals $-\frac{\zeta'(s)}{s\zeta(s)}-\frac1{s-1}$ [D, Ch.~17]; hence $\zeta'/\zeta$ has no pole in $\re s>\frac12$ other than $s=1$, which is RH. (a)$\Rightarrow$(d) and the asymptotic: under RH, $A(U)=\int_0^U\Delta^2=cU+o(U)$ by \eqref{eq:cramer}, and partial integration gives $\int_{L+a}^\infty\Delta^2u^{-2}du=-A(L+a)(L+a)^{-2}+2\int_{L+a}^\infty A(u)u^{-3}du=c/L+o(1/L)$. For the range $[L-a,L+a]$, which contributes $O(L^{-2})$, we use the bound $\int_V^{V+1}\Delta^2\ll1$ uniformly in $V$, that is, $\int_Y^{2Y}(\psi(x)-x)^2dx\ll Y^2$ under RH [MV2, Ch.~13]. \end{proof} The criterion is a form of the classical equivalence between RH and a polynomially bounded mean square of $\Delta$: (c)$\Rightarrow$(a) is the Mellin argument of [I], valid for any weight $u^{-A}$, and (a)$\Rightarrow$(b) is Cram\'er's theorem, valid for any weight $u^{-1-\varepsilon}$ but not for $u^{-1}$. We state it with the weight $u^{-2}$ because that is the weight the isometry below produces. \begin{theorem}[an isometry at $s_0=1$]\label{thm:isometry} Assume RH, let $X>e^{a+1}$, let $\log Z_X(s)=-\sum_\rho K_X(s-\rho)$ (analytic in $\re s>\frac12$, and defined on the line off the ordinates), and put $F_X(u)=G_X(u)\Delta(u)/u$, $\widehat F_X(t)=\int F_X(u)e^{-itu}du$ (the Fourier transform in $L^2$). Then, for almost every $t$, with $s=\frac12+it$, \[ \log Z_X(s)=s\,\widehat F_X(t)+R_X(t),\qquad R_X(t)=\int\Delta(u)\Big(\frac{G_X(u)}{u^2}-\frac{g_X(u)}u\Big)e^{-itu}\,du+\sum_{k\ge1}K_X(s+2k), \] and \[ \int\big|s\,\widehat F_X(t)\big|^2P_1(t)\,dt=\mathcal E_\Delta(X),\qquad\int|R_X(t)|^2P_1(t)\,dt=O\Big(\frac1{\log^2X}\Big). \] Consequently $\displaystyle\int|\log Z_X(\tfrac12+it)|^2P_1(t)\,dt=\mathcal E_\Delta(X)+O\big((\log X)^{-3/2}\big)=\frac c{\log X}+o\Big(\frac1{\log X}\Big)$. \end{theorem} \begin{proof} For $\re s>1$, \eqref{eq:Estar} and $\log\zeta(s)=\sum\Lambda(n)n^{-s}/\log n$ give, since $\int_1^\infty G_X(\log x)x^{-s}dx/\log x=K_X(s-1)$, \[ \log Z_X(s)=\int_1^\infty G_X(\log x)\frac{x^{-s}}{\log x}\,d\big(\psi(x)-x\big)+\sum_{k\ge1}K_X(s+2k). \] (Here $\log\zeta-\log E^*_X$ and $-\sum_\rho K_X(s-\rho)$ are both analytic logarithms of $Z_X$ in $\re s>1$ that tend to $0$ as $\re s\to+\infty$, by (L1) of Lemma~\ref{lem:kernel}, so they coincide.) Put $x=e^u$, $\psi(e^u)-e^u=e^{u/2}\Delta(u)$, and integrate by parts; there are no boundary terms, because $G_X(u)=0$ for $u1$. Since $\frac d{du}\big(G_Xe^{-su}/u\big)=\big(g_X/u-sG_X/u-G_X/u^2\big)e^{-su}$, this gives $\log Z_X(s)=s\int F_X(u)e^{-(s-\frac12)u}du+\int F_{2}(u)e^{-(s-\frac12)u}du+\sum_kK_X(s+2k)$, with $F_2=\Delta(G_X/u^2-g_X/u)$. Under RH, $F_X$ and $F_2$ lie in $L^2(0,\infty)$ (Proposition~\ref{prop:energy} and the bound $\int_V^{V+1}\Delta^2\ll1$ [MV2, Ch.~13]), so their Laplace transforms belong to the Hardy space $H^2$ of $\re s>\frac12$ (Paley--Wiener); $\log Z_X$ is analytic there because $Z_X$ has no zeros in $\re s>\frac12$, so the identity holds in $\re s>\frac12$ by uniqueness. As $\sigma\downarrow\frac12$ the $H^2$ functions converge in $L^2(dt)$ to $\widehat F_X$ and $\widehat F_2$, while $\log Z_X(\sigma+it)\to\log Z_X(\frac12+it)$ for every $t$ that is not an ordinate; along a subsequence the convergence is also a.e.\ on the right, which gives the identity a.e. Since $|s|^2P_1(t)=1/(2\pi)$, Plancherel gives $\int|s\widehat F_X|^2P_1=\frac1{2\pi}\int|\widehat F_X|^2=\int F_X^2=\mathcal E_\Delta(X)$. For the remainder, $P_1\le2/\pi$ and Plancherel give $\int|\widehat F_2|^2P_1\le4\int F_2^2\le8\int_{L-a}^\infty\Delta^2u^{-4}+8\|g_0\|_\infty^2(L-a)^{-2}\int_{L-a}^{L+a}\Delta^2=O(L^{-2})$, and $|K_X(s+2k)|\le\int g_X(u)e^{-(2k+\frac12)u}du/((2k+\frac12)u)$ is $O(X^{-5/2})$ after summation. The last statement follows by expanding the square, Cauchy--Schwarz for the cross term, and Proposition~\ref{prop:energy}. (The exponent $\frac32$ is not sharp: numerically both correction terms are below $10^{-5}/\log X$ from $\log X=10$ on.) \end{proof} \begin{corollary}[the bound against the energy of the primes]\label{cor:ratio} Assume RH. Then \[ \lim_{X\to\infty}\frac{B_X(1)}{\mathcal E_\Delta(X)}=\frac{\sum_\gamma\kappa_{m(\gamma)}C_2(\infty,\frac12+i\gamma)}{2\pi\sum_\gamma m(\gamma)^2C_2(\infty,\frac12+i\gamma)}, \] which equals $\kappa_1/(2\pi)=0.29092\ldots$ if all zeros are simple. \end{corollary} \begin{proof} Theorem~\ref{thm:law} at $s_0=1$ and Proposition~\ref{prop:energy}. \end{proof} \emph{What the shared kernel does not give.} Corollaries~\ref{cor:bound_energy} and~\ref{cor:ratio} and Theorem~\ref{thm:isometry} are statements under RH: two sums over the zeros with the same kernel, and an isometry whose proof uses $Z_X\ne0$ in $\re s>\frac12$. The bounds $\mathcal E_\Delta(X)=O(1/\log X)$ and $B_X(1)=O(1/\log X)$ are equivalent, but only because each is equivalent to RH (Proposition~\ref{prop:energy} and Corollary~\ref{cor:criteria}); the equivalence goes through RH, not through the kernel. Without RH, the shared kernel gives no inequality between $\mathcal E_\Delta(X)$ and $B_X(1)$ at a given $X$, nor between their constants. \begin{remark}[the wells and the primes compute the same number] Under RH, the function $\log Z_X=-\sum_\rho K_X(s-\rho)$ is a sum of wells, and $\int_{\mathbb R}|E_1(iv)|^2dv=2\int_0^\infty(\Ci^2+\operatorname{si}^2)=2\pi$, with $\operatorname{si}=\operatorname{Si}-\frac\pi2$; hence \[ \int_{\mathbb R}|K_X(iy)|^2\,dy=\frac{2\pi}{\log X}\Big(1+O\Big(\frac1{\log X}\Big)\Big). \] Summing over the zeros by dominated convergence (with (L2) of Lemma~\ref{lem:kernel} and $\sum_\gamma m(\gamma)^2P_1(\gamma)<\infty$), the diagonal part of $\int|\log Z_X|^2P_1$ is $\sum_\gamma m(\gamma)^2P_1(\gamma)\cdot2\pi/\log X+o(1/\log X)=c/\log X+o(1/\log X)$, the value given by the primes in Theorem~\ref{thm:isometry}. Consequently the interaction of the wells in this quadratic mean is $o(1/\log X)$, the counterpart, on the side of the zeros, of the vanishing of the off-diagonal terms in Cram\'er's theorem. Of the energy $2\pi/\log X$ of a simple well, the real part $\phi_X$ carries half, $\pi/\log X$; the Jensen gap does not see it quadratically but through $h(x)=e^{2x}-1-2x$, which turns $\pi$ into $\kappa_1=1.828$ because $\phi_X\to-\infty$ at the zero. This is the factor $\kappa_1/(2\pi)$ of Corollary~\ref{cor:ratio}: the bound is the energy of the primes beyond $X$, compressed by the logarithmic singularity of the wells. \end{remark} \textbf{Numerics.} Table~\ref{tab:energy} compares the two sides of Theorem~\ref{thm:isometry}, with the bump weight of half-width $a=1$. The primes side is computed from $\psi(x)$ for $x\le10^9$, on a grid of step $2\cdot10^{-4}$ in $u$: first $\mathcal E_\Delta(X)$ alone, then $\mathcal E_\Delta(X)$ with the contribution of $R_X$ (its cross term with $s\widehat F_X$ and its square, by a discrete Fourier transform on the same grid). The part $u>\log10^9$ of $\mathcal E_\Delta(X)$ cannot be computed and is replaced by its value $c/\log10^9$ under RH; it is shown in the last column and is a large part of the primes side for large $X$, so that there the comparison tests the identity only on $[\log X-a,\log10^9]$. (Moving the cut from $u=16$ to $u=20.72$ changes every entry by at most $0.0006$: $\int_0^U\Delta^2-cU$ is nearly constant on $[8,20.7]$, and a constant offset cancels in the partial integration.) The zeros side is $\int|\log Z_X(\frac12+it)|^2P_1(t)\,dt$ computed from the first $1000$ zeros, with the diagonal value for the zeros above them. With $R_X$ included the two sides agree to $1\%$ or better at every $X$, including $X=20$; $\mathcal E_\Delta(X)$ alone agrees within $2\%$ from $X\approx400$ on, the difference for small $X$ being mainly the cross term. \begin{table}[h]\centering\small \begin{tabular}{rrcccc} \toprule & & \multicolumn{2}{c}{primes} & zeros & \\ \cmidrule(lr){3-4}\cmidrule(lr){5-5} $\log X$ & $X$ & $L\,\mathcal E_\Delta(X)$ & with $R_X$ & $L\int|\log Z_X|^2P_1$ & tail $u>\log10^9$\\ \midrule $3$ & $20$ & $0.0727$ & $0.0434$ & $0.0437$ & $0.0067$\\ $4$ & $55$ & $0.0532$ & $0.0444$ & $0.0446$ & $0.0089$\\ $5$ & $148$ & $0.0470$ & $0.0443$ & $0.0444$ & $0.0111$\\ $6$ & $403$ & $0.0455$ & $0.0446$ & $0.0447$ & $0.0134$\\ $8$ & $2981$ & $0.0448$ & $0.0447$ & $0.0449$ & $0.0178$\\ $10$ & $2.2\cdot10^4$ & $0.0451$ & $0.0451$ & $0.0453$ & $0.0223$\\ $12$ & $1.6\cdot10^5$ & $0.0457$ & $0.0457$ & $0.0459$ & $0.0268$\\ $14$ & $1.2\cdot10^6$ & $0.0455$ & $0.0455$ & $0.0459$ & $0.0312$\\ \bottomrule \end{tabular} \caption{Theorem~\ref{thm:isometry}, $L=\log X$: the energy of the primes beyond $X$, alone and with the remainder $R_X$, against the mean square of $\log Z_X$ on the critical line with the weight $P_1$; the limit is $c=2+\gamma_E-\log4\pi=0.046191$. The last column is the part of the primes side that comes from $u>\log10^9$, where $\Delta$ was replaced by its mean square under RH.}\label{tab:energy} \end{table} \begin{remark}[what the coordinates say about the bound]\label{rem:phases} By Corollary~\ref{cor:criteria}(a) the bound is equivalent to the location of the zeros, and under RH its leading term $\Omega(s_0)/\log X$ involves only the ordinates and their multiplicities. The simultaneous alignment of the numbers $\gamma\log x$ modulo $2\pi$, which decides the size of $\psi(x)-x$ at a given $x$ {\cite[Rem.~2.10]{CE5}}, does not enter $\Omega(s_0)$, and the differences of the ordinates enter $B_X$ only at second order, through the interaction $\mathcal I_X$ of Theorem~\ref{thm:second}. Off the line, if $\beta>\frac12$, the coefficient $1/\rho$ sits in the normalized cross-energy at the reflected zero $1-\bar\rho$, and $\mathcal W(n,1-\bar\rho)\sim2n^{2\beta-1}/\bar\rho$ grows like the square of the factor $x^{\beta-\frac12}$ by which the term of $\rho$ outgrows $\sqrt x$ {\cite[Prop.~2.6 and Rem.~2.10]{CE5}}; this is the cross-energy counterpart of Theorem~\ref{thm:explode}, and the criterion of {\cite[Cor.