\documentclass[11pt]{amsart} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{longtable,booktabs,array,calc} \usepackage{newunicodechar} \usepackage{etoolbox} \AtBeginEnvironment{longtable}{\small} \usepackage[margin=1in]{geometry} \usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} \usepackage{xcolor}\usepackage{graphicx} \newunicodechar{−}{\ensuremath{-}} \newunicodechar{σ}{\ensuremath{\sigma}} \newunicodechar{½}{\ensuremath{\tfrac12}} \newunicodechar{–}{--} \newunicodechar{—}{---} \newunicodechar{≤}{\ensuremath{\le}} \newunicodechar{…}{\ldots} \DeclareRobustCommand{\S}{\ensuremath{\mathsection}} \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \setlength{\LTleft}{0pt} \newcommand{\doi}[1]{\href{https://doi.org/#1}{doi:#1}} \title[The cross-energy of Dirichlet partial sums]{The Cross-Energy of Dirichlet Partial Sums and the Zeros of the Riemann Zeta Function} \author{Pedro Caceres} \address{Universidad Europea de Valencia, Valencia, Spain} \email{pedrojesus.caceres@universidadeuropea.es} \subjclass[2020]{Primary 11M26; Secondary 11M06, 11N05, 11Y35} \keywords{Riemann hypothesis, partial sums, explicit formula, Weil positivity, monotonicity of $\xi$, zero density, Davenport--Heilbronn function, Landau's formula} \date{\today} \begin{document} \begin{abstract} Let $X_n(s)=\sum_{k\le n}k^{-s}$ and let $C_2(n,s)=|X_n(s)|^2-|X_{n-1}(s)|^2-n^{-2\sigma}$ be the cross-energy of consecutive partial sums. At every point of the critical strip the limit of $C_2(n,s)$ is $1/(\frac14+t^2)$, $0$ or $+\infty$ according as $\sigma=\frac12$, $\sigma>\frac12$ or $\sigma<\frac12$; this trichotomy is the trace of the pole at $s=1$. On the critical line the associated helix has a natural orientation whose signed radius is Hardy's $Z(t)$, and the zeros of the truncated functions spiral into the zeros of $\zeta$ at the rate $n^{-\beta}$. We prove that the mean energy of the normalized error $E(u)$ in the prime number theorem equals $\sum_\rho\lim_nC_2(n,\rho)$ in $[0,\infty]$; it is at least $2+\gamma_0-\log4\pi$, with equality if and only if the Riemann hypothesis (RH) holds and all zeros are simple, and an unconditional local version expresses the energy in a Gaussian window as a double sum over zeros. We then study the pointwise inequality $S(\sigma,t)=\partial_\sigma\log|\xi(\sigma+it)|>0$, which is equivalent to RH: an off-line pair lowers $S$ exactly inside a disc, the failure set at abscissa $\sigma$ up to height $T$ has measure at most $N(\sigma,T+1)$, and $S$ is Weil's functional at a Poisson kernel, whose truncation is an exact finite expression in the primes up to $e^L$. Every criterion is tested against the Davenport--Heilbronn function, which has zeros off the critical line, and against its Euler-product twin $L(s,\chi)$ mod $5$: the former violates each criterion in the predicted way, the latter behaves like $\zeta$. For $\zeta$, the primes up to $10^8$ give mean energy $0.0453$ against $0.0462$. On the circle of the Euler factor of a prime $p$ the zeros lean toward the angle $\pi$, and toward $\pi+\arg\chi(p)$ for $L(s,\chi)$. We do not prove the Riemann hypothesis. \end{abstract} \maketitle \section{Introduction} Let \(X_n(s)=\sum_{k\le n}k^{-s}\) be the partial sums of the Dirichlet series of \(\zeta\), with \(s=\sigma+it\). The \emph{cross-energy} of consecutive partial sums is \[C_2(n,s)=|X_n(s)|^2-|X_{n-1}(s)|^2-n^{-2\sigma}=2\operatorname{Re}\big(X_{n-1}(s)\,\overline{n^{-s}}\big).\] It records how the newest term \(n^{-s}\) interacts with all previous ones. In a series of preprints \cite{C1}--\cite{C8} the author studied this quantity at the zeros of \(\zeta\) and related it to several criteria for the Riemann hypothesis (RH). The present paper collects the results of that series which, in our view, add something to the existing literature, states them with complete proofs, and tests every criterion against the Davenport--Heilbronn function \(f\), which satisfies a functional equation of the same type as \(\zeta\) but has zeros off the critical line. Throughout, \(\rho=\beta+i\gamma\) denotes a nontrivial zero of \(\zeta\), \(\Theta=\sup\beta\), \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\), and \[C_B=\sum_\rho\frac1{\rho(1-\rho)}=2+\gamma_0-\log4\pi=0.0461914\ldots,\] where \(\gamma_0\) is Euler's constant and the sum, unconditional, counts multiplicity and is taken in symmetric order \cite{Da,C4}. Under RH, \(\rho(1-\rho)=|\rho|^2=\frac14+\gamma^2\). \textbf{Main results.} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Theorem A (Section 2).} For \(0<\sigma<1\), \[X_{n-1}(s)\,\overline{n^{-s}}=\frac{n^{1-2\sigma}}{1-s}+\zeta(s)\,n^{-\sigma}e^{it\log n}-\tfrac12n^{-2\sigma}+O\big(|s|n^{-2\sigma-1}\big).\] Consequently \(\lim_nC_2(n,s)\) equals \(1/(\frac14+t^2)\), \(0\) or \(+\infty\) according as \(\sigma=\frac12\), \(\sigma>\frac12\) or \(\sigma<\frac12\), at every point of the strip; the zeros of \(\zeta\) appear only in the second term. The first term is the trace of the pole at \(s=1\). On the critical line the helix described by \(X_{n-1}(s)\overline{n^{-s}}\) has a natural orientation, and its signed radius is Hardy's function \(Z(t)\); each zero on the line is a crossing of the axis, and an off-line pair is a missing crossing (Turing's method). At every zero, uniformly, \(C_2(\cdot,\rho)\) changes sign exactly once beyond \(|\gamma|^{3/2}\), at \(n^*=\gamma^2+\frac13+\frac1{180\gamma^2}+O(\gamma^{-4})\) when \(\beta=\frac12\) and at distance \((2\beta-1)\gamma^2/(2(1-\beta))+O(1)\) from it otherwise (Proposition 2.2), and the zeros of the truncated function \(X_{n-1}(s)-n^{1-s}/(1-s)\) spiral into their limits at the rate \(n^{-\beta}\) (Proposition 2.3). As applications we give a zero detector on the critical line, and we show that the zeros, placed on the circle \(|p^{-s}|=p^{-1/2}\) of the Euler factor of a prime \(p\), are biased toward the angle \(\pi\), whereas on the circle of \(e\) used in \cite{E24} they are uniformly distributed; for a Dirichlet \(L\)-function the bias points toward \(\pi+\arg\chi(p)\). For the finite sums \(\sum_{k\le N}k^{-s}\), the real parts of the zeros are governed by a polygon condition in \(\pi(\sqrt N)\) angles (Proposition 2.6); adding a prime never moves their right edge to the left (Proposition 2.5), adding a power of \(2\) moves it to the left for \(2^k\le16\) (Proposition 2.7), and \(11^2\) is an exception to the corresponding rule for prime powers. At a zero on the line, \(C_2(n,\rho)=1/(\gamma^2+\frac14)-\frac1n-\frac1{12n^2}+O(|\gamma|^5n^{-5})\) increases monotonically to its limit, and RH is equivalent to the boundedness of \(C_2(\cdot,\rho)\) at every zero (Corollary 2.1). \item \textbf{Theorem B (Section 3).} Let \(E(u)\) be the normalized error term of the prime number theorem. In \([0,\infty]\), \[\limsup_{U\to\infty}\frac1U\int_0^UE(u)^2\,du=\sum_\rho\lim_{n\to\infty}C_2(n,\rho)\] when all zeros are simple; in general both sides are finite or both are infinite, and RH holds if and only if they are finite. The exponential growth rate of the energy is \(2\Theta-1\) (Theorem 3.2). An unconditional local version expresses the energy in a Gaussian window as an absolutely convergent double sum over zeros (Theorem 3.4); under RH it stays in a band of explicit width around \(C_B\). The energy is at least \(C_B\) unconditionally, with equality if and only if RH holds and all zeros are simple (Corollary 3.7). Each zero contributes an orbit of the prime signal, a closed ellipse if \(\beta=\frac12\) and an expanding spiral if \(\beta>\frac12\), whose phase also determines \(\beta\). \item \textbf{Theorem C (Section 4).} Let \(S(\sigma,t)=\partial_\sigma\log|\xi(\sigma+it)|\), so that RH is equivalent to \(S>0\) for \(\sigma>\frac12\) \cite{La,SD}. An off-line pair \(\rho,1-\bar\rho\) contributes negatively to \(S(\sigma,t)\) exactly inside the disc \((\sigma-\frac12)^2+(t-\gamma)^2<(\beta-\frac12)^2\). \item \textbf{Theorem D (Section 4).} For \(\frac12<\sigma<1\), the set of \(t\in[0,T]\) where \(S(\sigma,t)\le0\) has measure at most \(2\sqrt{\sigma(1-\sigma)}\,N(\sigma,T+1)\), so zero-density estimates make it sparse. \item \textbf{Theorem E (Section 5).} \(S(\sigma,t)\) is Weil's functional evaluated at a Poisson kernel of width \(\sigma-\frac12\). Truncating the kernel to a window of length \(L\) gives an exact finite expression in the prime powers \(n\le e^L\), which under RH lies between \((1-e^{-\delta L})^2\) and \((1+e^{-\delta L})^2\) times \(2S(\sigma,t)\), \(\delta=\sigma-\frac12\). Consequently, under RH, primes up to \(P\) resolve \(S\) at distance \(\delta\) from the critical line once \(\delta\log P\gtrsim\log(1/\varepsilon)\) (Corollary 5.1); for the Davenport--Heilbronn function, an off-line zero at horizontal distance \(\beta-\sigma\) from the probe becomes visible in the truncated functional at \(L\approx4.5/(\beta-\sigma)\). \end{enumerate} Section 6 treats the Euler product inside the strip, identifies each partial Euler product with the \(C_2\) energy of the partial sums restricted to smooth integers, and shows that twisting by characters cannot remove the pole from linear positivity arguments: a corrected partial product converges to \(\zeta\) in \(\frac12<\sigma<1\) exactly when RH holds, and at a zero on the line it decays like \(|\zeta'(\rho)|e^{-\gamma_0}/\log x\). On the critical line the corrected partial products, read along \(u=\log x\), are a superposition of helices, one for each zero, whose mean energy at \(\frac12+it\) is the curvature \(-(\log|\xi|)''(t)\) (Remark 6.5). Computed with care, the helix built from the primes up to \(10^8\) fixes the real parts of known zeros to within about \(4\cdot10^{-5}\), a precision that improves with \(x\); symmetric off-line pairs enter only at second order (Remark 6.6). The amplification that proves the analogue of RH for curves becomes over \(\mathbb Q\) a one-sided upper bound for \(\psi(x)\); Selberg's sieve gives such bounds, and on short intervals near \(10^8\) it loses a factor close to \(2\) exactly along Liouville's function, the parity barrier (Remark 6.7). On the heat-flow deformation of \(\xi\), the mean energy of the Euler helix, which is the first Laguerre expression, changes sign exactly at the first collision of zeros; its variance is the second Laguerre expression, and on the side of the primes the whole Laguerre hierarchy reduces to the finiteness of the helix energy (Remark 6.8). In the square \(v=(\sigma-\frac12)^2\) of the distance to the line, RH is equivalent to the concavity of \(\log|\xi|\) in \(v\) at every height, a criterion that combines the functional equation, the \(C_2\) energy and the Euler helix (Proposition 4.3). In the variable \(\lambda=s(1-s)\) the crossing law reads \(n^*=\lambda+\frac1{12}+O(\lambda^{-1})\). Evaluated at its own crossing, the cross-energy reduces to \(Z(t)\cos\varphi(t)\) on the line, and the zeros of \(\zeta\) are the dislocations of its nodal pattern in the strip. In energy-slope form, the argument of de la Vallée Poussin gives \(S>0\) in a region of width \(c/\log t\) to the left of \(\sigma=1\) (Proposition 4.4). Finally, the sums over the zeros that see \(\beta\) are non-holomorphic and are reached from the primes only through energies; the damped energy of the prime signal has its first singularity at \(2\sup\beta-1\) and detects an off-line zero at height \(T\) only with the primes up to about \(T^{1/\delta}\) (Remark 6.9). Sieving the wave \(C_2+iS_2\) by the primes up to \(y\) and removing the pole correction turns its free amplitude into the Euler helix, so the partial sums of Section 2 and the partial products of Section 6 describe the same object (Remark 6.10). On the critical line each prime acts on the wave by the real operator \(1-p^{-1}T_p\), the same at every height, and the sieved functional equation shows why this rigidity does not combine with the reflection: it maps the half of the strip where the Euler product converges onto the half where it diverges (Remark 6.11). We also show that no combination of cosines without a constant term can be nonnegative at every \(\log p\) (Proposition 6.12): positivity prime by prime requires the pole, and without it positivity can only be quadratic. \textbf{The Davenport--Heilbronn control.} Each criterion above is equivalent to RH or implied by it, and none uses the Euler product beyond what is visible in the numerics. We therefore test each of them on \(f\), whose off-line zeros below height \(400\) are known \cite{C5}. For \(f\) the energy grows as predicted by its off-line zeros; the failure set of the pointwise inequality consists of one interval per off-line zero and fills \(86\)--\(90\%\) of the bound of Theorem D; the truncated Weil functional becomes negative, but outside the discs of Theorem C; and the spectrum of its zeros has spikes of both signs. The \(L\)-function of the character mod \(5\), which has the same conductor and gamma factor as \(f\) and an Euler product, behaves like \(\zeta\) in every test. For \(\zeta\) the corresponding quantities behave as RH predicts. \textbf{What is classical.} Cramér's theorem, the equivalence of bounded energy with RH, Hardy's function and Turing's method, Lagarias's monotonicity criterion, Weil's criterion, Landau's formula, the density theorems and the positivity argument behind Proposition 6.4 are known, and we cite them as such. The disc lemma, the crossing law and the spiral of a zero are elementary consequences of Euler--Maclaurin summation and Lagarias's pairing argument. To our knowledge the formulation of Theorem B, the local energy law (Theorem 3.4) and the characterization of equality in Corollary 3.7, the exceptional-set bound of Theorem D, the identification and truncation of Theorem E, the reading of Landau's formula on the circles of the Euler factors (including the twisted circles), and the systematic tests against the Davenport--Heilbronn function and its Euler-product twin \(L(s,\chi)\) are new. We do not prove the Riemann hypothesis. Appendix A collects the formulas used in the paper, with their status. \section{The cross-energy and its limits} \subsection{The expansion} \textbf{Theorem A.} Let \(s=\sigma+it\) with \(0<\sigma<1\) and put \(Z_n(s)=X_{n-1}(s)\overline{n^{-s}}\), so that \(C_2(n,s)=2\operatorname{Re}Z_n(s)\). As \(n\to\infty\), \[Z_n(s)=\frac{n^{1-2\sigma}}{1-s}+\zeta(s)\,n^{-\sigma}e^{it\log n}-\frac12n^{-2\sigma}+O\big(|s|\,n^{-2\sigma-1}\big).\qquad\text{(2.1)}\] In particular \[C_2(n,s)=\frac{2(1-\sigma)}{|1-s|^2}\,n^{1-2\sigma}+2|\zeta(s)|\,n^{-\sigma}\cos\big(t\log n+\arg\zeta(s)\big)-n^{-2\sigma}+O\big(|s|n^{-2\sigma-1}\big),\] and \(\lim_nC_2(n,s)\) equals \(1/(\frac14+t^2)\) if \(\sigma=\frac12\), \(0\) if \(\sigma>\frac12\), and \(+\infty\) if \(\sigma<\frac12\). \emph{Proof.} By Euler--Maclaurin summation, \(X_{n-1}(s)=\zeta(s)-\zeta(s,n)\) with \(\zeta(s,n)=\frac{n^{1-s}}{s-1}+\frac12n^{-s}+O(|s|n^{-\sigma-1})\). Multiply by \(\overline{n^{-s}}=n^{-\sigma}e^{it\log n}\) and use \(\operatorname{Re}\frac1{1-s}=\frac{1-\sigma}{|1-s|^2}\). \(\square\) Two consequences shape the rest of the paper. \emph{The limit sees the line, not the zeros.} The limit of \(C_2\) is decided by the first term of (2.1), which depends only on \(s\). At \(s=0.7+20i\), where \(\zeta(s)\ne0\), \(C_2\to0\); at \(0.3+20i\), \(C_2\to\infty\). A zero changes only the second term, of lower order. The first term is the tail \(n^{1-s}/(1-s)\) of the harmonic series, the trace of the pole of \(\zeta\) at \(s=1\). The Davenport--Heilbronn function has coefficients of mean zero, no pole, and partial sums that converge throughout the strip, so its analogue of \(C_2\) tends to \(0\) for every \(\sigma>0\): it has no trichotomy. \emph{At a zero the spiral collapses.} Geometrically, \(Z_n\) winds around the centre \(n^{1-2\sigma}/(1-s)\) with radius \(|\zeta(s)|n^{-\sigma}\) and angle \(t\log n\); on the critical line \(\sqrt n\,(Z_n-\frac1{1-s})\) is a helix of radius \(|\zeta(\frac12+it)|\) (Figure \ref{fig:spiral}). At a zero the helix collapses and \(\arg Z_n\to\arg\frac1{1-\rho}\). RH is the statement that every collapse occurs on a vertical axis, \(\sigma=\frac12\). The two properties come from different terms of (2.1), and the expansion alone does not link them: at a hypothetical zero with \(\beta>\frac12\) the sequence \(Z_n(\rho)\) converges to \(0\) along the tilted axis without any inconsistency. For \(t=17\), \(20\), \(14.1347\) and \(21.0220\) the expansion agrees with the computed \(Z_n\) to within \(2\cdot10^{-4}\) for \(100\frac12\) or \(\beta<\frac12\). On the critical line this is the straight-line law \(|X_n(\rho)|^2=(n+\frac12)/(\frac14+\gamma^2)+O(n^{-1})\) \cite{C1}. \item If \(\beta=\frac12\), then for every integer \(n\ge1\) \[C_2(n,\rho)=\frac1{\gamma^2+\frac14}-\frac1n-\frac1{12n^2}+\frac{\operatorname{Re}(\rho)_3}{360\,n^4}+R_n,\qquad|R_n|\le2\kappa\,|(\rho)_5|\,n^{-5},\] where \((\rho)_k=\rho(\rho+1)\cdots(\rho+k-1)\) and \(\kappa=\|B_5\|_\infty/540<4.6\cdot10^{-5}\). Consequently \(C_2(n+1,\rho)>C_2(n,\rho)\) for every \(n\) with \[\frac{n^4}{n+1}>K(\rho):=\frac{|(\rho)_3|}{90}+4\kappa\,|(\rho)_5|,\] so \(C_2(\cdot,\rho)\) is eventually strictly increasing, bounded, and converges to \(1/(\gamma^2+\frac14)\) from below; the threshold is of order \((4\kappa)^{1/3}|\gamma|^{5/3}\approx0.057|\gamma|^{5/3}\). \item The Riemann hypothesis holds if and only if \(\sup_nC_2(n,\rho)<\infty\) for every zero \(\rho\), and if and only if \(\lim_nC_2(n,\rho)\) is finite and positive for every zero \(\rho\). \item Unconditionally: (a) \(\lim_nC_2(n,\rho)=1/(\gamma^2+\frac14)\) for every zero with \(0<\gamma\le3\cdot10^{12}\); (b) the same holds for a proportion greater than \(\frac5{12}\) of all zeros, counted by height; (c) for every \(\delta>0\) the number of zeros with \(0<\gamma\le T\) at which \(C_2(n,\rho)\ne O(n^{2\delta})\) is \(O_\delta(T)=o(N(T))\); (d) at every zero, \(C_2(n,\rho)\le(A_\rho+o(1))\,n^{1-2/(R_0\log\gamma)}\) with \(A_\rho=2(1-\beta)/|1-\rho|^2\) and \(R_0=5.573412\). \end{enumerate} \emph{Proof.} (i) is Theorem A with \(\zeta(\rho)=0\). (ii) By Euler--Maclaurin, \(X_{n-1}(\rho)=-\zeta(\rho,n)\) and \(\zeta(\rho,n)=\frac{n^{1-\rho}}{\rho-1}+\frac12n^{-\rho}+\frac\rho{12}n^{-\rho-1}-\frac{(\rho)_3}{720}n^{-\rho-3}+R\), with \(|R|\le\frac{\|B_5\|_\infty}{5!