~3.5]{CE1}} (RH holds if and only if $C_2(\cdot,\rho)$ is bounded at every zero) is a restatement of $\beta=\frac12$ through this reflection. \end{remark} \section{What a proof of the bound would need}\label{sec:obstruction} By Corollary~\ref{cor:criteria} the bound $B_X(s_0)=O(1/\log X)$ is RH. In this section we ask what a proof of it would have to contain. We look at a function-field model, where RH is Weil's theorem and the bound holds (Section~\ref{sec:ff}); at the prime expansion over $\mathbb Q$, which is not absolutely convergent (Section~\ref{sec:overQ}); at the energy of $\zeta/E^*_X$ on vertical lines (Section~\ref{sec:abscissa}); at Chebyshev's identity evaluated at a zero (Section~\ref{sec:chebyshev}); at Beurling systems that share the zeros of $\zeta$ and have one more, real, zero, and at the reflection asymmetry of $Z_X$ (Section~\ref{sec:bdr}); and at Weil's quadratic form (Section~\ref{sec:weil}). The propositions are proved; the conclusions about what a proof would need are heuristic. \subsection{A function-field model}\label{sec:ff} \subsubsection*{Setting} Let $q$ be prime, $M\in\mathbb F_q[T]$ monic irreducible of degree $d$, and $\chi$ a character of $(\mathbb F_q[T]/M)^*$ that is nontrivial on $\mathbb F_q^*$. Then $L(u,\chi)=\sum_{f\text{ monic}}\chi(f)u^{\deg f}$ is a polynomial of degree $D=d-1$, \[ L(u,\chi)=\prod_{j=1}^D(1-\alpha_ju),\qquad \log L(u,\chi)=\sum_{n\ge1}\frac{c_n}nu^n,\qquad c_n=\sum_{\deg f=n}\Lambda(f)\chi(f)=-\sum_j\alpha_j^n, \] and RH, $|\alpha_j|=\sqrt q$, is Weil's theorem [W] (applied to the curve of the associated cover, whose zeta function has $L(u,\chi)$ as a factor; see [R]). With $u=q^{-s}$ the critical line is the circle $|u|=R:=q^{-1/2}$, the half-plane $\re s>\frac12$ is the disc $|u|N-a}(|\alpha_j|R)^n/n=O\big((\max_{|\alpha_j|<\sqrt q}|\alpha_j|R)^{N-a}\big)$ for the zeros outside the closed disc. (These bounds, and those for $c_N$ below, do not use Lemma~\ref{lem:sep}, which uses them.) With $T_M(\varphi)=\sum_{n\ge M}\cos(n\varphi)/n$, summation by parts against the non-decreasing $G_0$ gives $c_N(\varphi)=-\sum_M\big(G_0(M-N)-G_0(M-1-N)\big)T_M(\varphi)$, a convex combination of the $-T_M$ with $M>N-a$. For $0<|\varphi|\le\pi$, Abel summation with $|\sum_{k\le n}\cos k\varphi|\le\frac\pi{2|\varphi|}+\frac12$ gives $|T_M(\varphi)|\le(\pi/|\varphi|+1)/M$; and if $M|\varphi|<1$, the terms with $M\le n<1/|\varphi|$ are positive and add up to at most $\log\frac1{M|\varphi|}+1$. Hence, for $N>a+1$ and $0<|\varphi|\le\pi$, \[ c_N(\varphi)\le\pi+1,\qquad|c_N(\varphi)|\le\frac{\pi/|\varphi|+1}{N-a},\qquad|c_N(\varphi)|\le\pi+2+\big|\log\big((N-a)|\varphi|\big)\big|, \] the analogues of \eqref{eq:cbounds}, and $c_N(v/N)\to\Ci(|v|)$ for every $v\ne0$ (a Riemann sum for $-\int_1^\infty\cos(vy)\,dy/y$, since $G_0(n-N)\to\mathbf 1_{n>N}$ in the scale $n=Ny$). The Riemann sum converges because the tail bound $|T_M(\varphi)|\le(\pi/|\varphi|+1)/M$ controls the improper integral uniformly. With these four facts the proof of Theorem~\ref{thm:law} applies on the circle, with changed constants (for instance $|c_N(v/N)|\le(\pi N/(N-a)+1)/|v|\le5/|v|$ for $1\le|v|\le\pi N$ in place of $3/|v|$): there are finitely many wells, no tail, $\log|Z_N(u_0)|=O(\tau_*^N)$ for some $\tau_*<1$, and $\int h(\phi_N)P_{u_0}\,d\theta$ times $N$ tends to $2\sum_\theta\kappa_{m(\theta)}P_{u_0}(\theta)$. \emph{(c)$\Rightarrow$(d)} is trivial. \emph{Not (a) $\Rightarrow$ not (c).} (This also follows from (c)$\Rightarrow$(d)$\Rightarrow$(a), but it gives an explicit lower bound.) If $u_0$ is a zero, $B_N(u_0)=+\infty$. Otherwise, if some zero $u_j$ lies in $|u|1$, and let $J$ be the set of zeros with $R|\alpha_j|=\nu$, $\alpha_j=\nu\sqrt qe^{i\theta_j}$, with multiplicities $m_j$. Fix $r'\in(|u_0|,R)$ with $\rho':=\nu r'/R>1$ such that no zero of $L$ lies on the circle $|u|=r'$; then $\log|1-\alpha_ju|$ is bounded on that circle for every $j$. We may assume that $u_0$ is not a zero (otherwise $B_N(u_0)=+\infty$). \emph{Reduction to a point:} $|Z_N|^2$ is subharmonic and continuous on the closed disc, so $|Z_N(u_1)|^2\le\int|Z_N|^2P_{u_1}$ for $|u_1|=r'$, and $P_{u_1}\le HP_{u_0}$ with $H=\frac{(R+r')(R+|u_0|)}{(R-r')(R-|u_0|)}$ (Harnack); hence $B_N(u_0)\ge\log|Z_N(u_1)|-\log|Z_N(u_0)|-\frac12\log H$. \emph{The dominant zeros:} by \eqref{eq:KNclosed}, for $|z|=\rho'$, \[ -K_N(z)=\log(1-z)+\frac{z^N}N\big(\mathfrak r(z)+O(N^{-1})\big),\qquad\mathfrak r(z)=\sum_{k\le a}\big(1-G_0(k)\big)z^k, \] uniformly in $\arg z$, since $\frac N{N+k}=1+O(\frac{|k|}N)$ for $|k|\le\frac N2$ and the terms with $k<-\frac N2$ are $O(N\rho'^{-N/2})$ relative to $z^N/N$; $\mathfrak r$ is analytic in $|z|>1$. The zeros with $R|\alpha_j|<\nu$ contribute $O\big((\rho'')^N/N+1\big)$ to $\log|Z_N(u_1)|$ for some $\rho''<\rho'$, and every zero contributes $O\big((\nu|u_0|/R)^N/N+1\big)$ to $\log|Z_N(u_0)|$. \emph{Choice of $\arg u_1$:} with $u_1=r'e^{i\varphi}$, the main term is $\frac{\rho'^N}N\re\big(e^{iN\varphi}G_{a(N)}(\varphi)\big)$, where $G_a(\varphi)=\sum_{j\in J}m_ja_j\,\mathfrak r(\rho'e^{i(\theta_j+\varphi)})$ and $a(N)=(e^{iN\theta_j})_j$ lies on the torus $\mathbb T^J$. The functions $\varphi\mapsto\mathfrak r(\rho'e^{i(\theta_j+\varphi)})$ are linearly independent: a relation $\sum_jb_j\mathfrak r(\rho'e^{i(\theta_j+\varphi)})\equiv0$ gives $\sum_jb_je^{ik\theta_j}=0$ for every $k\le-a$, where $1-G_0(k)=1$, and the Vandermonde matrix of the distinct $e^{i\theta_j}$ forces $b_j=0$. So $\mu:=\min_{a\in\mathbb T^J}\max_\varphi|G_a(\varphi)|>0$, by continuity and compactness, and the $G_a$ are uniformly Lipschitz in $\varphi$. Taking $\varphi_0$ with $|G_{a(N)}(\varphi_0)|\ge\mu$ and then $\varphi\in[\varphi_0,\varphi_0+2\pi/N)$ with $e^{iN\varphi}G_{a(N)}(\varphi_0)>0$ gives $\re\big(e^{iN\varphi}G_{a(N)}(\varphi)\big)\ge\mu-O(1/N)\ge\mu/2$ for $N$ large. Since $\rho'>\max(1,\nu|u_0|/R,\rho'')$, this gives $B_N(u_0)\ge\frac{\mu\rho'^N}{3N}$ for $N$ large, so $B_N\to\infty$. \end{proof} \begin{lemma}[one frequency per degree]\label{lem:sep} Assume (b) and write $Z_N=\exp\Phi_N$, $\Phi_N(u)=\sum_nb_nu^n$, $b_n=(1-w_N(n))c_n/n$. With $u_0=\tau Re^{i\theta_0}$, $0\le\tau<1$, \[ \int|\Phi_N(Re^{i\theta})|^2P_{u_0}(\theta)\,d\theta=\sum_{m,n}b_m\bar b_nR^{m+n}\tau^{|m-n|}e^{i(m-n)\theta_0}, \] the double series converging absolutely, with sum at most $\frac{1+\tau}{1-\tau}D^2\sum_{n>N-a}n^{-2}=O(1/N)$. Consequently $\int|Z_N|^2P_{u_0}=1+O(1/N)$ and $0\le B_N(u_0)=O(1/N)$. \end{lemma} \begin{proof} The Fourier coefficients of $P_{u_0}$ are $\tau^{|k|}e^{-ik\theta_0}$, $|b_n|R^n\le D/n$, and $|b_mb_n|\le\frac12(|b_m|^2+|b_n|^2)$; sum the geometric series in $|m-n|$. For the energy, (a) gives the one-sided bound $\re\Phi_N=\log|Z_N|\le(\pi+1)D+o(1)$ on the circle (proof of Proposition~\ref{prop:ff}: $c_N\le\pi+1$), so with $h(x)=e^{2x}-1-2x\le C x^2$ for $x\le(\pi+1)D+1$, $\int|Z_N|^2P=1+2\re\Phi_N(u_0)+O(\int|\Phi_N|^2P)=1+O(1/N)$; Jensen gives $B_N\ge0$. \end{proof} \begin{remark}[where Weil enters] In the function-field case the whole bound reduces to (b), and Lemma~\ref{lem:sep} turns (b) into control of the energy. The mechanism is that all prime powers of degree $n$ have the same frequency $n\log q$, so the cancellation among them is already inside the single number $c_n$, and Weil's theorem bounds it. In the Stepanov--Bombieri method [R, Appendix], (b) comes from an auxiliary polynomial that vanishes to high order at the rational points of a curve, built with the Frobenius map, and from the passage to all extensions $\mathbb F_{q^k}$. No counterpart of these ingredients that yields RH is known over $\mathbb Q$. \end{remark} \subsubsection*{Numerical illustration} We took $q=3$, $M=T^8+T^5+2$ (irreducible, with $T$ primitive), and $\chi$ the character of order $3^8-1$ with $\chi(T)=e^{2\pi i/6560}$, which is odd; $L$ has degree $7$ and all its zeros on $|u|=3^{-1/2}$ (checked to $10^{-12}$). With $a=2.5$ and $u_0=R/3$ (the analogue of $\re s_0=\frac32$), $N B_N(u_0)=1.591$, $1.769$, $1.906$, $1.985$, $2.008$, $2.042$ for $N=10,20,40,80,160,320$, against the limit $\kappa_1\sum_jP_{u_0}(\theta_j)=2.063$; the analogue of Theorem~\ref{thm:second} predicts $1.985$ at $N=80$ and $2.043$ at $N=320$. Moving one zero into the disc ($|\alpha_1|=1.05\sqrt q$), the prime sums leave the Weil range from $n\approx50$ on and $B_N$ grows geometrically: $0.20$, $0.28$, $1.97$, $14.8$ for $N=10,20,40,80$ (Figure~\ref{fig:ff}). \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Function_field_bound.png} \caption{The Euler--Caceres bound over $\mathbb F_3[T]$, $\deg L=7$. (a) $N B_N$ for the true $L$-function (all zeros on the circle, by Weil's theorem). (b) $B_N$ with one zero moved into the disc. (c) The normalized prime sums $|c_n|q^{-n/2}$: bounded by $D$ for the true $L$, geometrically growing for the perturbed one.