}\frac{|(\rho)_5|}{\beta+4}n^{-\beta-4}\). Multiply by \(-2\overline{n^{-\rho}}=-2n^{-\bar\rho}\) and take real parts; since \(n^{-\rho}n^{-\bar\rho}=n^{-1}\) no oscillating factor remains, \(-2\operatorname{Re}\frac1{\rho-1}=\frac1{\gamma^2+1/4}\) and \(\operatorname{Re}\rho=\frac12\), which gives the expansion with \(|R_n|\le2n^{-1/2}|R|\). Then \(C_2(n+1,\rho)-C_2(n,\rho)\ge\frac1{n(n+1)}-\frac{|(\rho)_3|}{360}\big(n^{-4}-(n+1)^{-4}\big)-4\kappa|(\rho)_5|n^{-5}\), and \(n^{-4}-(n+1)^{-4}\le4n^{-5}\). (iii) If \(\beta<\frac12\) the limit is infinite by (i); if \(\beta>\frac12\), then \(1-\bar\rho\) is a zero with real part \(1-\beta<\frac12\). So all limits are finite (and then equal to \(1/(\gamma^2+\frac14)>0\)) exactly when every zero has \(\beta=\frac12\). (iv) By (i) the limit is \(1/(\gamma^2+\frac14)\) whenever \(\beta=\frac12\). (a) All zeros with \(0<\gamma\le3\cdot10^{12}\) lie on the line \cite{PT}. (b) More than five twelfths of the zeros lie on the line \cite{PRZZ}. (c) By (i), \(C_2(n,\rho)\ne O(n^{2\delta})\) requires \(\beta<\frac12-\delta\), and then \(1-\bar\rho\) has real part \(>\frac12+\delta\); the number of such zeros up to height \(T\) is \(O_\delta(T)\) (Proposition 4.5), while \(N(T)\sim\frac T{2\pi}\log T\). (d) There are no zeros with \(\sigma\ge1-1/(R_0\log|t|)\), \(|t|\ge2\) \cite{MoT}; applied to \(1-\bar\rho\) this gives \(\beta\ge1/(R_0\log\gamma)\), and (i) gives the bound. \(\square\) For the \(341\) zeros with \(\gamma<600\), \(C_2(20000,\rho)\) agrees with \(1/(\gamma^2+\frac14)-\frac1N-\frac1{12N^2}\) to \(1.4\cdot10^{-12}\). The last decrease of \(C_2(\cdot,\rho)\) occurs at \(n\le0.952\,\gamma/2\pi\) for every one of them (median \(0.932\,\gamma/2\pi\)), the threshold of the approximate functional equation, while the proven threshold in (ii) is \(n_0=6\), \(39\), \(513\), \(2413\) for \(\gamma=14.13\), \(49.77\), \(236.52\), \(599.55\). Part (iv) is the unconditional part of (iii): the boundedness is proved for all zeros up to height \(3\cdot10^{12}\) and for a positive proportion of all zeros, an exponent \(2\delta\) is proved for almost all of them, and for every zero only the exponent \(1-2/(R_0\log\gamma)\), barely below \(1\), is known. The step from these statements to every zero is RH. Part (iii) is a reformulation and not a constraint: at a zero, \(C_2(n,\rho)=-2\operatorname{Re}[n^{-\bar\rho}\zeta(\rho,n)]\) depends only on the position of \(\rho\) through the Hurwitz function, and boundedness at \(\rho\) is equivalent to \(\beta\ge\frac12\). A proof of RH along these lines would need an independent reason for the boundedness of \(C_2(\cdot,\rho)\) at the zeros. \textbf{Proposition 2.2} (crossing law). Extend \(C_2\) to real \(n\ge1\) by \(X_{n-1}(s)=\zeta(s)-\zeta(s,n)\), with \(\zeta(s,n)\) the Hurwitz zeta function, and for \(s=\sigma+it\) put \[g_s(n)=-\operatorname{Re}\big[n^s\zeta(s,n)\big],\qquad A(s)=\frac{|1-s|^2}{2(1-\sigma)},\qquad c(\sigma)=(1-\sigma)\Big(\frac{1-\sigma^2}{30}-\frac{\sigma^2}{18}\Big).\] There is an absolute constant \(t_0\) such that for \(0<\sigma<1\) and \(|t|\ge t_0\): \begin{enumerate} \def\labelenumi{(\roman{enumi})} \item \(g_s\) has exactly one zero \(n^*(s)\) in \(n\ge|t|^{3/2}\), it lies in \([0.99A,1.01A]\), and \(g_s\) changes sign there from negative to positive; \item uniformly in \(\sigma\), \[n^*(s)=A(s)+\frac\sigma6+\frac{c(\sigma)}{t^2}+O(t^{-4}).\] \end{enumerate} If \(\zeta(\rho)=0\) with \(\rho=\beta+i\gamma\), then \(C_2(n,\rho)=2n^{-2\beta}g_\rho(n)\) for every real \(n\ge1\). Hence, for every zero with \(|\gamma|\ge t_0\), \(C_2(\cdot,\rho)\) has exactly one sign change in \(n\ge|\gamma|^{3/2}\), at \(n^*(\rho)\), and \[n^*(\rho)-\Big(\gamma^2+\frac13\Big)=(2\beta-1)\Big(\frac{\gamma^2}{2(1-\beta)}-\frac16\Big)+\frac{c(\beta)}{\gamma^2}+O(\gamma^{-4}).\qquad\text{(2.2)}\] In particular \(n^*(\rho)=\gamma^2+\frac13+\frac1{180\gamma^2}+O(\gamma^{-4})\) if \(\beta=\frac12\), while for \(\beta\ne\frac12\) the deviation is \((2\beta-1)\gamma^2/(2(1-\beta))+O(1)\). For an off-line pair the two deviations add up to \((2\beta-1)^2\gamma^2/(2\beta(1-\beta))+O(\gamma^{-2})\ge0\). \emph{Proof.} Euler--Maclaurin summation of \(x^{-s}\) over \(x=n,n+1,\dots\) gives, for every \(m\ge1\), \[\zeta(s,n)=\frac{n^{1-s}}{s-1}+\frac12n^{-s}+\sum_{j=1}^m\frac{B_{2j}}{(2j)!}(s)_{2j-1}n^{-s-2j+1}+R_m,\qquad |R_m|\le\frac{\|B_{2m+1}\|_\infty}{(2m+1)!}\,\frac{|(s)_{2m+1}|}{\sigma+2m}\,n^{-\sigma-2m},\] where \((s)_k=s(s+1)\cdots(s+k-1)\) and \(\|B_{2m+1}\|_\infty\) is the maximum of the Bernoulli polynomial on \([0,1]\). With \(m=2\) and \(\operatorname{Re}\frac1{1-s}=\frac1{2A}\), \[g_s(n)=\frac n{2A}-\frac12-\frac\sigma{12n}+\frac{\operatorname{Re}(s)_3}{720\,n^3}+E(n),\qquad |E(n)|\le\kappa\,|(s)_5|\,n^{-4},\quad\kappa=\frac{\|B_5\|_\infty}{480}<5.1\cdot10^{-5}.\] For \(n\ge|t|^{3/2}\) the last three terms are bounded by \(\frac1{12}|t|^{-3/2}+(|t|+3)^3|t|^{-9/2}/720+\kappa(|t|+5)^5|t|^{-6}\), which is less than \(0.002\) for \(|t|\ge16\). Hence \(g_s<0\) for \(|t|^{3/2}\le n\le0.99A\) and \(g_s>0\) for \(n\ge1.01A\) (note \(A\ge t^2/2\)). On \([0.99A,1.01A]\) we have \(g_s'(n)=\frac1{2A}+\frac\sigma{12n^2}+O(|t|^3n^{-4}+|t|^6n^{-5})\), and since \(n\asymp A\ge t^2/2\) the error is \(O(t^{-2})\) times \(\frac1{2A}\); so \(g_s\) is increasing there and has one zero, which proves (i). For (ii) take \(m=4\), so that the remainder contributes \(O(|t|^9n^{-8})=O(t^{-7})\) to \(g_s\) near \(n=A\), and the Bernoulli terms with \(j=3,4\) contribute \(O(t^{-6})\). Write \(n^*=A+\frac\sigma6+\varepsilon\). Expanding \(\frac\sigma{12n}=\frac\sigma{12A}\big(1-\frac\sigma{6A}+O(\frac{1+|\varepsilon|}{A^2})\big)\) and using \(\operatorname{Re}(s)_3=-3(\sigma+1)t^2+O(1)\), the equation \(g_s(n^*)=0\) becomes \[\frac\varepsilon{2A}+\frac{\sigma^2}{72A^2}-\frac{(\sigma+1)t^2}{240A^3}=O(t^{-6}),\] so \(\varepsilon=-\frac{\sigma^2}{36A}+\frac{(\sigma+1)t^2}{120A^2}+O(t^{-4})\), and \(A^{-1}=2(1-\sigma)t^{-2}+O(t^{-4})\) gives \(\varepsilon=c(\sigma)t^{-2}+O(t^{-4})\). All constants are independent of \(\sigma\in(0,1)\), because \(A\ge t^2/2\). At a zero, \(X_{n-1}(\rho)=-\zeta(\rho,n)\), and multiplying by \(\overline{n^{-\rho}}=n^{\rho-2\beta}\) gives \(C_2(n,\rho)=2n^{-2\beta}g_\rho(n)\). Finally \(A(\rho)-\gamma^2-\frac14=(2\beta-1)\big(\frac{\gamma^2}{2(1-\beta)}-\frac14\big)\) and \(\frac\beta6-\frac1{12}=\frac{2\beta-1}{12}\), which gives (2.2); \(c(\frac12)=\frac1{180}\). Adding (2.2) for \(\beta\) and \(1-\beta\) gives the statement on pairs. \(\square\) The constant \(\frac13\) is the sum of \(\frac14\), from the leading term of Theorem A, and \(\frac1{12}\), from the first Bernoulli term; the next term, \(\frac1{180\gamma^2}\), comes from the first two Bernoulli terms together. For the first eight zeros of \(\zeta\) the root of \(C_2(\cdot,\frac12+i\gamma)\) agrees with \(\gamma^2+\frac13+\frac1{180\gamma^2}\) to within \(3\cdot10^{-7}\), against \(3\cdot10^{-5}\) for \(\gamma^2+\frac13\) (at \(\gamma=14.1347\) the root is \(200.1238157\), and \(\gamma^2+\frac13+\frac1{180\gamma^2}=200.1238160\)); with integer \(n\), \(C_2(n,\rho_1)\) is negative for \(n\le200\) and positive for \(n\ge201\). The range \(n\ge|t|^{3/2}\) in (i) comes from the use of a fixed number of Bernoulli terms. Numerically the sign change is unique from about \(n=|\gamma|/2\pi\) on, the threshold of the approximate functional equation: for \(\gamma=37.59\), \(79.34\) and \(165.54\) the function \(C_2(\cdot,\rho)\) has \(9\), \(28\) and \(98\) sign changes in \(10.\] The numbers \(n^*-\frac1{12}\) are therefore, up to \(O(\lambda^{-1})\), the real zeros of an entire function defined by positive moments, without reference to the zeros. We computed \(b_0,\dots,b_{400}\) by the trapezoidal rule on the whole line (step \(\frac1{400}\), decimal arithmetic with \(260\) digits); \(\Psi(\frac14)=0.4971207781883141=\xi(\frac12)\). The truncation of the series after \(K=50\), \(100\), \(200\), \(400\) terms has its first \(3\), \(10\), \(24\) and \(25\) zeros (all those with \(\lambda<8000\)) equal to \(\frac14+\gamma^2\) within \(10^{-8}\), the accuracy of our table of zeros, and \(n^*-\frac1{12}-\lambda\) is \(2.8\cdot10^{-5}\) at the first zero, the term \(1/(180\lambda)\). RH is equivalent to \(\Psi\) having only real zeros; in Pólya's formulation \cite{Po}, to the hyperbolicity of all the Jensen polynomials \(J^{d,n}(X)=\sum_{j\le d}\binom dj\hat\gamma(n+j)X^j\), \(\hat\gamma(j)=j!\,b_j/(2j)!\), which Griffin, Ono, Rolen and Zagier proved for each fixed \(d\) and all sufficiently large \(n\) \cite{GORZ}. All \(J^{d,n}\) with \(d\le150\) and \(n\in\{0,1,2,5,10\}\) are hyperbolic, as expected from the verification of RH to large height. Proposition 2.2 thus says that the crossing of the cross-energy measures the spectral variable \(\lambda\), with an explicit correction. It is a reading of a classical reformulation, not a new constraint. Status: the identity is proved; the computations are numerical. \textbf{The cross-energy at its own crossing.} Let \(0<\sigma<1\), \(|t|\ge t_0\), let \(n^*(s)\) be the crossing point of Proposition 2.2, and put \(K(s)=C_2(n^*(s),s)\), with \(C_2\) extended to real \(n\) as there. Then: \begin{enumerate} \def\labelenumi{(\roman{enumi})} \item \(K(s)=2\,n^*(s)^{-\sigma}\operatorname{Re}\big[\zeta(s)\,n^*(s)^{it}\big]\); \item on the critical line \(K(\frac12+it)\sqrt{n^*}/2=Z(t)\cos\varphi(t)\), with \(\varphi(t)=t\log n^*(\frac12+it)-\theta(t)\); \item \(\varphi'(t)=\frac32\log t+O(1)\), so the zeros of \(K\) on the critical line up to height \(T\) are the zeros of \(Z\) together with \(\varphi(T)/\pi+O(1)=\frac{3}{2\pi}T\log T+O(T)\) further zeros, asymptotically three for each zero of \(\zeta\); \item the zeros of \(\zeta\) in the strip are the common zeros of \(K\) and of \(K^\perp(s)=2\,n^*(s)^{-\sigma}\operatorname{Im}\big[\zeta(s)\,n^*(s)^{it}\big]\), and at a zero of multiplicity \(m\) the phase of \(K+iK^\perp\) turns by \(2\pi m\) along a small circle. \end{enumerate} \emph{Proof.} With \(X_{n-1}(s)=\zeta(s)-\zeta(s,n)\) we have \(C_2(n,s)=2\operatorname{Re}[\zeta(s)n^{-\bar s}]-2\operatorname{Re}[\zeta(s,n)n^{-\bar s}]\), and \(\operatorname{Re}[\zeta(s,n)n^{-\bar s}]=n^{-2\sigma}\operatorname{Re}[n^s\zeta(s,n)]=-n^{-2\sigma}g_s(n)\), which vanishes at \(n=n^*(s)\); since \(n^{-\bar s}=n^{-\sigma}n^{it}\), this is (i). On the line \(\zeta(\frac12+it)=e^{-i\theta(t)}Z(t)\), which gives (ii). By Proposition 2.2, \(n^*(\frac12+it)=t^2+O(1)\), so \(\frac d{dt}[t\log n^*]=2\log t+2+O(t^{-1})\), while \(\theta'(t)=\frac12\log\frac t{2\pi}+O(t^{-2})\); thus \(\varphi'(t)=\frac32\log t+O(1)>0\) for large \(t\), \(\cos\varphi\) has \(\varphi(T)/\pi+O(1)\) zeros up to \(T\), and \(N(T)\sim\frac T{2\pi}\log T\) gives the ratio \(3\), which is (iii). For (iv), \(K+iK^\perp=2n^{*-\sigma}n^{*it}\zeta(s)\), and \(n^*(s)\) is a positive smooth function of \(s\), so the factor in front of \(\zeta\) is smooth and does not vanish and the index of \(K+iK^\perp\) at a zero equals that of \(\zeta\). \(\square\) \(K\) can be read from the partial sums: at \(s=\frac12+20i\), \(n^*=400.33\), and \(C_2(400,s)=0.0849\) from \(X_{399}(s)\) against \(K(s)=0.0861\); the difference is the smooth part at the integer point. Identity (ii) holds numerically to \(10^{-13}\), and on \(10n_0=\big(2|\zeta'(\rho)|\,|\beta-\frac12|\big)^{-1/\beta}\). For the first \(18\) zeros of \(\zeta\) and \(10^4\le n\le2\cdot10^4\) the number of crossings differs from \(\gamma\log2/\pi\) by at most \(1\) (\(3\) against \(3.12\) at \(\rho_1\), \(16\) against \(15.90\) at \(\gamma=72.07\)). For the Davenport--Heilbronn function, with \(n\) a multiple of \(5\), the on-line zero at \(\gamma=80.33\) gives \(18\) crossings against \(17.7\) for \(N=500\), \(2000\) and \(8000\), and the two zeros of the off-line pair at height \(85.699\) give none (Figure \ref{fig:spiralcross}). Summed over the zeros up to height \(T\), the count is a version of Turing's method in which each zero carries the weight \(\gamma\), and an off-line pair removes about \(2\gamma\log2/\pi\) crossings; like Turing's method it verifies RH up to the computed height and no further. The threshold \(\log n_0\approx2\log\big(1/(2|\zeta'(\rho)|\,|\beta-\frac12|)\big)\) is much shorter than the detection length of the prime signal (Section 5), because \(F_n\) evaluates \(\zeta\) itself. For a close off-line pair \(|\zeta'(\rho)|\) is of order \(|\beta-\frac12|\), and \(n_0\) grows like \(|\beta-\frac12|^{-4}\). \begin{figure}[p]\centering\includegraphics[width=\textwidth]{Spiral_crossings.png} \caption{Crossings of the spiral. (a) $\operatorname{Re}\rho_n-\frac12$ for $2000\le n\le4000$: the first zero of $\zeta$, an on-line zero of the Davenport--Heilbronn function, and the two zeros of its off-line pair at height $85.699$. (b) Crossings of the critical line for $10^4\le n\le2\cdot10^4$ against $\gamma\log2/\pi$. (c) The length $\log n_0$ after which an off-line zero at distance $\delta$ from the line stops crossing, for $|\zeta'(\rho)|=1$ and for a close pair, against the detection length of the prime signal.}\label{fig:spiralcross}\end{figure} \subsection{A detector of zeros on the critical line} By Corollary 2.1, at a zero \(\frac12+i\gamma\) the quantity \(D_n(t)=|X_n(\frac12+it)|^2-(n+\frac12)/(\frac14+t^2)\) is \(O(1/n)\) for every \(n\), whereas at other points (2.1) shows that it oscillates in \(n\) with amplitude about \(2|\zeta(\frac12+it)|\sqrt{n/(\frac14+t^2)}\). A single value of \(n\) is not enough: the equation \(D_n(t)=0\) has many spurious solutions, because \(D_n(t)\) oscillates in \(t\) with period about \(2\pi/\log n\). Requiring smallness for several \(n\) removes them. With \(\max_{n\in\{2000,2500,3000,3500,4000\}}|D_n(t)|\) on a grid of step \(0.002\) in \(14\le t\le31\), the local minima below \(10^{-2}\) occur exactly at \(14.134\), \(21.022\), \(25.010\) and \(30.424\), with values between \(2\cdot10^{-4}\) and \(5\cdot10^{-3}\) (limited by the grid), while the median elsewhere is \(7.7\). The detector is valid but not efficient: each evaluation costs \(O(n)\) operations, against \(O(\sqrt t)\) for the Riemann--Siegel formula, which is itself a truncated \(C_2\) sum \cite{C1}. It finds zeros on the critical line only, and each zero is detected by its own identity; a known zero gives no information about the next one \cite{Er}. \subsection{The circle of a prime} The construction of \cite{E24} placed each zero \(\frac12+i\gamma\) on the circle \(|w|=e^{-1/2}\), \(w=e^{-s}\), at the angle \(-\gamma\bmod2\pi\), which is the circle of the Euler factor \((1-e^{-s})^{-1}\) of a ``prime'' of size \(e\). By Hlawka's theorem the angles \(\gamma\bmod2\pi\) are uniformly distributed \cite{Hl}, so the circle carries no arithmetic information \cite{Er}. With a true prime the situation changes. Landau's formula \cite{Lan} states that for fixed \(x>1\), \[\sum_{0<\gamma\le T}x^{\rho}=-\frac T{2\pi}\Lambda(x)+O_x(\log T).\qquad\text{(2.3)}\] Under RH, \(x^\rho=\sqrt x\,e^{i\gamma\log x}\), and (2.3) says that the angles \(\gamma\log p\bmod2\pi\) of the zeros on the circle \(|p^{-s}|=p^{-1/2}\) are biased toward \(\pi\), with mean cosine about \(-\Lambda(p)/(\sqrt p\log\frac T{2\pi e})\), while for bases that are not prime powers there is no bias. Results on the distribution of \(\gamma\alpha\bmod1\) in this direction are due to Ford and Zaharescu \cite{FZ}. For the \(202\) zeros with \(0<\gamma<400\) the mean of \(\cos(\gamma\log b)\) is: \begin{longtable}[]{@{}llllllll@{}} \toprule\noalign{} base \(b\) & \(2\) & \(3\) & \(5\) & \(7\) & \(e\) & \(6\) & \(10\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot measured & \(-0.147\) & \(-0.194\) & \(-0.226\) & \(-0.222\) & \(-0.003\) & \(+0.014\) & \(+0.009\) \\ from (2.3) & \(-0.155\) & \(-0.200\) & \(-0.227\) & \(-0.232\) & \(0\) & \(0\) & \(0\) \\ \end{longtable} The full spectrum \(\sum_{0<\gamma<400}\cos(\gamma\log n)\), \(2\le n\le40\), has spikes exactly at the prime powers, all negative and within a few percent of \(-(T/2\pi)\Lambda(n)/\sqrt n\) (Figure \ref{fig:primes}). For the Davenport--Heilbronn function the same sum over its zeros, \(\sum\operatorname{Re}n^{\rho-1/2}\), reproduces \(-(T/2\pi)b(n)/\sqrt n\), where \(b(n)\) are the Dirichlet coefficients of \(-f'/f\); these take both signs and are not supported on prime powers (for example \(+46.6\) at \(n=4\), \(-50.7\) at \(n=6\), \(+49.1\) at \(n=9\)). The one-signed spectrum is the Euler product seen from the zeros. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Zeros_know_primes.png} \caption{(a) $\sum_{0<\gamma<400}\cos(\gamma\log n)$ for $2\le n\le40$ (bars) and Landau's prediction $-(T/2\pi)\Lambda(n)/\sqrt n$ (marks). (b) The same sum for the Davenport--Heilbronn function and $-(T/2\pi)b(n)/\sqrt n$. (c) Histogram of $\gamma\log b\bmod2\pi$ for $b=2$ and for $b=e$, the circle of [E24]. (d) Fraction of the prime spikes ($p\le13$) kept after an independent Gaussian displacement of each zero, in units of the mean spacing; shaded band: GUE points of the same density.}\label{fig:primes}\end{figure} The arithmetic information lies in the exact positions of the zeros, not in their local statistics. Displacing each zero independently by a Gaussian amount of standard deviation \(\varepsilon\) times the local mean spacing keeps a fraction \(0.96\), \(0.89\), \(0.73\) and \(0.32\) of the prime spikes (\(p\le13\)) for \(\varepsilon=0.02\), \(0.1\), \(0.2\) and \(0.5\). Points with the GUE statistics and the same density keep \(0.04\pm0.06\). \textbf{Twisted circles.} The same holds for Dirichlet \(L\)-functions, with the character visible in the angle. Let \(\chi\) be the character mod \(5\) with \(\chi(2)=i\). Its \(L\)-function has \(303\) zeros with \(0<\gamma<400\) and \(303\) with \(-400<\gamma<0\), all on the critical line, against \(303.2\) for each sign predicted by the Riemann--von Mangoldt formula. Fitting \(\sum_{|\gamma|<400}n^{i\gamma}\), \(2\le n\le40\), by \(c\,\Lambda(n)\chi(n)/\sqrt n\) gives \(c=-126.6\), against \(-T/\pi=-127.3\), with relative residual \(3\%\); at integers that are not prime powers the sum is at most \(7.9\) in modulus, against a median of \(76\) at prime powers. On the circle of \(p\) the zeros therefore lean toward the angle \(\pi+\arg\chi(p)\). The measured mean directions are \(-0.496\pi\), \(0.504\pi\), \(-0.503\pi\) and \(-1.000\pi\) for \(p=2,3,7,11\), against \(-\frac\pi2,\frac\pi2,-\frac\pi2,\pi\). \textbf{Remark 2.4} (the angles do not see \(\beta\)). The angle of a zero \(\rho=\beta+i\gamma\) on the circle of \(p\) is \(\gamma\log p\), which does not depend on \(\beta\); an off-line zero differs from an on-line zero at the same height only through the modulus \(x^{\beta-1/2}\) of its term in (2.3). The joint distribution of the angles \((\gamma\log p)_p\) has Fourier coefficients \(\sum_\gamma(m/n)^{i\gamma}\), which Landau's formula determines unconditionally. Statistics of the angles alone are therefore consistent with off-line zeros. The quadratic version of the question, how small a Dirichlet polynomial \(\sum a_nn^{i\gamma}\) can be at all zeros simultaneously, is Weil's form in a basis of Dirichlet polynomials (Section 5). \textbf{Zeros of the sections.