}\label{fig:ff}\end{figure} \subsection{Over \texorpdfstring{$\mathbb Q$}{Q}: no frequency is shared}\label{sec:overQ} Over $\mathbb Q$ the analogue of $\Phi_N$ is the tail of the prime series, which, because $E^*_X$ contains the pole term, is an integral against the signed measure $d\psi(x)-dx$: \[ \log\zeta-\log E^*_X=\int_1^\infty\big(1-w_X(x)\big)\frac{x^{-s}}{\log x}\,\big(d\psi(x)-dx\big)+(\text{trivial zeros}), \] with $w_X(x)=\psi_X(\log x)=1-G_X(\log x)$ in the notation of \eqref{eq:Estar}. Each prime power has its own frequency $\log n$, and no two coincide. \begin{lemma}\label{lem:div} Let $b_n=\Lambda(n)n^{-1/2}/\log n$, $\eta>0$, $X\ge2$. Then, for $Y>4X$, \[ \sum_{X1$, so that $Z_X=\tilde D_X\,e^{-K_X(s-1)}e^{\sum_kK_X(s+2k)}$. Then, unconditionally: \begin{enumerate} \item[(i)] $\tilde b_X(n)\log n=\sum_{d\mid n}\Lambda(d)\,(1-w_X(d))\,\tilde b_X(n/d)$, $\tilde b_X(1)=1$; hence $\tilde b_X(n)\ge0$ for all $n$; \item[(ii)] $\tilde b_X(n)=0$ for $2\le n\le Xe^{-a}$, and $\tilde b_X(p)=1-w_X(p)$ for every prime $p$, so $\tilde b_X(p)=1$ for $p>Xe^a$; \item[(iii)] with $\tilde S_X(x)=\sum_{n\le x}\tilde b_X(n)$ and $\tilde c_X=e^{-A_X(1)}$ (the residue of $\tilde D_X$ at $1$), $\tilde S_X(x)-\tilde c_Xx$ is bounded for $x\ge1$; hence the energy $\int_1^\infty|\tilde S_X(x)-\tilde c_Xx|^2x^{-2}\,dx$ is finite for every $X$, and it equals $\frac1{2\pi}\int|\Psi(\frac12+it)|^2dt$ with $\Psi(s)=\tilde D_X(s)/s-\tilde c_X/(s-1)$. \end{enumerate} (i) is the logarithmic derivative of $\log\tilde D_X=\sum\Lambda(n)(1-w_X(n))n^{-s}/\log n$, a Dirichlet series with nonnegative coefficients; (ii) follows from (i), since $1-w_X(d)=0$ for $d\le Xe^{-a}$ and $=1$ for $d>Xe^a$. For (iii), $e^{-A_X(s)}=\sum_m\eta_X(m)m^{-s}$ is a finite product over $p\le Xe^a$ of exponentials of polynomials in $p^{-s}$, so $\sum_m|\eta_X(m)|<\infty$. Since $\tilde b_X=1*a$, $\tilde S_X(x)=\sum_m\eta_X(m)\lfloor x/m\rfloor=\tilde c_Xx-\sum_m\eta_X(m)\{x/m\}$, with $\sum_m\eta_X(m)/m=e^{-A_X(1)}=\tilde c_X$, so $|\tilde S_X(x)-\tilde c_Xx|\le\sum|\eta_X(m)|$. Then $u\mapsto(\tilde S_X(e^u)-\tilde c_Xe^u)e^{-u/2}$ lies in $L^2(0,\infty)$; its Laplace transform, $\int_1^\infty(\tilde S_X(x)-\tilde c_Xx)x^{-s-1}dx$, equals $\Psi(s)$ for $\re s>1$ and belongs to the Hardy space of $\re s>\frac12$, and Plancherel gives the identity. So the coefficients of $\zeta e^{-A_X}$ have no sign changes, and the analogue of Lemma~\ref{lem:div} holds for them a fortiori, for $Y>4Xe^a$. The energy of their summatory error is finite for every $X$ without any hypothesis, so it carries no information about RH. A finite energy for $\tilde D_X$ says nothing about $Z_X=\tilde D_Xe^{-K_X(s-1)}e^{\sum_kK_X(s+2k)}$, because on the critical line the explicit factor $e^{-K_X(s-1)}$ has size up to $\exp(O(\sqrt X/\log X))$ (near $t=0$, $-\re K_X(s-1)\approx\int g_X(u)\Ei(u/2)\,du$), and the bound for $Z_X$ requires a cancellation between the two factors. Status: (i)--(iii) are proved (elementary), and the vanishing, the values at primes and the boundedness of $\tilde S_X(x)-\tilde c_Xx$ were checked numerically for $X=100$; the remarks after them are heuristic. \begin{remark}[the signed form]\label{rem:signed} For the logarithmic derivative instead of the logarithm (coefficients $\Lambda(n)$ instead of $\Lambda(n)/\log n$), the analogue of the form in Lemma~\ref{lem:div} at $s_0=1$ ($\eta=\frac12$, $t_0=0$), with the range $1\le m,n\le N$ in place of $(X,Y]$, is \[ \sum_{m,n\le N}\frac{\Lambda(m)\Lambda(n)}{\sqrt{mn}}\Big(\frac{\min(m,n)}{\max(m,n)}\Big)^{1/2}=\sum_{m,n\le N}\frac{\Lambda(m)\Lambda(n)}{\max(m,n)}\ \sim\ 2N. \] Its signed form, with the pole removed, is an exact identity for every real $N\ge1$: expanding $\psi(x)^2=\sum_{m,n}\Lambda(m)\Lambda(n)\mathbf 1_{\max(m,n)\le x}$, \[ \sum_{m,n\le N}\frac{\Lambda(m)\Lambda(n)}{\max(m,n)}-\frac{\psi(N)^2}N-2\sum_{n\le N}\Lambda(n)\log\frac Nn+N-1=\int_1^N\frac{(\psi(x)-x)^2}{x^2}\,dx . \] Equivalently, with $\mathcal P_N$ as in Section~\ref{sec:c2} and $\mathfrak D_N(s)=\mathcal P_N(s)-\frac{N^{1-s}}{1-s}+\frac s{1-s}-N^{-s}(\psi(N)-N)=s\int_1^N(\psi(x)-x)x^{-s-1}dx$, Plancherel ($\int P_1(t)e^{i\xi t}dt=e^{-|\xi|/2}$ and $|s|^2P_1=1/2\pi$) gives \[ \int|\mathfrak D_N(\tfrac12+it)|^2P_1(t)\,dt=\int_1^N\frac{(\psi(x)-x)^2}{x^2}\,dx, \] without any hypothesis. This is the analogue, for $-\frac{\zeta'}\zeta(s)-\frac s{s-1}$ truncated at $x\le N$, of the Plancherel step in the proof of Theorem~\ref{thm:isometry}; it is unconditional because the support is compact. The averaged arithmetic cross-energy is elementary, $\int C_2^\Lambda(n,\frac12+it)P_1(t)\,dt=2\Lambda(n)\psi(n-1)/n$. Under RH the right side is $c\log N+o(\log N)$ by \eqref{eq:cramer}; if RH fails, $\limsup_N\log\big(\int_1^N\big)/\log N=2\Theta-1>0$, by the Mellin argument of Proposition~\ref{prop:energy} with the weight $x^{-\delta}$ and by $\psi(x)-x\ll x^{\Theta+\varepsilon}$. So the cancellation that the bound requires is this: a sum of positive terms asymptotic to $2N$, minus single prime sums of total size $2N$ whose individual fluctuations are of order $\sqrt N$, must be $O(N^\varepsilon)$ for every $\varepsilon>0$; this is equivalent to RH, hence to the bound (Corollary~\ref{cor:criteria}). Numerically, with the primes up to $10^6$, the two sides of the identity agree to nine digits, and their difference from $c\log N$ is $2.481$, $2.504$, $2.510$, $2.513$, $2.511$ for $N=10^2,\dots,10^6$. \end{remark} \begin{remark}[positivity alone pays the pole]\label{rem:positivity} The positivity of $\Lambda(n)$ does give a pointwise bound, but not a useful one. Let $\Sigma_X(s)=\sum_n\Lambda(n)\psi_X(\log n)n^{-s}/\log n$ be the prime sum of $\log E^*_X$ in \eqref{eq:Estar}. For each $n$, $3+4\cos\theta+\cos2\theta\ge0$, so $3\re\Sigma_X(\frac12)+4\re\Sigma_X(\frac12+it)+\re\Sigma_X(\frac12+2it)\ge0$, which bounds $-\re\Sigma_X(\frac12+it)$, and hence $\log|Z_X(\frac12+it)|=\log|\zeta(\frac12+it)|-\re\Sigma_X(\frac12+it)-\re K_X(-\frac12+it)+\sum_k\re K_X(\frac12+2k+it)$, from above. The bound is useless: the constant term of the trigonometric polynomial sits at $t=0$, on the pole, where $\Sigma_X(\frac12)\asymp\sqrt X/\log X$. Numerically, with the Gaussian weight of Table~\ref{tab:barrier}, $\Sigma_X(\frac12)=6.6$, $14.0$, $30.8$ for $X=10^2,10^3,10^4$, the resulting upper bound for $\log|Z_X(\frac12+it)|$ at $t=20,30,40$ is about $5$, $10$ and $23$ for these $X$, and the true values lie between $-0.2$ and $0.5$ (script \texttt{mollifier\_tables.py}). Every nonnegative trigonometric polynomial $\sum_ja_j\cos j\theta$ has $a_0>0$, so every argument of this type pays a term of size $X^{1/2}/\log X$ on the critical line; this is a heuristic statement about a class of arguments, not a theorem. \end{remark} \subsection{The energy abscissa}\label{sec:abscissa} For $\sigma\ge\frac12$ let $\mathcal E_\sigma(X)=\int|\zeta/E^*_X(\sigma+it)|^2\,\frac{dt}{\pi(1+t^2)}$ and \[ \sigma_E=\inf\Big\{\sigma\ge\tfrac12:\ \limsup_{X\to\infty}\mathcal E_\sigma(X)<\infty\Big\}. \] \begin{proposition}\label{prop:abscissa} (i) Unconditionally, $\mathcal E_\sigma(X)\to1$ for every $\sigma>\Theta$; hence $\sigma_E\le\Theta$. (ii) If $\Theta>\frac12$ is attained, then $\mathcal E_\sigma(X)\to\infty$ for every $\sigma\in(\frac12,\Theta)$; hence $\sigma_E=\Theta$. In particular, when $\Theta$ is attained, RH holds if and only if $\sigma_E=\frac12$. \end{proposition} \begin{proof} (i) For $\re s=\sigma>\Theta$ every zero has $\re(s-\rho)\ge\sigma-\Theta=:\eta>0$, so by (L1) of Lemma~\ref{lem:kernel}, $|\log|Z_X(\sigma+it)||\le C_\eta X^{-\eta}L^{-1}\sum_\rho(1+|t-\gamma|)^{-2}\le C_\eta X^{-\eta}\log(|t|+3)$; for large $X$, $|Z_X|^2\le(|t|+3)^{1/2}$, which is integrable against $dt/(1+t^2)$, and dominated convergence gives $\mathcal E_\sigma\to1$. Here the pole term of $E^*_X$ is essential: without it the factor $X^{1-\sigma}$ of the pole would make $\mathcal E_\sigma$ explode for $\sigma<1$. (ii) Fix $\sigma\in(\frac12,\Theta)$ and $X$. \emph{Polynomial growth.} $Z_X=\zeta/E^*_X$ is entire (Lemma~\ref{lem:factor}). On $\re s\ge2$, $\zeta$ and $\log E^*_X$ are bounded (for $K_X(s-1)$ and the trivial-zero terms use $|E_1(z)|\le e^{-\re z}/|z|$, $\re z>0$). On $\sigma\le\re s\le2$, $|t|\ge1$: $|\zeta(s)|\ll|t|^{1/2}$ (convexity), the finite prime sum in $\log E^*_X$ is bounded, $K_X(s-1)$ is bounded by (L2) (with $x_{\max}=1$) if $\re s\ge1$ and by the first part of (L3) if $\sigma\le\re s<1$, and the trivial-zero terms are bounded (proof of Lemma~\ref{lem:outer}); the rest is compact and $Z_X$ is continuous there. So $|Z_X(s)|\le C_X(|s|+2)^{1/2}$ on $\re s\ge\sigma$. \emph{Sub-mean-value inequality.} Then $f=Z_X/(s+1)$ is bounded and analytic on $\re s>\sigma$ and continuous up to the line, so $\log|f(s_1)|\le\int\log|f(\sigma+it)|P^{(\sigma)}_{s_1}(t)\,dt$ for $\re s_1>\sigma$, where $P^{(\sigma)}_{s_1}$ is the Poisson kernel of $\re s>\sigma$; and $\log|s+1|$ is the Poisson integral of its boundary values, because $s+1$ has no zeros in $\re s\ge\sigma$ and grows at most linearly. Hence $\log|Z_X(s_1)|\le\int\log|Z_X(\sigma+it)|P^{(\sigma)}_{s_1}(t)\,dt\le\frac12\log\int|Z_X(\sigma+it)|^2P^{(\sigma)}_{s_1}(t)\,dt$, by Jensen's inequality. Since $P^{(\sigma)}_{s_1}(t)\le C_{s_1}/(1+t^2)$, $\mathcal E_\sigma(X)\ge c_{s_1}|Z_X(s_1)|^2$. \emph{Growth at one point.} Take $\rho_0=\Theta+i\tau_0$, and choose $r$ as in part (b) of the proof of Theorem~\ref{thm:explode} and moreover $r<\Theta-\sigma$, so that $s_1=\rho_0-r$ has $\re s_1>\sigma$. Part (c) of that proof uses only Lemma~\ref{lem:kernel} and the position of the zeros, not the half-plane, and gives $\log|Z_X(s_1)|\ge\frac12\Ei(r(\log X-a))-C\log\log X\to\infty$. Hence $\mathcal E_\sigma(X)\to\infty$. The last statement follows: under RH, $\Theta=\frac12$ is attained and (i) gives $\sigma_E=\frac12$; if $\Theta>\frac12$ is attained, $\sigma_E=\Theta>\frac12$. \end{proof} \begin{remark} When $\Theta$ is attained, Proposition~\ref{prop:abscissa} is a mean-square analogue of the classical fact that $\Theta$ is the abscissa of convergence of $\sum(\Lambda(n)-1)n^{-s}$: the energy of the Euler--Caceres quotient on vertical lines sees exactly the real parts of the zeros. It does not make the bound easier: for $\frac12<\sigma<1$ the absolute series of the arithmetic expansion of $\mathcal E_\sigma$ diverges as in Lemma~\ref{lem:div}, so in the whole region where the question lives only cancellation can help. \end{remark} \textbf{Numerics.