} Remark 2.4 has a counterpart for the finite sums \(S_N(s)=\sum_{k\le N}k^{-s}\). With \(x_p=p^{-s}\) for \(p\le N\), \(S_N\) becomes a polynomial \(P_N\) in \(\pi(N)\) variables, and since the numbers \(\log p\) are linearly independent, Kronecker's theorem makes \((p^{-it})_{p\le N}\) dense in the torus \(\mathbb T^{\pi(N)}\). Hence the closure of the set of real parts of the zeros of \(S_N\) is the set \(A_N\) of \(a\) for which \(P_N(p^{-a}w_p)=0\) for some \(w\in\mathbb T^{\pi(N)}\). A zero is a closed polygon with sides of lengths \(k^{-a}\) in which only the directions of the prime sides are free. For \(N=2\) the zeros are the points \(i(2m+1)\pi/\log2\). For \(N=3\) the condition is that \(1\), \(2^{-a}\) and \(3^{-a}\) form a triangle, and \(A_3=[-1,0.7879]\), the right end being the root of \(2^{-a}+3^{-a}=1\). Let \(e(N)=\sup A_N\) and \(m_N(a)=\min_{w}|P_N(p^{-a}w_p)|\). \textbf{Proposition 2.5} (the prime step). If \(p\) is prime, then \(e(p)=\sup\{a\ge0:\ m_{p-1}(a)\le p^{-a}\}\); in particular \(e(p)\ge e(p-1)\). \emph{Proof.} Let \(a\ge0\). The image of the torus under \(w\mapsto P_{p-1}(q^{-a}w_q)\) is connected, and at \(w=1\) it takes the value \(\sum_{kp^{-a}\), then \(|P_{p-1}|>|p^{-a}w_p|\) on the torus and \(P_p\) has no zero. Since \(m_{p-1}=0\) on \(A_{p-1}\), \(e(p)\ge e(p-1)\). \(\square\) For a composite \(n\) the direction of \(n^{-s}\) is fixed by those of its prime factors, and no such statement holds. We computed \(e(N)\) for \(3\le N\le60\) by minimizing \(|P_N|\) on the torus; each value is a lower bound, since the zero found is genuine. In all \(15\) cases where \(N\) is prime, \(e(N)>e(N-1)\), as Proposition 2.5 requires. In all \(8\) cases where \(N\) is a prime power (\(4,8,9,16,25,27,32,49\)), \(e(N)1\) and all large \(N\) would imply RH \cite{Tu}; Montgomery showed that \(S_N\) has zeros with \(\sigma>1+(\frac4\pi-1-\varepsilon)\log\log N/\log N\) \cite{Mo83}, and Montgomery and Vaughan gave the matching bound from the other side \cite{MV}. The configuration in which all primes have direction \(\pi\), which turns \(P_N\) into the Liouville sum \(\sum_{n\le N}\lambda(n)n^{-a}\) of Turán's problem, is not extremal: its largest root is \(0.901\) at \(N=19\), against \(e(19)=1.007\). The dimension of the problem can be reduced in general. Two primes larger than \(\sqrt N\) have product larger than \(N\), so every \(n\le N\) contains at most one of them, to the first power. \textbf{Proposition 2.6} (polygon reduction). For \(N\ge2\) and \(a\ge0\) write \[P_N=R_N(w')+\sum_{\sqrt NN/2\)). Then \(a\in A_N\) if and only if there is \(w'\in\mathbb T^{\pi(\sqrt N)}\) for which the largest of the numbers \(|R_N(w')|\) and \(p^{-a}|C_p(w')|\), \(\sqrt N\sqrt N\) appear in no \(R_N\) and no \(C_q\), and each appears in exactly one term. For fixed \(w'\), the sums \(\sum_pw_p\ell_p\) with free \(|w_p|=1\) and lengths \(\ell_p=p^{-a}|C_p(w')|\) fill the annulus \(\max(0,2\max_p\ell_p-\sum_p\ell_p)\le|z|\le\sum_p\ell_p\). Hence \(P_N\) vanishes for some choice of these coordinates if and only if \(|R_N(w')|\) lies in that annulus, which is the polygon condition. \(\square\) By Bertrand's postulate there is always a prime in \((N/2,N]\), a side of length \(p^{-a}\) independent of \(w'\); Proposition 2.5 is the case in which this is the only change from \(N-1\) to \(N\). For \(4\le N\le8\) the problem has one variable, \(w_2\), and for \(9\le N\le24\) two. For \(N=4\) the sides are \(|1+z+z^2|\), \(z=2^{-a}w_2\), and \(3^{-a}\); the minimum of \(|1+z+z^2|\) on \(|w_2|=1\) is \(\frac{\sqrt3}2(1-4^{-a})\) when \(2^{-a}\ge2-\sqrt3\), so \(e(4)=0.6263\) is the root of \(3^{-a}=\frac{\sqrt3}2(1-4^{-a})\), and \(e(4)2/\sqrt3-1\), which holds. \textbf{Proposition 2.7.} \(e(2^k)1}n^{-a}\) exceeds the sum of the other sides (\(0.857\) against \(0.053\) for \(N=8\), \(0.814\) against \(0.014\) for \(N=16\)). \(\square\) The computation is in double precision with explicit error bounds; a version in interval arithmetic would make it a computer-assisted proof in the strict sense. With the reduction we extended the computation of \(e(N)\) to \(N\le128\). The powers \(64\), \(81\), \(125\) and \(128\) again lower the edge, from \(1.0320\), \(1.0328\), \(1.0392\), \(1.0383\) to \(1.0277\), \(1.0300\), \(1.0380\), \(1.0366\), but \(121=11^2\) raises it slightly, from \(1.03714\) to \(1.03720\). At \(a=1.03719\) the polygon for \(N=121\) closes (defect \(-2\cdot10^{-5}\)), while \(1600\) independent minimizations for \(N=120\) return the same positive defect \(1.8\cdot10^{-4}\); this last statement is numerical. The regularity of prime powers is therefore not a general law. The powers of \(2\) behave differently because of a mechanism that can be read off the extremal configurations. In \(P_{2^k-1}\) the powers of \(2\) form the geometric sum \(1+z+\dots+z^{k-1}=(1-z^k)/(1-z)\), and at the edge the direction of \(w_2\) sits near a \(k\)-th root of unity, where this sum nearly cancels: \(k\theta_2\bmod2\pi\) is \(0.008\pi\), \(0.067\pi\), \(0.128\pi\), \(0.227\pi\) for \(k=3,\dots,6\). The new term \(z^k\) completes the geometric sum and undoes the cancellation. In the form of Proposition 2.6 the edge of \(N-1\) is the point where the minimum of the polygon defect reaches \(0\). If this minimum is attained at a single point \(w^*\), where the side \(R_{N-1}\) is the largest, adding \(\varepsilon q^{-a}w_p^k\), \(q=p^k\), changes the defect by \(\varepsilon q^{-a}\operatorname{Re}\big(\overline{R_{N-1}(w^*)}\,w_p^{*k}\big)/|R_{N-1}(w^*)|+O(\varepsilon^2)\), so the edge moves to the left when this projection is positive. For \(q=2^k\) the normalized projection is \(0.93\), \(0.92\), \(0.94\), \(0.99\) for \(k=3,\dots,6\). For \(11^2\) the powers of \(11\) give only \(1+z_{11}\), the projection is \(0.10\), and the edge moves to a second local minimum. We expect \(e(2^k)0}\frac{\cos\gamma a}{\frac14+\gamma^2}\] exists. It is positive-definite, because it is an autocorrelation. For every test function \(\varphi\), \[\lim_{U\to\infty}\frac1U\int_0^U|(\varphi*E)(u)|^2\,du=\sum_\rho\frac{|\hat\varphi(\gamma_\rho)|^2}{\frac14+\gamma_\rho^2}\ \ge0.\] In particular \(R(0)=\sum_\rho|\rho|^{-2}=2+\gamma_0-\log4\pi=C_B\), the constant of the introduction \cite[\S 8.3]{C4}. \item Conversely, if \(\frac1U\int_0^UE(u)^2\,du\) is bounded as \(U\to\infty\), then RH holds. \end{enumerate} \emph{Proof.} (a) Under RH, \(E\) is a Besicovitch almost periodic function with Fourier--Bohr coefficients \(-1/\rho\) at the frequencies \(\gamma_\rho\) (Cramér \cite{Cr}; see also \cite[Ch. V]{In}). Parseval's identity for almost periodic functions gives \(R\) and the filtered energy. For simple zeros, \(\sum_\rho|\rho|^{-2}=\sum_\rho\frac1{\rho(1-\rho)}=2+\gamma_0-\log4\pi\). \begin{enumerate} \def\labelenumi{(\alph{enumi})} \setcounter{enumi}{1} \tightlist \item Suppose the running energy is bounded. By summation by parts, \(\int_0^\infty E(u)^2e^{-2\varepsilon u}\,du<\infty\) for every \(\varepsilon>0\). By Cauchy--Schwarz, the Laplace transform \(\int_0^\infty E(u)e^{-(s-1/2)u}\,du\) then converges absolutely for \(\operatorname{Re}s>\tfrac12\) and defines an analytic function there. For \(\operatorname{Re}s>1\) it equals \[\int_1^\infty\big(\psi(x)-x\big)x^{-s-1}\,dx+(\text{terms analytic in }\operatorname{Re}s>0)=-\frac{\zeta'(s)}{s\,\zeta(s)}-\frac1{s-1}+(\text{analytic}).\] So \(\zeta'/\zeta\) continues analytically to \(\operatorname{Re}s>\tfrac12\), and \(\zeta\) has no zeros there. \(\square\) \end{enumerate} So a Weil-type positivity, with the weight \(1/(\frac14+\gamma^2)\), holds \emph{by construction} as soon as the prime signal has finite energy. The sign problem becomes a size problem, and the size problem is exactly RH. The weight \(1/(\frac14+\gamma^2)\) is the limit of \(C_2\) at the zeros and the harmonic measure seen from the pole \cite[\S\S 2, 4]{C4}. So the \(C_2\) weight is precisely what converts Weil's distribution into the autocorrelation of a real arithmetic signal. \textbf{The growth rate of the energy.} Proposition 3.1(b) has a quantitative form. Let \(\Theta=\sup_\rho\operatorname{Re}\rho\), so that \(\frac12\le\Theta\le1\) and RH is the statement \(\Theta=\frac12\), and write \(F(U)=\int_0^UE(u)^2\,du\). \textbf{Theorem 3.2.} If \(\Theta>\frac12\), then \[\limsup_{U\to\infty}\frac1U\log F(U)=2\Theta-1.\] If \(\Theta=\frac12\), then \(F(U)\sim C_BU\). In particular RH holds if and only if \(F(U)=O(U)\), and if and only if \(F\) grows subexponentially. \emph{Proof.} The case \(\Theta=\frac12\) is Proposition 3.1(a). Let \(\Theta>\frac12\) and let \(\delta_c\) be the abscissa of convergence of \(I(\delta)=\int_0^\infty E(u)^2e^{-2\delta u}\,du\). Since \(F\) is nondecreasing and unbounded, the Laplace--Stieltjes abscissa formula gives \(\limsup_U\frac1U\log F(U)=2\delta_c\), and it remains to show \(\delta_c=\Theta-\frac12\). The classical bound \(\psi(x)-x\ll x^\Theta\log^2x\) \cite[Ch. IV]{In} gives \(E(u)\ll u^2e^{(\Theta-1/2)u}\), so \(I(\delta)<\infty\) for \(\delta>\Theta-\frac12\). Conversely, if \(I(\delta)<\infty\) for some \(\delta<\Theta-\frac12\), the Cauchy--Schwarz argument in the proof of Proposition 3.1(b) shows that \(\zeta'/\zeta\) continues analytically to \(\operatorname{Re}s>\frac12+\delta\). This contradicts the existence of a zero with \(\beta>\frac12+\delta\). \(\square\) In the language of the series, the energy of the prime signal is a sum of \(C_2\) slopes. At a zero \(\rho\) on the critical line the partial sums satisfy the straight-line law \(|X_n(\rho)|^2=(n+\frac12)/|\rho|^2+O(n^{-1})\) of \cite{C1} (Corollary 2.1), so \(1/|\rho|^2=1/(\frac14+\gamma^2)\) is the slope of the \(C_2\) energy at \(\rho\). Proposition 3.1(a) states that the mean energy of the primes is the sum of these slopes over all zeros, and Theorem 3.2 states that the exponential growth rate of the same energy is \(2\Theta-1\), twice the distance from the critical line of the zero farthest from it. The same argument applies to the Davenport--Heilbronn signal \(E_f\), whose Laplace transform is \(-f'(s)/(sf(s))\) without a pole term: the growth rate of its energy is \(2\Theta_f-1\). Davenport and Heilbronn proved that \(f\) has zeros with real part greater than \(1\) \cite{DH}, so \(2\Theta_f-1>1\). These zeros lie high in the strip and have small weight \(1/|\rho|^2\), so in any computable range the growth of \(E_f\) is governed by the low off-line zeros listed in \cite{C5}. \textbf{The \(C_2\) energy law.} Proposition 3.1 and Theorem 3.2 combine with Corollary 2.1 into a single identity. \textbf{Theorem B} (\(C_2\) energy law). In \([0,\infty]\), \[\limsup_{U\to\infty}\frac1U\int_0^UE(u)^2\,du\;=\;\sum_\rho\lim_{n\to\infty}C_2(n,\rho)\qquad\text{if all zeros are simple,}\] and in general both sides are finite or both are infinite. RH holds if and only if they are finite, and then the right side equals \(\sum_\rho|\rho|^{-2}=C_B\). \emph{Proof.} Suppose RH fails, and let \(\rho\) be a zero with \(\beta>\frac12\). Then \(\Theta>\frac12\), so \(\limsup_U\frac1U\log F(U)>0\) by Theorem 3.2 and the left side is \(+\infty\). The reflected zero \(1-\bar\rho\) has real part \(1-\beta<\frac12\), so by Corollary 2.1 its term on the right is \(+\infty\); all other terms are nonnegative. Suppose RH holds. By Corollary 2.1 the right side is \(\sum_\rho(\frac14+\gamma^2)^{-1}=C_B\), counted with multiplicity. By Cramér's theorem (Proposition 3.1(a)) the left side is a limit, equal to \(\sum_{\rho\ \mathrm{distinct}}m_\rho^2/|\rho|^2\), where \(m_\rho\) is the multiplicity. This is finite because \(m_\rho\ll\log|\gamma|\), and it equals \(C_B\) when all zeros are simple. \(\square\) Thus the mean energy of the primes is the sum over the zeros of the limiting values of \(C_2\), and RH is the statement that no zero has an infinite \(C_2\) limit. This is the trichotomy of Theorem A read through the primes. The identity is a reformulation, not a new constraint: proving that the right side is finite is RH. \textbf{The Nyman--Beurling distance.} The same constant appears in an approximation problem in which the positivity comes from a Gram matrix built from the integers alone. Let \(\rho_k(x)=\{1/(kx)\}\), \(\chi\) the indicator of \((0,1)\), and \[d_N^2=\inf_{c_1,\dots,c_N}\int_0^\infty\Big|\chi(x)-\sum_{k\le N}c_k\rho_k(x)\Big|^2dx=1-b^{\mathsf T}G^{-1}b,\] with \(b_k=\langle\chi,\rho_k\rangle=(\log k+1-\gamma_0)/k\) and \(G_{jk}=\langle\rho_j,\rho_k\rangle\), given in closed form by Vasyunin's formula \cite{Va}. The following are known, written with Corollary 2.1. \begin{enumerate} \def\labelenumi{(\roman{enumi})} \item RH holds if and only if \(d_N\to0\) \cite{Ny,Be,BD}. If RH fails, \(d_N\) is bounded below by a positive constant, so \(d_N^2\log N\to\infty=\sum_\rho\lim_nC_2(n,\rho)\). \item Unconditionally, \(\liminf_Nd_N^2\log N\ge\sum_{\rho:\,\beta=1/2}m_\rho^2\lim_nC_2(n,\rho)\), where \(m_\rho\) is the multiplicity \cite{Bu02}. \item It is conjectured \cite{BBLS} that \(d_N^2\log N\to\sum_\rho|\rho|^{-2}=C_B\) under RH with simple zeros; in the notation of Corollary 2.1 this reads \(\lim_Nd_N^2\log N=\sum_\rho m_\rho^2\lim_nC_2(n,\rho)\), which by (i) also holds when RH fails. \end{enumerate} We computed \(G\) from Vasyunin's formula, checked against direct integration to \(10^{-9}\), and \(d_N^2\) in double precision for \(N\le1200\); the condition number of \(G\) is \(5\cdot10^6\) at \(N=1200\). The values of \(d_N^2\log N\) are \(0.0464\), \(0.0469\), \(0.0462\), \(0.0453\) at \(N=50\), \(100\), \(400\), \(1000\), and lie in \([0.0449,0.0464]\) with mean \(0.0456\) for \(300\le N\le1200\) (Figure \ref{fig:nyman}). The sum of the limits of \(C_2\) over the \(341\) zeros with \(\gamma<600\) is \(0.04324\), and with the tail from the zero density \(0.046193\), against \(C_B=2+\gamma_0-\log4\pi=0.046191\); the mean square of the prime error up to \(10^8\) is \(0.0466\) (Figure \ref{fig:signal}). The three quantities coincide because each is \(\sum_\rho|\rho|^{-2}\), by Parseval applied to the explicit formula, so the coincidence is not a new constraint. It shows, however, that \(\lim_nC_2(n,\rho)\) is the contribution of \(\rho\) to the Nyman--Beurling distance, in a problem where the positivity is that of a Gram matrix on the integers and the open part, \(d_N\to0\), is an upper bound. Proving it requires good approximating coefficients \(c_k\); the natural ones, of Möbius type, lead back to the behaviour of \(1/\zeta\) on the critical line. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Nyman_Beurling_C2.png} \caption{The Nyman–Beurling distance and the limits of $C_2$. (a) $d_N^2\log N$ for $5\le N\le1200$; dashed: $\sum_\rho\lim_nC_2(n,\rho)=C_B$; dotted: mean square of the prime error for $x\le10^8$. (b) $\sum_{|\gamma|\le T}\lim_nC_2(n,\rho)$ over the zeros with $\gamma<600$.}\label{fig:nyman}\end{figure} \textbf{The optimal coefficients.} The minimizing coefficients \(c=G^{-1}b\) have a simple structure (Figure \ref{fig:nymancoef}a, b). For \(N=1200\) their sign is \(-\mu(k)\) at every squarefree \(k\), \(95.6\%\) of \(\sum c_k^2\) lies on the squarefree \(k\), and the profile \(w(u)=-c_k\mu(k)\), \(u=\log k/\log N\), decreases from about \(1\) at \(u=0\) to about \(0.2\) at \(u=1\); its means over the ten intervals of \(u\) agree within \(0.02\) for \(N=200\), \(600\), \(1200\). Restricting \(c\) to the squarefree \(k\) raises \(d_N^2\) by \(2\) to \(5\%\). Coefficients built from the primes alone come close to the optimum when the profile is chosen in the norm of the distance: with \(c_k=\mu(k)P(\log k/\log N)\) and \(P\) a polynomial of degree \(10\) optimized for each \(N\), \(d_N^2\log N\) lies between \(0.051\) and \(0.058\) for \(50\le N\le1200\), within a factor \(1.13\) to \(1.24\) of the optimum, while \(c_k=\mu(k)\) and \(c_k=\mu(k)(1-\log k/\log N)\), each with the best multiple, are worse by factors \(3.5\) to \(16\) (Figure \ref{fig:nymancoef}c). A profile fitted pointwise to \(w\) and used without reoptimization is worse by a factor \(5.6\) to \(10\); the quadratic form is sensitive to the fine structure of the coefficients. The reason is the Mellin form of the distance. Since the Mellin transform of \(\rho(1/(kx))\) is \(-k^{-s}\zeta(s)/s\) for \(0<\sigma<1\), Plancherel's theorem gives \[d_N^2=\frac1{2\pi}\int_{-\infty}^{\infty}\big|1+\zeta(s)C_N(s)\big|^2\frac{dt}{|s|^2},\qquad s=\tfrac12+it,\quad C_N(s)=\sum_{k\le N}c_kk^{-s},\] which we checked numerically: for \(N=50\) the integral over \(|t|<3000\) is \(0.011686\) against \(d_N^2=0.011869\), the difference being the omitted tail. The approximation problem is therefore a mollification problem: \(C_N\) is a Dirichlet polynomial approximating \(-1/\zeta\) on the critical line, and the structure of the optimal coefficients is that of the mollifiers of Levinson and Conrey. At each zero \(\zeta C_N=0\), so the integrand rises to \(1/|s|^2\) (Figure \ref{fig:nymancoef}d); a zero contributes about \(1/|\rho|^2=\lim_nC_2(n,\rho)\) times a width of order \(1/\log N\), which is the content of (ii) and (iii). An estimate \(d_N^2\ll1/\log N\) for \(c_k=\mu(k)P(\log k/\log N)\) amounts to a mean value theorem for \(\int|1+\zeta C_N|^2\) over \(|t|\le N^{1/\theta}\) with mollifiers of length \(t^\theta\) for arbitrarily large \(\theta\). Asymptotic formulas of this kind are known for \(\theta<\frac47\) \cite{Co89}; for large \(\theta\) they would require the zeros to lie on the line, since a zero off the line leaves \(|1+\zeta C_N|=1\) on a window of fixed width. Status: the Mellin identity is classical; the rest is numerical. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Nyman_Beurling_coefficients.png} \caption{The coefficients of the Nyman–Beurling approximation. (a) Optimal $c_k$ for $N=1200$, on squarefree and non-squarefree $k$. (b) The profile $-c_k\mu(k)$ against $\log k/\log N$ for $N=200$, $600$, $1200$. (c) $d_N^2\log N$ for coefficients $\mu(k)$, $\mu(k)(1-u)$, $\mu(k)P_3(u)$, $\mu(k)P_{10}(u)$, arbitrary coefficients on the squarefree $k$, and the optimum. (d) The integrand $|1+\zeta(s)C_N(s)|^2/|s|^2$ of the Mellin form for $N=1200$, and $1/|s|^2$; dotted: zeros.}\label{fig:nymancoef}\end{figure} \textbf{The law on vertical lines.} The Laplace transform of \(E\) gives an equivalent form in terms of \(\zeta'/\zeta\), the logarithmic gradient of the \(C_2\) energy \(|\zeta|^2\). Put \(s=\frac12+\delta+it\) and \[\mathcal J(\delta)=\delta\int_{-\infty}^{\infty}\Big|\frac{\zeta'}{\zeta}(s)+\frac s{s-1}\Big|^2\frac{dt}{|s|^2}.\] \textbf{Proposition 3.3.} If RH holds and the zeros are simple, then \(\mathcal J(\delta)\to\pi C_B\) as \(\delta\to0^+\). If \(\rho=\beta+i\gamma\) is a zero with \(\frac12<\beta<1\), then \(\mathcal J(\beta-\frac12)=\infty\). \emph{Proof.