} Table~\ref{tab:abscissa} gives $\log\mathcal E_\sigma(X)$ for $\zeta$ and for the Davenport--Heilbronn function $f$ (Gaussian weight $g_0$ of standard deviation $0.4$, which is not compactly supported, with the prime sum truncated at $4.5$ standard deviations; $|t|\le300$). For $\zeta$ the energy tends to $1$ on every line, as Proposition~\ref{prop:abscissa}(i) predicts under RH for $\sigma>\frac12$ and Corollary~\ref{cor:energy} on the critical line, where $\mathcal E_{1/2}=\mathcal E/\pi$. For $f$ it explodes on $\sigma=0.5$ and $\sigma=0.6$, below its off-line zero $\rho_f=0.8085+85.699i$. On $\sigma\ge0.75$ nothing is visible up to $X=10^4$, although $f$ has zeros to the right of these lines: the volcano of $\rho_f$ must first overcome the weight $w=1/(\pi(1+\gamma_f^2))\approx4.33\cdot10^{-5}$ of its height. Using the lower bound $\Ei(y)$, $y=(\beta_f-\sigma)\log X$, of Lemma~\ref{lem:kernel}(L4) for the volcano, a heuristic onset is $2\Ei(y)\approx\log\frac1{w(\beta_f-\sigma)}$, that is, $\log X\approx7.0$, $10.6$ and $40$ for $\sigma=0.5$, $0.6$ and $0.75$; this ignores the width of the lobe and the shift $\log X-a$, and is consistent in order of magnitude with the table. The third value is out of numerical reach. The energy abscissa is an asymptotic notion, and a finite computation sees it only for zeros that are low or far from the line. \begin{table}[h]\centering\small \begin{tabular}{lcccc} \toprule & $X=10$ & $10^2$ & $10^3$ & $10^4$\\ \midrule $\zeta$, $\sigma=0.5$ & $0.018$ & $0.010$ & $0.007$ & $0.005$\\ $\zeta$, $\sigma=0.6$ & $0.009$ & $0.002$ & $0.001$ & $0.000$\\ $\zeta$, $\sigma=0.75,\,0.9,\,1,\,1.25$ & $\le0.003$ & $0.000$ & $0.000$ & $0.000$\\ \midrule $f$, $\sigma=0.5$ & $0.080$ & $0.067$ & $1.876$ & $12.097$\\ $f$, $\sigma=0.6$ & $0.039$ & $0.014$ & $0.034$ & $0.950$\\ $f$, $\sigma=0.75$ & $0.014$ & $0.002$ & $0.001$ & $0.000$\\ $f$, $\sigma=0.9,\,1,\,1.25$ & $\le0.005$ & $0.000$ & $0.000$ & $0.000$\\ \bottomrule \end{tabular} \caption{$\log\mathcal E_\sigma(X)$ on vertical lines.}\label{tab:abscissa} \end{table} \subsection{Chebyshev's identity at a zero}\label{sec:chebyshev} A proof has to use the integers and the primes together: the Davenport--Heilbronn function has the additive structure and a functional equation, and Beurling systems have an Euler product with $\Lambda\ge0$, and both can have zeros off the line; Beurling systems whose integers satisfy $N(x)=cx+O(x^\theta)$ with $\frac12<\theta<1$ can even have zeros approaching $\re s=1$ [DMV]. The identity $\log=\Lambda*1$ joins the two structures. With $L_N(s)=\sum_{n\le N}\log n\,n^{-s}$, $D_y(s)=\sum_{k\le y}k^{-s}$ and $\mathcal P_N$ as in Section~\ref{sec:c2}, \[ L_N(s)=\sum_{m\le N}\Lambda(m)\,m^{-s}D_{N/m}(s),\quad\text{equivalently}\quad 2L_N(s)N^{-\bar s}=\sum_{m\le N}\Lambda(m)\,m^{-2\sigma}\,\mathcal W^+(N/m,s), \] where $\mathcal W^+(y,s)=2D_y(s)y^{-\bar s}$ is the cross-energy wave of the integers at the scale $y$: on the line, the logarithmically weighted wave at the scale $N$ is the sum, with the weights $\Lambda(m)/m$, of the integer waves at the scales $N/m$. The identity also holds in every discrete Beurling system; what distinguishes $\zeta$ is the integer side $D_y$, whose counting error is $O(1)$, and a proof along these lines must use more of it than $N(x)=cx+O(x^\theta)$ for any fixed $\theta>\frac25$ (Table~\ref{tab:ingredients}). At a zero the identity takes a special form. For a nontrivial zero $\rho=\beta+i\gamma$ and $y\ge1$ write $D_y(\rho)=A_\rho(y)+E_\rho(y)$, with $A_\rho(y)=\frac{y^{1-\rho}}{1-\rho}-\psi_1(y)\,y^{-\rho}$, $\psi_1(y)=\{y\}-\frac12$, the first two terms of the Euler--Maclaurin expansion (there is no constant term because $\zeta(\rho)=0$). Let $M(N)=\sum_{m\le N}\Lambda(m)/m=\log N-\gamma_E+r(N)$. \begin{proposition}[Chebyshev's identity at a zero]\label{prop:chebzero} For every nontrivial zero $\rho$ and every real $N\ge1$, with $\log\lfloor N\rfloor!$ in place of $\log N!$ when $N$ is not an integer, \[ \sum_{m\le N}\Lambda(m)\,m^{-\rho}E_\rho(N/m)=L_N(\rho)-\frac\rho{1-\rho}N^{1-\rho}M(N)-N^{-\rho}\Big(\frac{\psi(N)}2+\log N!\Big), \] and, as $N\to\infty$ through the integers, with $c_\rho=\frac12+\gamma_E\frac\rho{1-\rho}-\frac1{(1-\rho)^2}$, \begin{multline*} \sum_{m\le N}\Lambda(m)\,m^{-\rho}E_\rho(N/m)=c_\rho N^{1-\rho}-\zeta'(\rho)-\frac\rho{1-\rho}N^{1-\rho}r(N)\\-\frac12N^{-\rho}\big(\psi(N)-N\big)-\frac{\log2\pi}2N^{-\rho}+O_\rho\big(N^{-\beta-1}\log N\big). \end{multline*} \end{proposition} \begin{proof} In $\sum_m\Lambda(m)m^{-\rho}A_\rho(N/m)$ the power term gives $\frac{N^{1-\rho}}{1-\rho}M(N)$, and in the sawtooth term the twist cancels, $m^{-\rho}(N/m)^{-\rho}=N^{-\rho}$, leaving $-N^{-\rho}\sum_m\Lambda(m)\psi_1(N/m)$. Since $\lfloor y\rfloor=y-\psi_1(y)-\frac12$, Chebyshev's identity $\sum_{m\le N}\Lambda(m)\lfloor N/m\rfloor=\log N!$ gives $\sum_m\Lambda(m)\psi_1(N/m)=NM(N)-\frac12\psi(N)-\log N!$. Subtracting from $L_N(\rho)=\sum_m\Lambda(m)m^{-\rho}D_{N/m}(\rho)$ gives the identity. For the asymptotic, $L_N(\rho)=-\zeta'(\rho)+\frac{N^{1-\rho}\log N}{1-\rho}-\frac{N^{1-\rho}}{(1-\rho)^2}+\frac12N^{-\rho}\log N+O_\rho(N^{-\beta-1}\log N)$ (differentiate the Euler--Maclaurin expansion of $D_N$; the error is $O(|\rho|N^{-\beta-1}\log N)$ for $N\ge|\rho|^2$), $\log N!=N\log N-N+\frac12\log2\pi N+O(1/N)$, and $\psi(N)=N+(\psi(N)-N)$; the terms $N^{1-\rho}\log N$ cancel because $\frac1{1-\rho}-\frac\rho{1-\rho}-1=0$, and the terms $\frac12N^{-\rho}\log N$ from $L_N$ and from Stirling's formula cancel. \end{proof} The zero removes the twist from the sawtooth and from $\zeta(s)\mathcal P_N(s)$, and what remains of the primes is untwisted: $r(N)=r_1(N)-r_2(N)$ with $r_1(N)=(\psi(N)-N)/N$ and $r_2(N)=\int_N^\infty(\psi(x)-x)x^{-2}dx$, since $1+\int_1^\infty(\psi(x)-x)x^{-2}dx=-\gamma_E$. Let $T_\rho(N)=N^{\rho-1}\sum_{m\le N}\Lambda(m)m^{-\rho}E_\rho(N/m)$, a quantity computed from the primes and the integers up to $N$. \begin{corollary}[a criterion at one zero]\label{cor:onezero} Let $\rho_1=\frac12+i\gamma_1$ be the first zero (which lies on the critical line). Then RH holds if and only if $T_{\rho_1}(N)=c_{\rho_1}+O(N^{-1/2+\varepsilon})$ for every $\varepsilon>0$. \end{corollary} \begin{proof} By Proposition~\ref{prop:chebzero}, $T_{\rho_1}(N)-c_{\rho_1}=-(\omega+\frac12)r_1(N)+\omega r_2(N)+O(N^{-1/2})$ with $\omega=\rho_1/(1-\rho_1)=\rho_1/\bar\rho_1$. Since $r_1,r_2$ are real and $\im\omega=\gamma_1/|\rho_1|^2\ne0$, the imaginary part controls $r_1-r_2$ and then the real part, $-\frac12r_1-\re\omega\,(r_1-r_2)$, controls $r_1$; so the condition gives $\psi(N)-N=O(N^{1/2+\varepsilon})$, which is RH. Conversely RH gives $r_1,r_2=O(N^{-1/2}\log^2N)$. The same proof works at any zero known to lie on the line. \end{proof} The criterion is, by this linear algebra, the classical estimate $\psi(x)-x=O(x^{1/2+\varepsilon})$ in another form, and the zero plays no decisive role; we do not claim more. What the identity adds is a description of the twisted sum. At large scales $E_\rho(y)=-\frac\rho2\tilde B_2(y)y^{-\rho-1}+O(|\rho|^3y^{-\beta-2})$, with $\tilde B_2$ the periodic Bernoulli function (a nontrivial statement for $y\gg|\rho|^2$; numerically the error is much smaller), and there the twist cancels again, $m^{-\rho}(N/m)^{-\rho-1}=N^{-\rho-1}m$: the primes $m\le2\pi N/|\gamma|$ contribute an untwisted sum, of trivial size about $\frac{\pi^2}{6|\gamma|}N^{1/2}$, which is itself a sum of the type of $\psi(x)-x$. At the scales $1\le y<|\gamma|/2\pi$ the approximate functional equation [T, Thm.~4.13] with $x=y$, $y'=|\gamma|/(2\pi y)$, and $\zeta(\rho)=0$ give, on the line, \[ E_\rho(y)=-\chi(\rho)D_{|\gamma|/(2\pi y)}(1-\rho)+O\big(y^{-1/2}+(y/|\gamma|)^{1/2}\big),\qquad\zeta(s)=\chi(s)\zeta(1-s), \] so there the kernel is the dual Dirichlet polynomial of the functional equation (the error is $O(1)$ near $y=|\gamma|/2\pi$). Hence, for large $|\gamma|$, the twisted sum is carried, up to an untwisted part of relative size $O(1/|\gamma|)$, by the primes $m\in(2\pi N/|\gamma|,N]$, twisted by $m^{-i\gamma}$ and weighted by the dual sum at the length $|\gamma|m/(2\pi N)$; for $\rho_1$ the untwisted part is still visible ($17.7$ against a total of modulus $82.0$ in Table~\ref{tab:chebzero}), for $\rho_{100}$ it is $0.23$. RH is the statement that this sum, of size about $|c_\rho|N^{1/2}$, is determined by $c_\rho$ and $\zeta'(\rho)$ up to $O(N^\varepsilon)$. For a single twist $m^{-i\gamma}$ the best known cancellation in sums over primes is that of the Vinogradov--Korobov zero-free region; combinatorial identities, bilinear forms and the large sieve give power savings only for additive twists or on average over characters or heights. Numerically, Vaughan's identity with $U=V=N^{1/3}$ splits the sum at $\rho_1$ into pieces whose moduli, after the main term $c_\rho N^{1-\rho}$ is removed from the first, are $34.5$, $15.6$ and $19.3$ at $N=10^7$, while the total differs from $c_\rho N^{1-\rho}$ by a number of modulus $0.74$, mostly the term $-\zeta'(\rho_1)$, of modulus $0.79$: each piece carries the resonance $\sum_{d\le U}\mu(d)d^{-\rho}\approx\log U/\zeta'(\rho)$ of $1/\zeta$ at the zero (its modulus is $6.27$ at $U=10^2$ and $14.9$ at $U=10^5$, against $\log U/|\zeta'(\rho_1)|=5.81$ and $14.5$), and the information is in the cancellation among the pieces, not in any one of them. Whether the structure of the kernel can be used is the question this formulation raises. \begin{table}[h]\centering\small \begin{tabular}{lcc} \toprule & $\rho_1=\frac12+14.1347i$ & $\rho_{100}=\frac12+236.524i$\\ \midrule both sides of Prop.~\ref{prop:chebzero} & $-43.6987+69.4008i$ & $-70.9887+38.9334i$\\ $|c_\rho|N^{1/2}$ & $81.5$ & $77.2$\\ right side: part from $r(N)$, $\psi(N)-N$ & $-0.157+0.118i$ & $-0.176+0.086i$\\ $|$left side, $m>2\pi N/\gamma|$ & $92.1$ & $80.7$\\ $|$left side, $m\le2\pi N/\gamma|$ & $17.7$ & $0.23$\\ $\sum_{m\le N}\Lambda(m)m^{-1/2}|E_\rho(N/m)|$ & $707$ & $2300$\\ \bottomrule \end{tabular} \caption{Proposition~\ref{prop:chebzero} at $N=10^6$. The identity holds to all printed digits; the part of the prime sum that carries the error term of the prime number theorem is of order $1$, against a main term $c_\rho N^{1-\rho}$ of order $10^2$ and a sum of absolute values of order $10^3$.}\label{tab:chebzero} \end{table} \textbf{Which part of the integers the kernel uses.