} For \(\operatorname{Re}s>\Theta\) the Laplace transform \(L(s)=\int_0^\infty E(u)e^{-(s-1/2)u}\,du\) equals \(-\frac1s\big(\frac{\zeta'}{\zeta}(s)+\frac s{s-1}\big)+H(s)\), where \(H(s)=O(\log|s|/|s|)\) comes from the lower-order terms of \(E\). By Plancherel, \(I(\delta)=\int_0^\infty E(u)^2e^{-2\delta u}\,du=\frac1{2\pi}\int|L(\frac12+\delta+it)|^2\,dt\). Under RH, \(F(U)\sim C_BU\), and an Abelian argument gives \(I(\delta)\sim C_B/(2\delta)\). The contribution of \(H\) changes \(\mathcal J\) by \(O(\delta^{1/2})\), so \(\mathcal J(\delta)=2\pi\delta I(\delta)+o(1)\to\pi C_B\). If the line \(\operatorname{Re}s=\beta\) contains a zero, the integrand behaves like \(|t-\gamma|^{-2}\) near \(t=\gamma\) and the integral diverges. \(\square\) Near each zero on the critical line, \(\zeta'/\zeta(s)\approx(s-\rho)^{-1}\), and \(\delta\int|s-\rho|^{-2}|s|^{-2}\,dt\to\pi/|\rho|^2\): each zero contributes \(\pi\) times its \(C_2\) slope through a Poisson kernel of width \(\delta\), the kernel that reappears in Sections 4 and 5. The mean square of \(\zeta'/\zeta\) near the critical line is also the object that Goldston, Gonek and Montgomery relate to primes in short intervals \cite{GGM}. We computed \(\mathcal J(\delta)\) from \(\zeta\) on \(|t|\le T\) (\(T=600\), and \(400\), \(300\) for the two smallest \(\delta\)), adding the estimate \((\log\frac T{2\pi}+1)/T\) for the zeros above \(T\): \begin{longtable}[]{@{}lllllll@{}} \toprule\noalign{} \(\delta\) & 0.2 & 0.1 & 0.05 & 0.025 & 0.0125 & 0.00625 \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(\mathcal J(\delta)\) & 2.220 & 1.407 & 0.849 & 0.518 & 0.337 & 0.243 \\ \end{longtable} The differences halve with \(\delta\), consistent with \(\mathcal J(\delta)=\pi C_B+O(\delta)\). Richardson extrapolation from the last two values gives \(0.148\), against \(\pi C_B=0.1451\). \textbf{Numerics.} We computed \(E(u)\) from all prime powers up to \(10^8\) and took time averages over \(10\) put \[\mathcal E(U,H)=\frac1H\int_{-\infty}^{\infty}E(u)^2\,e^{-\pi(u-U)^2/H^2}\,du.\] \textbf{Theorem 3.4} (local energy law). For all \(U\) and \(H>0\), \[\mathcal E(U,H)=\sum_{\rho,\rho'}\frac{e^{zU+z^2H^2/(4\pi)}}{\rho\,\overline{\rho'}},\qquad z=\rho+\overline{\rho'}-1,\] where the sum runs over all pairs of nontrivial zeros, counted with multiplicity, and converges absolutely. \emph{Proof.} By the truncated explicit formula \cite{In,T}, \(E_T(u)=-\sum_{|\gamma|R\) negligible uniformly in \(T\). For finite \(T\), \(\int e^{zu}e^{-\pi(u-U)^2/H^2}du/H=e^{zU+z^2H^2/(4\pi)}\) gives the identity with the sum restricted to \(|\gamma|,|\gamma'|0\) and \(B(H)=\sum_{\rho\neq\rho'}e^{-(\gamma-\gamma')^2H^2/(4\pi)}/|\rho\rho'|\), the sum over pairs of distinct indices. \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item If RH holds, then \(|\mathcal E(U,H)-C_B|\le B(H)\) for all \(U\). \item RH holds if and only if \(\sup_U\mathcal E(U,H)<\infty\). \end{enumerate} \emph{Proof.} (a) Under RH, \(z=i(\gamma-\gamma')\); the terms with \(\rho=\rho'\) give \(\sum_\rho|\rho|^{-2}=C_B\), and the others have modulus at most \(e^{-(\gamma-\gamma')^2H^2/(4\pi)}/|\rho\rho'|\). (b) If \(\sup_U\mathcal E(U,H)\le M\), then \(\int_{U-H/2}^{U+H/2}E^2\le e^{\pi/4}HM\) for all \(U\), so \(\int_0^UE^2=O(U)\), and Theorem 3.2 gives \(\Theta=\frac12\). The converse is (a). \(\square\) A zero with \(\beta>\frac12\) enters \(\mathcal E\) through its diagonal term \(e^{(2\beta-1)U+(2\beta-1)^2H^2/(4\pi)}/|\rho|^2>0\), which no other term can cancel for large \(U\), since all other terms involving \(\rho\) grow at most like \(e^{(\beta-\frac12)U}\). It becomes visible only for \(U\gtrsim2\log|\gamma|/(2\beta-1)\), so the law does not by itself exclude zeros above the computed range. We computed \(\mathcal E(U,H)\) from the prime powers up to \(10^8\) and from the double sum over the \(682\) zeros with \(|\gamma|<600\), adding the diagonal contribution \(C_B-\sum_{|\gamma|<600}|\rho|^{-2}=0.00295\) of the higher zeros; the two agree to within \(3.3\cdot10^{-4}\) for \(H=1,2\) and \(6\le U\le14\). On \(5\le U\le14.5\), \(|\mathcal E(U,H)-C_B|\) is at most \(0.0074\) for \(H=1\) and \(0.0023\) for \(H=2\), against \(B(1)=0.044\) and \(B(2)=0.0123\) computed from the same zeros. For the Davenport--Heilbronn signal (\(H=1\)) the window energy grows from \(0.22\) at \(U=6\) to \(2.06\) at \(U=13\); the double sum over its zeros with \(\gamma<400\) accounts for \(84\)--\(96\%\) of it, the diagonal terms of its eight off-line zeros grow from \(0.02\) to \(1.52\), and the remainder increases with \(U\), as expected from off-line zeros above height \(400\). The Euler-product twin \(L(s,\chi)\) of Section 4, which has the same conductor and gamma factor as \(f\), behaves like \(\zeta\): the mean of \(|E_\chi(u)|^2\), with \(E_\chi(u)=e^{-u/2}\sum_{n\le e^u}\Lambda(n)\chi(n)\), is \(0.2052\) on \(3\le u\le18.4\), against \(\sum_\rho|\rho|^{-2}=0.2032\) over its zeros (those with \(|\gamma|<400\) plus a density estimate for the rest), and its window energy with \(H=2\) stays between \(0.189\) and \(0.221\), inside the band \(0.203\pm0.077\) of Corollary 3.5. \textbf{Remark 3.6} (the harmonic offset of \cite{E24}). The harmonic-series construction of \cite[\S 7.2]{E24} compares, at two zeros, the quantity \(H(\tfrac12,\gamma,n)=H_n-|X_n(\frac12+i\gamma)|^2+n/(\frac14+\gamma^2)\), where \(H_n\) is the harmonic number, and claimed that its difference at two zeros tends to \(0\). By the straight-line law of Corollary 2.1, at each zero on the critical line \[H_n-H(\tfrac12,\gamma,n)\longrightarrow\frac12\cdot\frac1{\frac14+\gamma^2}=\frac12\lim_{n\to\infty}C_2(n,\rho),\] so the difference at two zeros tends to \(\frac12[L(\gamma_2)-L(\gamma_1)]\), \(L(\gamma)=(\frac14+\gamma^2)^{-1}\), which is \(-0.001369\) for the first two zeros \cite{Er}. The harmonic offset of each zero is half its \(C_2\) limit and depends only on its own height, so it does not relate one zero to another. The offsets are related collectively: by Theorem B their sum over all zeros is half the energy of the prime signal, \(C_B/2=0.0231\) under RH, and an off-line pair would make the sum infinite. \textbf{Corollary 3.7} (the energy attains its lower bound exactly under RH). Let \[\mathcal E=\limsup_{U\to\infty}\frac1U\int_0^UE(u)^2\,du\in[0,\infty].\] Then \[\mathcal E\ \ge\ 2+\gamma_0-\log4\pi=0.0461914\ldots,\] with equality if and only if RH holds and all zeros are simple. \emph{Proof.} The identity \(\sum_\rho\frac1{\rho(1-\rho)}=2+\gamma_0-\log4\pi\), with multiplicity and in symmetric order, holds unconditionally \cite[Ch. 12]{Da}. If RH fails, \(\mathcal E=\infty\) by Theorem B. If RH holds, \(\rho(1-\rho)=|\rho|^2\), and \(E(u)=-\sum_\gamma m_\gamma e^{i\gamma u}/\rho\) over the distinct zeros, \(m_\gamma\) denoting the multiplicity; as in Proposition 3.1, the mean of the cross terms vanishes and \(\mathcal E=\sum_\gamma m_\gamma^2/|\rho|^2\ge\sum_\gamma m_\gamma/|\rho|^2=2+\gamma_0-\log4\pi\), with equality exactly when every \(m_\gamma=1\). \(\square\) Thus one side of the criterion is an explicit constant and the other is computed from the primes alone: the measured value \(0.0453\) (primes up to \(10^8\)) lies slightly below the limit, by an amount consistent with the finite range of \(u\). A finite computation cannot decide the equality, since an off-line zero at height \(\gamma\) affects the energy only for \(u\gtrsim2\log\gamma/(2\beta-1)\). \textbf{The non-holomorphic gap.} The constant \(C_B\) is the sum over the zeros of a holomorphic function of \(\rho\); the diagonal of Theorem B is \(\sum_\rho|\rho|^{-2}\), which is not. The difference is a positive form in the abscissae. Pair \(\rho\) with \(1-\bar\rho\), a bijection of the zeros preserving multiplicity, and put \(a=|\rho|^2\), \(b=|1-\rho|^2=a+1-2\beta\). Then \(\frac\beta a+\frac{1-\beta}b-\frac12\big(\frac1a+\frac1b\big)=-\frac{2(\beta-\frac12)^2}{ab}\), and summing over the zeros, with multiplicity and in symmetric order, \[\sum_\rho\frac1{|\rho|^2}=2+\gamma_0-\log4\pi+\mathcal Q,\qquad \mathcal Q=\sum_\rho\frac{(\beta-\frac12)^2}{|\rho|^2\,|1-\rho|^2}\ \ge0.\] Both series converge absolutely, since \(\sum_\rho|\rho|^{-2}<\infty\), and \(\mathcal Q=0\) if and only if RH holds. The Möbius map \(w_\rho=1-1/\rho\) sends the critical line onto the unit circle, the functional equation gives \(|w_{1-\bar\rho}|=|w_\rho|^{-1}\), and \(\frac{(\beta-\frac12)^2}{|\rho|^2|1-\rho|^2}=\frac14\big(|w_\rho|-|w_\rho|^{-1}\big)^2\); an off-line pair has reciprocal radii \(r\), \(1/r\) and contributes \(\frac12(r-r^{-1})^2>0\), so the symmetry does not cancel the defect. By Corollary 2.1 and Theorem A, \(1-2\beta=\lim_n\log C_2(n,\rho)/\log n\) for \(0<\beta<1\), so \(\mathcal Q\) is also a sum of squared growth exponents of the cross-energy, weighted by \(\frac14|\rho|^{-2}|1-\rho|^{-2}\). The identity explains why the criterion cannot be closed from the primes alone. The explicit formula evaluates \(\sum_\rho F(\rho)\) only for \(F\) holomorphic in a neighbourhood of the critical strip, and gives \(\sum_\rho\frac1{\rho(1-\rho)}=C_B\) without information on \(\beta\); the gap \(\mathcal Q\) is the part of \(\sum_\rho|\rho|^{-2}\) that depends on \(|\rho|\) and not on \(\rho\), and it is reached only through a positivity statement on the spectral side, such as the mean square of Theorem B. This is the obstruction of Proposition 6.21 in its simplest form. Status: the identity is elementary, and the equivalence of RH with \(\sum_\rho|\rho|^{-2}=2+\gamma_0-\log4\pi\) that it contains is known; we include it for the interpretation of Theorem B. \textbf{The orbit of each zero.