} At a zero the kernel has the exact form $E_\rho(y)=\rho\int_y^\infty\psi_1(u)u^{-\rho-1}du$ (from $D_y(s)-\zeta(s)=\frac{y^{1-s}}{1-s}-\psi_1(y)y^{-s}+s\int_y^\infty\psi_1(u)u^{-s-1}du$, valid for $\sigma>0$), and $\psi_1(u)=-\frac1\pi\sum_{k\ge1}\sin(2\pi ku)/k$. Let $E^{(K)}_\rho$ be the kernel built from the first $K$ modes of the sawtooth and $S^{(K)}(N)$ the corresponding twisted sum. When $y<|\gamma|/(2\pi k)$ the mode $k$ has a stationary point at $u=|\gamma|/(2\pi k)$, which produces the $k$-th term of the dual sum; the modes with $k>|\gamma|/2\pi$ have no stationary point on $[1,\infty)$ and contribute endpoint terms only. Since $S$ grows like $N^{1/2}$, we write $S-S^{(K)}=\delta_KN^{1-\rho}+R_K(N)$. Replacing $\Lambda$ by its mean shows that the omitted modes shift the constant $c_\rho$ of the main term by \[ \delta_K=c_\rho-c_\rho^{(K)},\qquad c_\rho^{(K)}=\frac\rho{\pi(1-\rho)}\sum_{k\le K}\frac1k\int_1^\infty\sin(2\pi kv)\big(v^{-2}-v^{-\rho-1}\big)dv; \] summed over all $k$ this gives $c_\rho$ (with $2000$ modes, to $10^{-11}$ at $\rho_1$ and $3\cdot10^{-9}$ at $\rho_{100}$), and we compute $\delta_K$ in this way, without fitting. $R_K$ is the part that can reach the terms of order $1$. These carry the prime error, the part $-\frac\rho{1-\rho}N^{1-\rho}r(N)-\frac12N^{-\rho}(\psi(N)-N)$ of Proposition~\ref{prop:chebzero}, whose modulus is about $0.085$, $0.09$, $0.20$, $0.22$ at $N=10^4,\dots,10^7$. Table~\ref{tab:modes} shows the same pattern at both zeros: $R_K$ is of the size of the prime error for $K\lesssim|\gamma|/2\pi$ ($K\le2$ for $\rho_1$, $K\le38$ for $\rho_{100}$), below it by a factor of about $3$ or more from $K\approx1.3|\gamma|/2\pi$ ($K=3$ and $K=50$), and below it by about an order of magnitude for $K=10$ and $K=100$. The count of about $|\gamma|/2\pi$ modes follows from the stationary-phase argument above, whose stationary points $u=|\gamma|/(2\pi k)$ depend only on the height; since the same kernel $s\int_y^\infty\psi_1(u)u^{-s-1}du$ appears at every $s$, it is a property of the height rather than of $\rho$ being a zero (the computations were done at zeros). \begin{table}[h]\centering\small \begin{tabular}{lrccccc} \toprule & $K$ & $|\delta_K|$ & $N=10^4$ & $10^5$ & $10^6$ & $10^7$\\ \midrule $\rho_1$ ($|\gamma|/2\pi=2.25$) & $1$ & $3.3\cdot10^{-2}$ & $0.162$ & $0.054$ & $0.425$ & $0.268$\\ & $2$ & $7.3\cdot10^{-3}$ & $0.062$ & $0.066$ & $0.124$ & $0.170$\\ & $3$ & $2.4\cdot10^{-3}$ & $0.021$ & $0.011$ & $0.069$ & $0.079$\\ & $5$ & $5.8\cdot10^{-4}$ & $0.006$ & $0.019$ & $0.011$ & $0.051$\\ & $10$ & $7.9\cdot10^{-5}$ & $0.009$ & $0.010$ & $0.002$ & $0.017$\\ & $20$ & $1.0\cdot10^{-5}$ & $0.008$ & $0.001$ & $0.005$ & $0.004$\\ & \multicolumn{2}{r}{prime error} & $0.085$ & $0.094$ & $0.197$ & $0.223$\\ \midrule $\rho_{100}$ ($|\gamma|/2\pi=37.6$) & $20$ & $3.0\cdot10^{-3}$ & $0.167$ & $0.275$ & $0.240$ & $0.276$\\ & $38$ & $1.3\cdot10^{-3}$ & $0.170$ & $0.054$ & $0.082$ & $0.098$\\ & $50$ & $2.9\cdot10^{-4}$ & $0.029$ & $0.009$ & $0.005$ & $0.050$\\ & $100$ & $2.6\cdot10^{-5}$ & $0.006$ & $0.013$ & $0.012$ & $0.011$\\ & $200$ & $3.0\cdot10^{-6}$ & $0.001$ & $0.003$ & $0.004$ & $0.010$\\ & \multicolumn{2}{r}{prime error} & $0.085$ & $0.093$ & $0.196$ & $0.222$\\ \bottomrule \end{tabular} \caption{Truncating the sawtooth at $K$ modes in the kernel of Proposition~\ref{prop:chebzero}: the shift $\delta_K$ of the main-term constant, computed from the omitted modes, and the remainder $|R_K(N)|=|S-S^{(K)}-\delta_KN^{1-\rho}|$; the rows ``prime error'' give the modulus of the part of Proposition~\ref{prop:chebzero} that carries $r(N)$ and $\psi(N)-N$. For comparison $|S(N)|$ is $82.0$ and $81.0$ at $N=10^6$ and $257$ and $241$ at $N=10^7$. Quadrature accurate to about $0.003$; smaller entries are not significant.}\label{tab:modes} \end{table} \begin{remark}[Selberg's formula at a zero]\label{rem:selberg} The same computation works at every order. Let $\Lambda_2=\Lambda\log+\Lambda*\Lambda=\mu*\log^2\ge0$, so that $\log^2=\Lambda_2*1$, and let $Q_N(s)=\sum_{n\le N}\log^2n\,n^{-s}$, $M_2(N)=\sum_{m\le N}\Lambda_2(m)/m$, $\psi_2(N)=\sum_{m\le N}\Lambda_2(m)$. Since $\sum_{m\le N}\Lambda_2(m)\lfloor N/m\rfloor=\sum_{n\le N}\log^2n$, the proof of Proposition~\ref{prop:chebzero} gives, at a zero, \[ \sum_{m\le N}\Lambda_2(m)\,m^{-\rho}E_\rho(N/m)=Q_N(\rho)-\frac\rho{1-\rho}N^{1-\rho}M_2(N)-N^{-\rho}\Big(\frac{\psi_2(N)}2+\sum_{n\le N}\log^2n\Big) \] (both sides are $-1115.7505+1838.7384i$ at $\rho_1$, $N=10^6$): the twist cancels again, and what remains of the primes is Selberg's sums $\psi_2$, $M_2$ without twist. The positivity $\Lambda_2\ge0$, together with Selberg's formula $\psi_2(x)=2x\log x+O(x)$, is the engine of the elementary proof of the prime number theorem [Se], [Er]. But the identities $\log^k=\Lambda_k*1$ and the positivity $\Lambda_k=\mu*\log^k\ge0$ hold in every discrete Beurling system, where RH can fail. \end{remark} \textbf{What the ingredients give.} Table~\ref{tab:ingredients} collects what is known about the ingredients used in this section, for discrete Beurling systems $10$, $\psi(x)=x+O(x^{\alpha+\varepsilon})$ and $N(x)=cx+O(x^{\beta+\varepsilon})$ for every $\varepsilon>0$ and for no $\varepsilon<0$; the ordinary integers form a $[\Theta,0]$-system, $\Theta=\sup\re\rho$, and RH is the statement $\Theta=\frac12$. In this and the next subsection we keep the notation of [BDR]: $\beta$ is the exponent of the integers, not the real part of a zero, which we write $\re\rho$ (in {\cite{CE3}} the same exponent is called $b$). We write $\Lambda_k=\mu*\log^k$, which is $\ge0$ in every such system by $\Lambda_{k+1}=\Lambda_k\log+\Lambda*\Lambda_k$. \begin{table}[h]\centering\small \begin{tabular}{p{0.5\textwidth}p{0.42\textwidth}} \toprule ingredients & do they force RH?\\ \midrule additive structure and functional equation & no: the Davenport--Heilbronn function\\ Euler product, $\Lambda_k\ge0$, $\log^k=\Lambda_k*1$ for every $k$, integers $N(x)=cx+O(x^{\beta+\varepsilon})$ with $\beta\ge\frac12$ & no: $[\alpha,\beta]$-systems exist for every $\alpha\in[0,1)$, $\beta\in[\frac12,1)$ [BDR, Thm.~1.1]; zeros near $\re s=1$ [DMV]\\ the same with $\frac25<\beta<\frac12$ & no: under RH, [BDR, Thm.~1.3] gives $[\alpha,\beta]$-systems with $\frac12<\alpha<\frac23$, $\frac{2\alpha}{\alpha+2}\le\beta<\frac12$, whose zeta function has a zero at $s=\alpha$ and every zero of $\zeta$ (Section~\ref{sec:bdr}); if RH fails, the integers themselves\\ the same with $\beta\le\frac25$ & open, even under RH; [BDR] conjecture that every pair with $\max\{\alpha,\beta\}\ge\frac12$ occurs\\ every discrete system & $\max\{\alpha,\beta\}\ge\frac12$ [Hi]\\ \bottomrule \end{tabular} \caption{Ingredients and what is known about them.}\label{tab:ingredients} \end{table} So, unconditionally, no estimate $N(x)=cx+O(x^\theta)$ with $\theta>\frac25$ gives RH together with the Euler product and the identities above, and if the conjecture of [BDR] holds, no estimate with a fixed $\theta>0$ does; a proof must then use the integers more precisely: through the sawtooth $\psi_1$, which by Poisson summation yields the functional equation, together with the Euler product. Chebyshev's identity at a zero uses $\lfloor N/m\rfloor$ exactly, but the identity for all $N$ is just $\Lambda*1=\log$, and Chebyshev's argument from it (Stirling's formula and positivity) gives only $\psi(x)\asymp x$ and Mertens' estimate. Table~\ref{tab:modes} shows which modes of the sawtooth the kernel at height $\gamma$ needs for the part of order $1$: about $|\gamma|/2\pi$ of them, the dual sum of the functional equation, the others changing only the constant of the main term. A mechanism that couples these modes with the positivity of the Euler product is what this section points to, and we do not have one. \subsection{Beurling systems with the zeros of \texorpdfstring{$\zeta$}{zeta}}\label{sec:bdr} The systems of [BDR, Thm.~1.3] in the third row of Table~\ref{tab:ingredients}, which are constructed under RH, are not built from a continuous model: they modify the rational primes [BDR, Sect.~5]. For $\frac12<\alpha<\frac23$ and $\frac{2\alpha}{\alpha+2}\le\beta<\frac12$ the numbers $p^{1/\beta}$, $p$ prime, are added to the primes, and a set $\mathcal S$ of rational primes with $\pi_{\mathcal S}(x)$ asymptotic to $\operatorname{Li}(x^\alpha)$ is removed (the precise condition is below), so that \[ \zeta_B(s)=\frac{\zeta(s)\,\zeta(s/\beta)}{\zeta_{\mathcal S}(s)},\qquad \zeta_{\mathcal S}(s)=\zeta(s+1-\alpha)\,e^{H(s)} . \] For the set $\mathcal S$ that we use below (a prime $p$ is in $\mathcal S$ when $\lfloor F\rfloor$ jumps at $p$, $F(x)=\sum_{p\le x}p^{\alpha-1}$, so that $|\pi_{\mathcal S}(x)-F(x)|<1$), $H(s)=\sum_p\big(-\mathbf 1_{\mathcal S}(p)\log(1-p^{-s})+\log(1-p^{-(s+1-\alpha)})\big)$ is analytic in $\re s>\frac\alpha2$: the terms of first order are $\int u^{-s}d(\pi_{\mathcal S}-F)$, which converges for $\re s>0$ by partial summation, and the others converge absolutely, because $\pi_{\mathcal S}(x)\ll x^\alpha$ and $2(\re s+1-\alpha)>1$. (In [BDR] the set is chosen at random, with a bound for $H$; we do not prove their bound for our deterministic set.) Hence, in $\re s>\frac\alpha2$, $\zeta_B$ has poles at $1$ and $\beta$, a simple zero at the real point $\alpha$, from the pole of $\zeta(s+1-\alpha)$, and the zeros of $\zeta$ there, since under RH $\zeta(s/\beta)$ and $\zeta(s+1-\alpha)$ do not vanish in that half-plane. In particular such a system has the zero $\rho_1=\frac12+14.1347i$ of $\zeta$ (unconditionally, since $\re\rho_1/\beta>1$ and $\zeta$ has no zeros off the line below height $3\cdot10^{12}$), it has $\Lambda\ge0$ and $\log=\Lambda*1$, and, under RH and for the random sets of [BDR], its integers satisfy $N(x)=cx+O(x^{\beta+\varepsilon})$ with $\beta<\frac12$ [BDR, Thm.~1.3], the pole at $\beta$ contributing a term $c'x^\beta$. The question of Section~\ref{sec:chebyshev} at $\rho_1$ is therefore asked of a system that shares $\rho_1$ with $\zeta$ and has an off-line zero. The following proposition shows what the identity at a zero contains in any Beurling system. \begin{proposition}[Chebyshev's identity at a zero of a Beurling zeta function]\label{prop:beurlingzero} Let a discrete Beurling system have integer counting function $N_B(x)=cx+R(x)$ with $c>0$ and $|R(x)|\le C_Rx^\beta$ for $x\ge1$, $0\le\beta<1$, so that $\zeta_B(s)=\int_{1^-}^\infty x^{-s}dN_B(x)$ is meromorphic in $\re s>\beta$ with a single pole, at $s=1$. Let $\rho\ne1$ be a zero of $\zeta_B$ with $\re\rho>\beta$. With $D_y(s)=\sum_{b\le y}b^{-s}$, $L_N(s)=\sum_{b\le N}\log b\,b^{-s}$ and $T(N)=\sum_{b\le N}\log b$ over the integers $b$ of the system, $M(N)=\sum_{m\le N}\Lambda_B(m)/m$ and $E_\rho(y)=-\rho\int_y^\infty R(u)u^{-\rho-1}du$: \begin{enumerate} \item[(i)] for every $y\ge1$, $D_y(\rho)=\frac{c\,y^{1-\rho}}{1-\rho}+R(y)y^{-\rho}+E_\rho(y)$, and \[ \sum_{m\le N}\Lambda_B(m)\,m^{-\rho}E_\rho(N/m)=L_N(\rho)-\frac\rho{1-\rho}\,c\,N^{1-\rho}M(N)-N^{-\rho}T(N); \] \item[(ii)] if $\int_1^\infty(\psi_B(x)-x)x^{-2}dx$ converges, write $M(N)=\log N+\kappa_B+r(N)$ with $r(N)\to0$; then $T^B_\rho(N)$, $N^{\rho-1}$ times the sum in (i), satisfies \[ T^B_\rho(N)=C_\rho-\frac\rho{1-\rho}\,c\,r(N)-\zeta_B'(\rho)\,N^{\rho-1}+O_\rho\big(N^{\beta-1}\log(N+1)\big),\qquad C_\rho=c\Big(1-\frac1{(1-\rho)^2}-\frac{\rho\,\kappa_B}{1-\rho}\Big); \] \item[(iii)] if $\beta<\frac12$ and $\re\rho=\frac12$, then $T^B_\rho(N)-C_\rho=O(N^{-1/2+\varepsilon})$ for every $\varepsilon>0$ if and only if $\psi_B(x)-x=O(x^{1/2+\varepsilon})$ for every $\varepsilon>0$. \end{enumerate} \end{proposition} \begin{proof} (i) Since $N_B(1^-)=0$, partial summation gives, for $\re s>1$ and then by continuation for $\re s>\beta$, $\zeta_B(s)=c+\frac c{s-1}+s\int_1^\infty R(u)u^{-s-1}du$ and \[ D_y(s)=\zeta_B(s)+\frac{c\,y^{1-s}}{1-s}+R(y)y^{-s}-s\int_y^\infty R(u)u^{-s-1}du ; \] put $s=\rho$. Then $\log=\Lambda_B*1$ gives $L_N(\rho)=\sum_m\Lambda_B(m)m^{-\rho}D_{N/m}(\rho)$; the power terms give $\frac{c\,N^{1-\rho}}{1-\rho}M(N)$, and in the terms $R(N/m)(N/m)^{-\rho}$ the twist cancels, $m^{-\rho}(N/m)^{-\rho}=N^{-\rho}$, leaving $N^{-\rho}\sum_m\Lambda_B(m)R(N/m)=N^{-\rho}\big(T(N)-cNM(N)\big)$ by $\sum_m\Lambda_B(m)N_B(N/m)=T(N)$. (ii) Differentiating the formula for $D_N(s)$ at $s=\rho$, \[ L_N(\rho)=-\zeta_B'(\rho)+\frac{c\,N^{1-\rho}\log N}{1-\rho}-\frac{c\,N^{1-\rho}}{(1-\rho)^2}+R(N)N^{-\rho}\log N+\int_N^\infty\frac{R(u)}{u^{\rho+1}}du-\rho\int_N^\infty\frac{R(u)\log u}{u^{\rho+1}}du, \] and $T(N)=c(N\log N-N+1)+R(N)\log N-\int_1^NR(u)u^{-1}du$. Substituting in (i) with $M(N)=\log N+\kappa_B+r(N)$, the terms $N^{1-\rho}\log N$ cancel because $\frac1{1-\rho}-\frac\rho{1-\rho}-1=0$, the terms $R(N)N^{-\rho}\log N$ cancel, and what remains is $C_\rho N^{1-\rho}-\frac\rho{1-\rho}cN^{1-\rho}r(N)-\zeta_B'(\rho)-cN^{-\rho}$ plus $N^{-\rho}\int_1^NR/u$ and the two tail integrals, which are $O(N^{\beta-\re\rho}\log(N+1))$ by $|R(u)|\le C_Ru^\beta$. Multiply by $N^{\rho-1}$. (iii) By (ii), $T^B_\rho(N)-C_\rho=-\frac\rho{1-\rho}c\,r(N)+O(N^{-1/2})$, and $\rho\ne0$. If $r(N)=O(N^{-1/2+\varepsilon})$, then $\psi_B(N)=NM(N)-\int_1^NM(x)\,dx=N+Nr(N)-\int_1^Nr(x)\,dx+O(1)=N+O(N^{1/2+\varepsilon})$; conversely, $\psi_B(x)-x=O(x^{1/2+\varepsilon})$ gives $r(N)=\frac{\psi_B(N)-N}N-\int_N^\infty\frac{\psi_B(x)-x}{x^2}dx=O(N^{-1/2+\varepsilon})$. \end{proof} For the integers ($c=1$, $R(u)=-\psi_1(u)-\frac12$, $\beta=0$) the sum in (i) is that of Proposition~\ref{prop:chebzero} plus $\frac12N^{-\rho}\psi(N)$, $C_\rho=c_\rho+\frac12$, and (iii) is Corollary~\ref{cor:onezero} at any zero on the line. What (iii) adds is that, as soon as the integers are better than $\sqrt x$, they enter the remainder of the identity only at order $N^{\beta-1}\log N$, below $N^{-1/2}$: in that range the identity at a zero on the line is a rewriting of the prime error, whatever the integers are. The size of the integer error can therefore not be what forces RH through this identity. What the identity does contain is the kernel $E_\rho$, which is built from the exact shape of $R$, and a proof along these lines has to bound the left side directly. \textbf{A computation.} We built the system above with $\alpha=0.6$ and $\beta=0.47\ge\frac{2\alpha}{\alpha+2}=0.4615$ up to $N=3\cdot10^7$. The set $\mathcal S$ has $6360$ primes below $10^8$, against $\operatorname{li}(10^{8\alpha})=6364$, beginning $3,7,13,23,41,61,\dots$; there are $20\,859\,734$ integers below $3\cdot10^7$, the products $n\,k^{1/\beta}$ with $n$ free of the primes of $\mathcal S$. From the primes up to $10^8$, $c=\zeta(1/\beta)\prod_{p\in\mathcal S}(1-p^{-1})=0.69529746$, which agrees with $\zeta(1/\beta)/(\zeta(2-\alpha)e^{H(1)})$ to $8\cdot10^{-10}$; $c'=\zeta(\beta)/(\zeta(\beta+1-\alpha)e^{H(\beta)})=0.23799$, $\zeta_B'(\rho_1)=\zeta'(\rho_1)\zeta(\rho_1/\beta)/\zeta_{\mathcal S}(\rho_1)=0.89145-0.23690i$ and $\zeta_B'(\alpha)=\zeta(\alpha)\zeta(\alpha/\beta)e^{-H(\alpha)}=-11.5733$. The identity (i) holds at $N=10^6$ to ten digits at $\rho_1$ and at $\alpha$. Table~\ref{tab:bdr} and Figure~\ref{fig:bdr} compare the system with the integers at the same zero $\rho_1$. For the integers $N^{1/2}|T_{\rho_1}(N)-C_{\rho_1}|$ stays near $0.8$, and (ii) predicts it to $3\cdot10^{-4}N^{-1/2}$. For the system it grows like $N^{\alpha-1/2}$: the prime remainder $r(N)$ follows $N^{\alpha-1}/(1-\alpha)-N^{\beta-1}/(1-\beta)$, the contribution of the removed and of the added primes, and (ii) predicts $T^B_{\rho_1}$ to within $0.21N^{-1/2}$, of the size of the error of $\kappa_B$, for which $\psi_B$ beyond $3\cdot10^7$ is replaced by its main terms. The integers of the system follow $cx+c'x^\beta$, with $(N_B(x)-cx)/x^\beta$ fluctuating around $c'$ by about $0.1$ at $x=10^4$ and $0.01$ at $3\cdot10^7$. \begin{table}[h]\centering\small \begin{tabular}{rcccccc} \toprule & \multicolumn{2}{c}{$N^{1/2}|T_{\rho_1}(N)-C_{\rho_1}|$} & \multicolumn{2}{c}{$r(N)$ of the system} & \\ \cmidrule(lr){2-3}\cmidrule(lr){4-5} $N$ & integers & system & computed & $\frac{N^{\alpha-1}}{1-\alpha}-\frac{N^{\beta-1}}{1-\beta}$ & $\frac{N_B(N)-cN}{N^\beta}$\\ \midrule $10^5$ & $0.712$ & $2.251$ & $0.02105$ & $0.02067$ & $0.260$\\ $10^6$ & $0.845$ & $3.210$ & $0.00879$ & $0.00878$ & $0.238$\\ $10^7$ & $0.784$ & $5.576$ & $0.00356$ & $0.00361$ & $0.245$\\ $3\cdot10^7$ & $0.789$ & $7.136$ & $0.00236$ & $0.00235$ & $0.248$\\ \bottomrule \end{tabular} \caption{Chebyshev's identity at $\rho_1=\frac12+14.1347i$ for the integers and for a Beurling system of the type of [BDR, Thm.~1.3] with $\alpha=0.6$, $\beta=0.47$, which has the same zero $\rho_1$ and a real zero at $0.6$. Values at the nearest of $160$ logarithmically spaced $N$; $c'=0.238$.}\label{tab:bdr} \end{table} \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Beurling_real_zero.png} \caption{A Beurling system with the zeros of $\zeta$ and a real zero at $\alpha=0.6$ ($\beta=0.47$). (a) $N^{1/2}|T_{\rho_1}(N)-C_{\rho_1}|$ for the system and for the integers, with the prediction of Proposition~\ref{prop:beurlingzero}(ii) (dashed). (b) At the real zero: $(T^B_\alpha(N)-C_\alpha)/N^{\alpha-1}$, the prediction and its limit $K$. (c) $(N_B(x)-cx)/x^\beta$ and $c'$.}\label{fig:bdr}\end{figure} At the real zero the identity has no twist at all: every factor $m^{-\alpha}(N/m)^{-\alpha}$ is real, there is no dual sum (the count $|\gamma|/2\pi$ of modes of Section~\ref{sec:chebyshev} is $0$), and the zero is seen directly in the untwisted part. By (ii) the remainder is $KN^{\alpha-1}$ up to lower-order terms, with $K=-\frac{\alpha c}{(1-\alpha)^2}-\zeta_B'(\alpha)=-2.607+11.573=8.966$, the first term from the removed primes and the second from the zero; the convergence is slow, with relative corrections of order $N^{\beta-\alpha}$, and the data follow the full prediction to $1\%$ (Figure~\ref{fig:bdr}b). At $\rho_1$ the same removed primes enter through $r(N)$, which is untwisted, so the system violates the condition of (iii) at a zero that it shares with $\zeta$, although its integers are better than $\sqrt x$ and $\Lambda_B\ge0$. \textbf{Three controls.} Table~\ref{tab:controls} sets the system beside the other two controls of the paper. The Davenport--Heilbronn function has the functional equation and no Euler product; the Beurling systems have the Euler product, $\Lambda\ge0$ and Chebyshev's identity, and those of [BDR] even have integers better than $\sqrt x$ and every zero of $\zeta$, but no functional equation of the type of $\zeta$ is known for them. Each control has an off-line zero. A proof of the bound, or of RH, must therefore use the functional equation and the Euler product together; by Proposition~\ref{prop:beurlingzero}(iii) the size of the integer error, however small, cannot replace the functional equation. The controls are studied in more detail in {\cite{CE46}}. \begin{table}[htbp]\centering\small \begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.15\textwidth}>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.08\textwidth}} \toprule & functional equation & Euler product, $\Lambda\ge0$, $\log=\Lambda*1$ & integer error $O(x^\beta)$, $\beta<\frac12$ & zeros of $\zeta$ on the line & off-line zero\\ \midrule Davenport--Heilbronn & yes & no & --- & no & yes\\ Beurling [DMV] & none known & yes & no & no & yes\\ Beurling [BDR, Thm.~1.3] & none known & yes & yes & yes (under RH) & yes\\ $\zeta$ & yes & yes & yes ($\beta=0$) & --- & RH\\ \bottomrule \end{tabular} \caption{The three controls and $\zeta$ (see {\cite{CE46}}).}\label{tab:controls} \end{table} \textbf{A quantity in which $\zeta$ cancels.} The functional equation suggests a combination of the two ingredients that does not contain $\zeta$ at all. For $s^*=1-\bar s$, $|\zeta(s^*)|=|\zeta(s)|/|\chi(s)|$, where $\zeta(s)=\chi(s)\zeta(1-s)$, so the \emph{reflection asymmetry} \[ D_X(s):=\log|Z_X(s)|-\log|Z_X(s^*)|=\log|\chi(s)|-\log|E^*_X(s)|+\log|E^*_X(s^*)| \] is built from the primes and the Gamma factor alone; numerically the two expressions agree to $2\cdot10^{-13}$, and also for the Davenport--Heilbronn function (Figure~\ref{fig:reflect}d). It might seem to couple the Euler product with the functional equation, but by Lemma~\ref{lem:factor}, reindexing the zeros by $\rho\mapsto1-\bar\rho$ (legitimate by absolute convergence), \[ D_X(s)=\sum_\rho h_X(s-\rho),\qquad h_X(w)=-\re\big[K_X(w)-K_X(-\bar w)\big], \] the same kernel for every zero and for every function with such a factorization. So $D_X$ records the zeros in the same way for $\zeta$, for $f$ and for a Beurling-type toy without functional equation (Figure~\ref{fig:reflect}): an off-line zero appears in it through the same kernel whatever the function, and the identity alone gives no reason for the zeros to be on the line. Its prime side contains $E^*_X$ at $\re s^*<\frac12$, where the prime sum is large and the pole barrier of Remark~\ref{rem:positivity} reappears. (The identity is proved for $\zeta$; the figure is numerical, script \texttt{reflection\_asymmetry.py}; for $f$ we do not prove the factorization, and for the toy, whose only zeros are $\rho_B$ and $\bar\rho_B$, it is exact by construction.) Weil's form, next, is the classical exact form of the coupling. \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Reflection_asymmetry.png} \caption{The reflection asymmetry $D_X(s)$, $X=10^3$, Gaussian weight of standard deviation $0.4$. (a) $\zeta$, from the primes and $\chi$ only; circles: zeros. (b) The Davenport--Heilbronn function near its off-line zero (star), from its coefficients and its $\chi$. (c) A Beurling-type toy with the nonnegative prime measure $(1-2\re x^{\rho_B-1})\,dx/\log x$ on $[16,\infty)$, $\rho_B=0.75+20i$ (star), and no functional equation, from its two zeros. (d) Difference between the primes-only formula and the direct one on $\sigma=0.7$.}\label{fig:reflect}\end{figure} \subsection{Weil's positivity: the primes against the Gamma factor}\label{sec:weil} The coupling that Table~\ref{tab:ingredients} asks for has a classical exact form, Weil's criterion [W2]: RH holds if and only if $Q(f)\ge0$ for every $f\in C_c^\infty(\mathbb R)$. For real $f$, with $G=f*\tilde f$ ($\tilde f(u)=f(-u)$), which is even, and $\Phi(z)=\hat f(z)\hat f(-z)$, the entire function equal to $|\hat f(r)|^2$ for real $r$, \[ Q(f)=\Phi(\tfrac i2)+\Phi(-\tfrac i2)+\mathcal A_\infty(G)-2\sum_{n\ge2}\Lambda(n)n^{-1/2}G(\log n), \] and, by the explicit formula, $Q(f)=\sum_\rho \Phi(t_\rho)$, $t_\rho:=(\rho-\frac12)/i$ (so $t_\rho=\gamma_\rho$ for a zero $\rho=\frac12+i\gamma_\rho$ on the line, and $t_\rho=\gamma-i\delta$ for a zero $\rho=\beta+i\gamma$ off it, $\gamma>0$, $\delta=\beta-\frac12$). Real $f$ suffice: $Q$ is real and symmetric on real functions and $Q(f_1+if_2)=Q(f_1)+Q(f_2)$. The archimedean term, which carries the functional equation, is $\mathcal A_\infty(G)=\frac1{2\pi}\int \Phi(r)\big(\re\frac{\Gamma'}\Gamma(\frac14+\frac{ir}2)-\log\pi\big)dr$, and for $G$ supported in $[-\ell,\ell]$ it equals \[ \mathcal A_\infty(G)=G(0)\big(-\gamma_E-\log\pi-\log(1-e^{-2\ell})\big)+\int_0^\ell\big(G(0)e^{-2x}-e^{-x/2}G(x)\big)\frac{2\,dx}{1-e^{-2x}}, \] from $\re\frac{\Gamma'}{\Gamma}(\tfrac14+\tfrac{ir}2)=-\gamma_E+\int_0^\infty\frac{e^{-t}-e^{-t/4}\cos(rt/2)}{1-e^{-t}}dt$. With these normalizations and a Gaussian $G$ of width $\frac12$, for which the prime term is $0.51$, the explicit formula agrees with the first $1000$ zeros to $3\cdot10^{-16}$. Two limiting cases connect this with the rest of the paper. The kernel of the cross-energy is $\hat G$ for $G(u)=e^{-|u|/2}$: $C_2(\infty,\frac12+it)=\frac1{\frac14+t^2}=\hat G(t)$, and unconditionally $\hat G(t_\rho)=\frac1{\rho(1-\rho)}$, so the lower bound $\sum_\rho C_2(\infty,\rho)=2+\gamma_E-\log4\pi$ of \eqref{eq:cramer}, which is Cram\'er's constant if all zeros are simple, is $\sum_\rho\hat G(t_\rho)$. This $G$ is not admissible in Weil's criterion ($\hat G$ has poles at $\pm\frac i2$, and $\sum\Lambda(n)n^{-1/2}G(\log n)=\sum\Lambda(n)/n$ diverges); it is the limit of $e^{-(\frac12+\varepsilon)|u|}$. Likewise the law of Theorem~\ref{thm:law} is $\sum_\gamma\kappa_{m(\gamma)}\hat G_{s_0}(\gamma)$ with $\hat G_{s_0}=P_{s_0}$, $G_{s_0}(u)=\frac1{2\pi}e^{-(\sigma_0-\frac12)|u|-it_0u}$, admissible in the explicit formula only for $\sigma_0>1$; it weighs an ordinate by $\kappa_m$ rather than by $m$, and it is $\kappa_1$ times the zero side of the explicit formula only if all zeros are simple. We computed $Q$ from the pole, the Gamma factor and the primes, without the zeros, on functions supported in $[-\frac \ell2,\frac \ell2]$ (so that $G$ is supported in $[-\ell,\ell]$ and the primes up to $e^\ell$ enter), in the basis $e_j(u)=\sin(j\pi(u+\frac \ell2)/\ell)$, $j\le J$: for $J=40$ with $80$ digits and for $J=60$ with $130$ digits (mpmath). The pole and prime entries are in closed form; the archimedean entries are computed by Gauss--Legendre quadrature ($40$ nodes on panels of width $0.05$). Table~\ref{tab:weil} gives the smallest eigenvalue of $Q/\|f\|^2$ on the span of $e_1,\dots,e_J$, with and without the prime term. Since these spans increase with $J$, every entry is an upper bound for the infimum over all $f$ supported in $[-\frac \ell2,\frac \ell2]$. \begin{table}[h]\centering\small \begin{tabular}{rrcccccc} \toprule & & \multicolumn{2}{c}{pole $+\ \Gamma$ + primes, $J=40$} & \multicolumn{2}{c}{pole $+\ \Gamma$ + primes, $J=60$} & \multicolumn{2}{c}{pole $+\ \Gamma$ only, $J=40$}\\ \cmidrule(lr){3-4}\cmidrule(lr){5-6}\cmidrule(lr){7-8} $\ell$ & $e^\ell$ & $f$ even & $f$ odd & $f$ even & $f$ odd & $f$ even & $f$ odd\\ \midrule $1.00$ & $2.7$ & $1.06\cdot10^{-6}$ & $2.30\cdot10^{-4}$ & $1.04\cdot10^{-6}$ & $2.20\cdot10^{-4}$ & $-0.030$ & $-0.313$\\ $1.25$ & $3.5$ & $3.06\cdot10^{-10}$ & $1.18\cdot10^{-7}$ & $3.03\cdot10^{-10}$ & $1.16\cdot10^{-7}$ & $-0.152$ & $-0.576$\\ $1.50$ & $4.5$ & $5.71\cdot10^{-15}$ & $4.41\cdot10^{-12}$ & $4.41\cdot10^{-15}$ & $3.81\cdot10^{-12}$ & $-0.301$ & $-0.819$\\ $2.00$ & $7.4$ & $4.80\cdot10^{-28}$ & $5.24\cdot10^{-25}$ & $1.08\cdot10^{-29}$ & $3.29\cdot10^{-26}$ & $-0.577$ & $-1.307$\\ $2.50$ & $12.2$ & $3.89\cdot10^{-38}$ & $2.64\cdot10^{-35}$ & $5.08\cdot10^{-46}$ & $3.74\cdot10^{-43}$ & $-0.811$ & $-1.877$\\ $3.00$ & $20.1$ & $2.70\cdot10^{-45}$ & $4.69\cdot10^{-43}$ & $9.19\cdot10^{-58}$ & $5.73\cdot10^{-55}$ & $-1.019$ & $-2.607$\\ $3.50$ & $33.1$ & $1.08\cdot10^{-51}$ & $6.52\cdot10^{-49}$ & $6.33\cdot10^{-67}$ & $2.36\cdot10^{-64}$ & $-1.211$ & $-3.573$\\ $4.00$ & $54.6$ & $5.64\cdot10^{-56}$ & $5.33\cdot10^{-54}$ & $1.36\cdot10^{-74}$ & $4.44\cdot10^{-72}$ & $-1.394$ & $-4.862$\\ $5.00$ & $148.4$ & $3.96\cdot10^{-64}$ & $3.87\cdot10^{-62}$ & $4.80\cdot10^{-87}$ & $1.58\cdot10^{-84}$ & $-1.750$ & $-8.844$\\ \bottomrule \end{tabular} \caption{Smallest eigenvalue of Weil's form on the span of $e_1,\dots,e_J$, functions supported in $[-\frac \ell2,\frac \ell2]$. The entries are upper bounds for the infimum, and for $\ell\ge2$ they are far from converged in $J$. At $\ell=\log2$, where no prime enters, the smallest eigenvalue is $1.42\cdot10^{-3}$ for $f$ even with $J=40$ ($1.36\cdot10^{-3}$ with $J=120$, double precision).}\label{tab:weil} \end{table} The pole and the Gamma factor alone stay positive somewhat beyond the entry of the prime $2$: in double precision with $J=40$, $80$ and $120$ their smallest eigenvalue changes sign at $\ell\approx0.745$ for odd $f$ and at $\ell\approx0.82$ for even $f$, and it is negative and decreasing from there on (Table~\ref{tab:weil}). With the prime term the smallest eigenvalue on these spans is positive; it falls from $10^{-6}$ at $\ell=1$ to $4.0\cdot10^{-64}$ with $J=40$ and $4.8\cdot10^{-87}$ with $J=60$ at $\ell=5$, and keeps decreasing as $J$ grows. These near-null directions are consistent with the very small eigenvalues of Weil's form on a fixed interval found by Connes and Consani, whose eigenvectors are built from prolate spheroidal wave functions [CC2] (we have not compared our eigenvectors with theirs). For the archimedean place alone and supports that admit no prime, [CC] give a conceptual reason for positivity, and [CCM] relate the zeros to a semilocal prolate wave operator. These numbers are not evidence for RH beyond what is known: $Q(f)$ for such $f$ is governed by the low zeros, which are known to lie on the line, and positivity on a finite-dimensional subspace at fixed $\ell$ is not Weil's criterion. They illustrate what the criterion demands: on these directions the prime sum must cancel the polar and archimedean terms, the latter encoding the functional equation, to within these margins, for every $\ell$. The positivity of $\Lambda$ alone does not provide this (Table~\ref{tab:ingredients}); what matters are the exact values and positions of the prime powers. Weil's form at the windows of the zeros, at large height and with two windows, is studied numerically in {\cite{CE19}}. \section{Conclusion}\label{sec:conclusion} The function-field model shows what makes the bound true where RH is known: a square-root estimate for the prime sums of each degree (Weil, via Frobenius), turned into control of the energy by an absolutely convergent form because all primes of one degree share one frequency. A proof over $\mathbb Q$ that follows this route cannot work: the bound is equivalent to RH, and the corresponding quadratic form in the primes is not absolutely convergent (Lemma~\ref{lem:div}). Under RH its convergence is a cancellation phenomenon among primes in windows of bounded logarithmic length, and we do not know how to control that cancellation without the zeros. This suggests, but does not prove, that a proof would need a mean-value principle for Dirichlet series whose length is unbounded compared with the height. Chebyshev's identity at a zero (Section~\ref{sec:chebyshev}) gives the same requirement a concrete form: square-root cancellation in one twisted sum over the primes in $(2\pi N/|\gamma|,N]$, weighted by the dual polynomial of the functional equation. The Beurling systems of Section~\ref{sec:bdr} share the zeros of $\zeta$, have integers better than $\sqrt x$ and $\Lambda\ge0$, and still have an off-line zero: the identity at a zero only restates the prime error, and what separates $\zeta$ from them is the functional equation. Weil's form (Section~\ref{sec:weil}) makes the demand quantitative on a fixed interval: on its near-null directions the primes up to $e^\ell$ must cancel the pole and the Gamma factor to within less than $10^{-86}$ at $\ell=5$, which the positivity of $\Lambda$ alone cannot provide. The reflected cross-energy, which separates the two controls and gives a finite-matrix form of the restricted Weil criterion, is studied in {\cite{CE3}}. Theorem~\ref{thm:landau} removes the last hypothesis from the equivalence: RH holds if and only if $B_X(s_0)\ll X^\varepsilon$ at one point with $\re s_0>1$, equivalently if and only if $\log|Z_X(s)|\le C_sX^\varepsilon$ on compact subsets of $\re s>\frac12$, a one-sided bound for the tail of the Euler product beyond $X$. By Landau's theorem the one-sided bound is not weaker than the two-sided one, just as $\psi(x)-x\le x^{\frac12+\varepsilon}$ is already equivalent to RH; the problem is unchanged in strength, but it no longer depends on whether $\Theta$ is attained. Coupling the bound with the reflected cross-energy of {\cite{CE3}} does not change this: its oscillating part is $\zeta K_N$ with an explicit carrier $K_N$, so the Euler--Caceres bound of $\zeta K_N/E^*_X$ is that of $\zeta/E^*_X$ plus the explicit Blaschke term of the zeros of $K_N$. \section{Open problems}\label{sec:open} \textbf{1. The bound.} Prove $B_X(s_0)=O(1/\log X)$ for one $s_0$. By Corollary~\ref{cor:criteria} this is RH; Section~\ref{sec:obstruction} explains why the function-field route does not transfer. Unconditionally only $0\le B_X(s_0)\ll\sqrt X/\log X$ is known for $\re s_0>1$ (Theorem~\ref{thm:uncond}); by Corollary~\ref{cor:landau} already a bound $B_X(s_0)\ll_\varepsilon X^\varepsilon$ for every $\varepsilon>0$, at one such $s_0$, is equivalent to RH. In the form of Corollary~\ref{cor:energy} it is a mean-value statement for $|\zeta/E^*_X|^2$ on the critical line with a fixed weight and an Euler factor of growing length, the opposite regime of the mollifier methods of Levinson [Le] and Conrey [Co], and related to the hybrid Euler--Hadamard products of Gonek, Hughes and Keating [GHK]. A proof must distinguish $\zeta$ from the Davenport--Heilbronn function, for which the bound grows, and from Beurling systems with an off-line zero, which have an Euler product but not the integers; positivity of $\Lambda(n)$ alone is not enough, since every nonnegative trigonometric polynomial carries a constant term that meets the pole (Remark~\ref{rem:positivity}). \textbf{2. The interaction of the wells.} Bound $\mathcal I_X(s_0)$ in Theorem~\ref{thm:second} by $O(1)$; under the gap hypothesis of Section~\ref{sec:gaps} it is $O(\log\log X)$ (Theorem~\ref{thm:rate}), and $O(1)$ would follow from $E(d)=O(1/d)$ for the two-well function with a bound for the interactions of three or more wells. Prove the asymptotic of $E(d)$ and identify its constants. Make \eqref{eq:pair} a theorem for $\alpha<1$ under RH, explain the excess near $\alpha=1$ in Figure~\ref{fig:pair}a, and find the nonlinear part of the interaction. \textbf{3. Simple zeros at height $T$.} With the centre at height $T$, Corollary~\ref{cor:criteria}(b) becomes a local test for multiple zeros near $T$. Quantify the dependence of $X$ on $T$ needed to detect a double zero. \textbf{4. $\Theta$ not attained.} Theorem~\ref{thm:landau} gives growth along a sequence without attainment. Remove the attainment hypothesis in Theorem~\ref{thm:explode}, that is, prove growth for all large $X$, which would give the $\liminf$ form of Corollary~\ref{cor:energy} unconditionally; Remark~\ref{rem:cluster} describes the cluster obstruction heuristically. \textbf{5. The exponent.} Prove $\limsup_X\log B_X(s_0)/\log X=\Theta-\frac12$ for $\re s_0>1$. The lower bound is Theorem~\ref{thm:landau}; the upper bound $B_X(s_0)\ll X^{\Theta-\frac12+\varepsilon}$ would need control of clusters of off-line zeros at large height, which zero-density estimates do not give pointwise. \textbf{6. Integers better than $x^{2/5}$.} Assume RH. Is there a discrete Beurling system with $N(x)=cx+O(x^\beta)$ for some $\beta\le\frac25$ (in particular with $O(x^\varepsilon)$ for every $\varepsilon>0$) whose zeta function has a zero with real part $>\frac12$? For $\frac25<\beta<\frac12$ there is [BDR, Thm.~1.3], with every zero of $\zeta$ and a real zero (Section~\ref{sec:bdr}), and [BDR] conjecture that there is for every $\beta$. In the form of Section~\ref{sec:chebyshev}: by Proposition~\ref{prop:beurlingzero}(iii), for integers better than $\sqrt x$ the identity at a zero on the line is a rewriting of the prime error, and a system of the type of [BDR] violates it at the zero $\rho_1$ that it shares with $\zeta$. The question is therefore whether the left side of Proposition~\ref{prop:chebzero} at $\rho_1$, the twisted sum of $\Lambda$ with the kernel $E_{\rho_1}$ of the integers, can be shown to be $c_{\rho_1}N^{1-\rho_1}+O(N^{1/2+\varepsilon})$ from $\Lambda\ge0$ and the structure of that kernel (its first $\asymp|\gamma_1|$ sawtooth modes, the dual sum of the functional equation), without the zeros; the size of the integer error alone is not enough. \appendix \section{The Dirichlet floor}\label{sec:floor} The following barrier, stated in the introduction, is why the mollifier of this paper is not a Dirichlet polynomial. Let $\mu$ be the probability measure $dt/(2\pi(\frac14+t^2))$ on the critical line. \begin{proposition}[the Dirichlet floor]\label{prop:floor} Let $M(s)=\sum_{n\le N}a_nn^{-s}$ with $M(1)\ne0$. Then \[ \sum_{\re\rho>\frac12}\log\Big|\frac{\rho}{1-\rho}\Big|\le\frac12\log R(M),\qquad R(M)=\frac{\int|\zeta M(\frac12+it)|^2\,d\mu(t)}{|M(1)|^2}\ \ge\ 1+c_0, \] \[ c_0=\sum_{k\ge1}\Big(1-k(k+1)\log^2\big(1+\tfrac1k\big)\Big)=0.0803270\ldots \] In particular Jensen's inequality with a Dirichlet-polynomial mollifier cannot bound the left side, which vanishes under RH, by less than $\frac12\log(1+c_0)=0.0386$. \end{proposition} \begin{proof} Let $F(s)=(s-1)\zeta(s)/s$. Poisson--Jensen for $FM$ in the half-plane $\re s>\frac12$ at $s=1$ gives $\int\log|FM|\,d\mu\ge\log|M(1)|+\sum_{\re\rho>\frac12}\log|\rho/(1-\rho)|$, since the zeros of $M$ and a possible singular factor only increase the left side; and $\int\log|FM|\,d\mu\le\frac12\log\int|FM|^2d\mu$ by Jensen's inequality, with $|F|=|\zeta|$ on the line. For the lower bound, $\zeta(s)/s=-\int_0^\infty\{1/x\}x^{s-1}\,dx$ for $0<\re s<1$, so by Plancherel $\int|\zeta M|^2d\mu=\int_0^\infty|h(u)|^2u^{-2}\,du$ with $h(u)=\sum_na_n\{u/n\}=M(1)u-S(u)$, $S(u)=\sum_{m\le u}(a*1)(m)$. On $(0,1)$, $h=M(1)u$ contributes $|M(1)|^2$. On $[k,k+1)$ the step function $S$ is constant, and \[ \int_k^{k+1}\frac{|M(1)u-S|^2}{u^2}\,du\ge|M(1)|^2\min_{c}\int_k^{k+1}\Big(1-\frac cu\Big)^2du=|M(1)|^2\Big(1-k(k+1)\log^2\big(1+\tfrac1k\big)\Big). \] Summing over $k$ gives $R(M)\ge1+c_0$. \end{proof} \noindent Status: proved (elementary); it first appeared in version~3 of {\cite{CP}}. The terms of $c_0$ are $\frac1{12k^2}+O(k^{-3})$. With $M(1)=1$ the mollifier must approximate the line $u$ by a step function with jumps at the integers, and the sawtooth costs $c_0$: the loss comes from the passage from $\log|\cdot|$ to $|\cdot|^2$ and from the discreteness of the integers, not from the arithmetic of $\zeta$. It is the analogue, for the whole critical line, of the constant $c(\theta)>1$ that limits the method of Levinson and Conrey. The coefficients $a=\mu*b$ with $b_m=S_m-S_{m-1}$, $S_m=m(m+1)\log(1+\frac1m)$, attain the minimum on each interval up to $N$; the truncated choice gives $R-1=0.102$, $0.088$, $0.0995$, $0.0894$, $0.0872$ for $N=10$, $30$, $100$, $300$, $1000$ (\texttt{anc/dirichlet\_floor.py}). The Nyman--Beurling mollifiers $\mu(n)(1-\log n/\log N)$ give $R$ growing with $N$: they approximate $1/\zeta$, while the normalization at $s=1$ requires $1/F$. The mollifier $1/P$ of this paper is not a Dirichlet polynomial, because $P$ carries the pole through $e^{K_X(s-1)}$, and with it the bound falls below the floor (Table~\ref{tab:barrier}). \section*{Data and code} The ancillary Python scripts (NumPy, SciPy, Matplotlib; mpmath for the high-precision Weil computations) reproduce all tables and figures (the exact values of the best Dirichlet polynomial at $s_0=1$ come from \texttt{dirichlet\_exact\_gram.py}, added on 10 November 2026); two scripts new in version~2, which need only NumPy and SciPy, reproduce Table~\ref{tab:law} and Tables~\ref{tab:kpair}--\ref{tab:planted}: \texttt{ec\_bound\_numerics.py} and \texttt{ec\_phase2\_controls.py}. The file \texttt{README.txt} lists, for each script, the section, table or figure it reproduces. The computations are in double precision unless stated otherwise, without interval arithmetic; they illustrate the theorems and are not used in their proofs. \paragraph{Provenance.} This paper supersedes the preprint [C2]: its theorems are here in the same or a stronger form (Theorems~\ref{thm:decomp} and~\ref{thm:explode}, Corollaries~\ref{cor:criteria} and~\ref{cor:energy}), its heuristic law is Theorem~\ref{thm:law}, and its computations and remaining remarks are in Sections~\ref{sec:decomp}, \ref{sec:law}, \ref{sec:explode}, \ref{sec:numerics}, \ref{sec:geometry}, \ref{sec:overQ} and~\ref{sec:bdr} (Tables~\ref{tab:identity}, \ref{tab:barrier} and~\ref{tab:abscissa}, Figures~\ref{fig:BX}, \ref{fig:geom} and~\ref{fig:reflect}). The material on the Euler--Caceres bound in version~3 of {\cite{CP}} is also here, including the zero factor seen through the cross-energy (Section~\ref{sec:zerofactor}); the explicit formula in cross-energy coordinates of earlier versions of this paper is in {\cite[Sect.~2]{CE5}}, and their reflected cross-energy is in {\cite{CE3}}. Version~2 adds Theorems~\ref{thm:landau}, \ref{thm:omega} and~\ref{thm:rate}, Lemma~\ref{lem:laplace}, Corollary~\ref{cor:landau}, the unconditional $\limsup$ form of Corollary~\ref{cor:energy}, Section~\ref{sec:gaps}, Tables~\ref{tab:kpair}, \ref{tab:pairtest}, \ref{tab:dhdecomp} and~\ref{tab:planted}, the strict positivity of $J_X$, and the comparison with the Nyman--Beurling problem. \paragraph{Use of AI tools.} The proofs, computations and drafting of this paper, and of the preprint [C2] that it absorbs, were carried out with the assistance of an AI system (Claude, Anthropic), under the author's direction; an independent AI review agent refereed the proofs. The author checked the results and takes responsibility for them. \begin{thebibliography}{DLMF} \bibitem[BSY]{BSY} M. Balazard, E. Saias, M. Yor, Notes sur la fonction $\zeta$ de Riemann, 2, \emph{Adv. Math.} 143 (1999), 284--287. \bibitem[BK]{BK} E.~B. Bogomolny, J.~P. 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