} The energy can be resolved zero by zero. For a frequency \(\gamma\) put \[A_\gamma(u)=\int E(v)\,e^{-i\gamma v}\,w_s(u-v)\,dv,\qquad E_\gamma(u)=2\operatorname{Re}\big(A_\gamma(u)e^{i\gamma u}\big),\] with \(w_s\) a Gaussian window of width \(s\), wide enough that \(s\,\Delta\gg1\), where \(\Delta\) is the distance from \(\gamma\) to the neighbouring zeros. If \(\frac12+i\gamma\) is a simple zero and the neighbouring zeros are resolved by the window, the explicit formula gives \(A_\gamma(u)\approx-1/\rho\), so the curve \(u\mapsto(E_\gamma(u),E_\gamma(u+\frac\pi{3\gamma}))\) is a closed ellipse of semi-amplitude \(2/|\rho|\), traversed once for every increase of \(u\) by \(2\pi/\gamma\). A zero \(\rho=\beta+i\gamma\) with \(\beta>\frac12\) contributes \(-e^{(\beta-\frac12)u}e^{i\gamma u}/\rho\) to \(E\), and the same curve is a spiral whose amplitude grows at the rate \(\beta-\frac12\). In this language RH states that every orbit of the prime signal is closed, and Theorem B states that the mean squares of the orbits, \(2|A_\gamma|^2=2/|\rho|^2\) for each \(\gamma>0\), add up to the energy \(\sum_\rho\lim_nC_2(n,\rho)\). For \(\zeta\) at \(\gamma=14.135\) (\(s=1\), primes up to \(10^8\)) the measured mean of \(|A_\gamma|\) is \(0.0706\), against \(1/|\rho|=0.0707\), and \(\log|A_\gamma|\) has slope \(0.0001\) on \(4.6\le u\le14.5\): the ellipse is traversed about \(22\) times and closes on itself (Figure \ref{fig:ellipses}a). For the Davenport--Heilbronn signal (\(s=1.5\), coefficients up to \(2\cdot10^7\)) the isolated on-line zero at \(\gamma=43.08\) gives a closed ellipse with \(|A_\gamma|=0.0233\), against \(0.0232\), and slope \(0.007\) (Figure \ref{fig:ellipses}b). The off-line zero \(0.8085+85.699i\) gives a spiral with growth rate \(0.3057\), against \(\beta-\frac12=0.3085\) (Figure \ref{fig:ellipses}c,d), and the zero \(0.8695+240.40i\) gives \(0.3716\), against \(0.3695\). The curve depends on \(\beta\) through the growth of its amplitude, in contrast with the circles of Section 2, which record only \(\gamma\). \textbf{The phase of each orbit.} The complex amplitude \(A_\gamma(u)\approx-e^{(\beta-\frac12)u}/\rho\) carries \(\beta\) twice: in the growth of its modulus and in its phase, \(\arg(-A_\gamma)=-\arg\rho=-\arctan(\gamma/\beta)\). On the critical line all orbits are phase-locked, and \[E(u)=-\sum_{\gamma>0}\frac{\cos(\gamma u)+2\gamma\sin(\gamma u)}{\frac14+\gamma^2},\] so the prime signal is close to a sine series. The phase can be read in a finite window, without waiting for any growth. From the primes up to \(10^8\) it gives \(\beta=0.499\pm0.023\), \(0.480\pm0.029\) and \(0.505\pm0.059\) for the first three zeros of \(\zeta\), where the error is \(\gamma\) times the standard deviation of the measured phase along the window. The sensitivity decreases with the height, since the phase difference between \(\beta\) and \(\frac12\) is about \((\beta-\frac12)/\gamma\): the error is \(0.23\) at \(\gamma=30.4\) and \(0.44\) at \(\gamma=49.8\), and for the off-line zero \(0.8085+85.699i\) of the Davenport--Heilbronn function the phase gives \(0.93\pm0.05\), while the growth rate gives \(0.8057\). The two readings are complementary: the phase sees \(\beta\) in a short window but only at low height, and the growth sees it at every height but only over a long range of \(u\). Neither constrains a zero that has not been located. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Prime_signal_ellipses.png} \caption{Orbits of single zeros in the prime signal: the component of $E$ at frequency $\gamma$ at time $u$ against its value at $u+\pi/(3\gamma)$, coloured by $u$. (a) $\zeta$, zero at $\gamma=14.135$, from the primes up to $10^8$: a closed ellipse. (b) Davenport--Heilbronn function, on-line zero at $\gamma=43.081$: a closed ellipse. (c) Its off-line zero $0.8085+85.699i$: an expanding spiral. (d) Amplitude $|A_\gamma(u)|$; the dotted line has the predicted slope $\beta-\frac12$.}\label{fig:ellipses}\end{figure} \textbf{Spiral and harmonic.} Proposition 2.3 and the orbit amplitude are two readings of the same exponential: the zero of the truncated function satisfies \(\rho_n-\rho\approx n^{-\rho}/(2\zeta'(\rho))\), and the component of the prime signal at frequency \(\gamma\) is \(h_\rho(u)=A_\gamma(u)e^{i\gamma u}\approx-e^{(\rho-\frac12)u}/\rho\). Their product at \(u=\log n\) does not depend on \(\beta\): \[-2\sqrt n\,(\rho_n-\rho)\,h_\rho(\log n)\to\frac1{\rho\,\zeta'(\rho)},\] the residue of \(1/(s\zeta(s))\) at \(\rho\), which is the coefficient of \(\rho\) in the explicit formula for \(M(x)=\sum_{n\le x}\mu(n)\). The integers supply \(1/\zeta'(\rho)\) and the primes supply \(1/\rho\). For the first three zeros of \(\zeta\) the product, computed from \(F_n\) with \(n\le2\cdot10^5\) and from the primes up to \(10^8\), agrees with \(1/(\rho\zeta'(\rho))\) to within \(1\)--\(3\%\) (Figure \ref{fig:mobius}a). Summed over the zeros, the modes rebuild \((M(x)+2)/\sqrt x\): with the \(341\) zeros with \(0<\gamma<600\) the correlation with the values computed from \(\mu(n)\) up to \(10^8\) is \(0.995\) (Figure \ref{fig:mobius}b). The energy is the Möbius analogue of Theorem B: under RH, with simple zeros and a hypothesis on the sums \(\sum_{0<\gamma0}|\rho\zeta'(\rho)|^{-2}\) \cite{Ng}. We obtain \(0.0290\) from \(\mu(n)\) up to \(10^8\) and \(0.0287\) from the zeros with \(\gamma<600\). For the Davenport--Heilbronn function, the summatory function of the Dirichlet inverse of its coefficients, computed up to \(10^7\), has window energies that grow at the rate \(0.61\) in \(u\), against \(2\beta-1=0.617\) for its off-line zero (Figure \ref{fig:mobius}c). The product reads \(1/\zeta'(\rho)\) at a located zero; it does not involve \(\beta\) and adds no constraint on it. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Spiral_times_harmonic.png} \caption{Spiral times harmonic. (a) $-2\sqrt n(\rho_n-\rho)h_\rho(\log n)$ for $200\le n\le2\cdot10^5$ at the first three zeros of $\zeta$; stars: $1/(\rho\zeta'(\rho))$. (b) $(M(x)+2)/\sqrt x$ from $\mu(n)$ and the sum of the $341$ modes. (c) Energy of the Mertens signal in windows of length $2$, for $\zeta$ and for the Davenport--Heilbronn function; the dashed line has the slope $2\beta-1=0.617$.}\label{fig:mobius}\end{figure} \section{\texorpdfstring{The energy slope: a pointwise \(C_2\) inequality}{The energy slope: a pointwise C\_2 inequality}} Theorem B is a statement about an average over \(u=\log x\). This section formulates an inequality at each fixed \(t\), describes its local structure, and determines which parts of it can be proved. \textbf{The criterion.} Let \(\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)\) and define the \emph{energy slope} \[S(\sigma,t)=\partial_\sigma\log|\xi(\sigma+it)|=G(\sigma,t)+\operatorname{Re}\frac{\zeta'}{\zeta}(\sigma+it),\qquad G(\sigma,t)=\operatorname{Re}\Big[\frac1s+\frac1{s-1}-\frac12\log\pi+\frac12\psi\Big(\frac s2\Big)\Big],\] where \(\psi=\Gamma'/\Gamma\). The term \(G\), which we call the gamma pressure, equals \(\frac12\log\frac t{2\pi}+O(t^{-1})\). The term \(\log|\zeta|\) is half the logarithm of the \(C_2\) energy: for \(\sigma>1\), \(|\zeta(s)|^2=\zeta(2\sigma)+\sum_{n\ge2}C_2(n,s)\), and for \(0<\sigma<1\), \(|\zeta(s)|\) is the radius of the spiral of Section 2. The slope can be evaluated from a finite \(C_2\) sum: with \(E_N(s)=\big|X_N(s)+\frac{N^{1-s}}{s-1}-\frac12N^{-s}+\frac s{12}N^{-s-1}\big|^2\), the quantity \(\frac12\partial_\sigma\log E_N+G\) agrees with \(S\) to six digits at \(N=200\) for the points we tested (\(\sigma=0.55,0.7,0.9\); \(t=27.7,100\)). \textbf{Theorem 4.1} (Lagarias \cite{La}; Sondow and Dumitrescu \cite{SD}). RH holds if and only if \(S(\sigma,t)>0\) for all \(\sigma>\frac12\) and all real \(t\), that is, if and only if \(|\xi(\sigma+it)|\) is strictly increasing in \(\sigma\) on \(\sigma>\frac12\) for every fixed \(t\). The proof rests on the Hadamard product, which gives \(S(\sigma,t)=\sum_\rho\operatorname{Re}\frac1{s-\rho}\) with the zeros summed in symmetric order \cite{La}. The next lemma makes the contribution of each zero explicit. \textbf{Theorem C} (disc lemma). Let \(\sigma>\frac12\). A zero on the critical line contributes \((\sigma-\frac12)/((\sigma-\frac12)^2+(t-\gamma)^2)>0\) to \(S(\sigma,t)\). A pair of zeros \(\rho=\beta+i\gamma\) and \(\iota(\rho)=1-\beta+i\gamma\) with \(\beta>\frac12\) contributes a negative amount if and only if \[\Big(\sigma-\tfrac12\Big)^2+(t-\gamma)^2<\Big(\beta-\tfrac12\Big)^2,\] that is, if and only if \(s\) lies in the open disc \(D_\rho\) with centre \(\frac12+i\gamma\) passing through \(\rho\) and \(\iota(\rho)\). \emph{Proof.} Put \(a=\beta-\sigma\), \(b=\sigma-1+\beta\) and \(d=t-\gamma\). The pair contributes \[-\frac a{a^2+d^2}+\frac b{b^2+d^2}=\frac{(a-b)(ab-d^2)}{(a^2+d^2)(b^2+d^2)}.\] Here \(a-b=1-2\sigma<0\) and \(ab=(\beta-\frac12)^2-(\sigma-\frac12)^2\), so the contribution is negative exactly when \(ab>d^2\). \(\square\) \textbf{Corollary 4.2.} If \(\sigma>\frac12\) and \(s=\sigma+it\) lies in no disc \(D_\rho\), then \(S(\sigma,t)>0\). In particular, the inequality \(S>0\) can fail only within distance \(\beta-\frac12<\frac12\) of the ordinate of an off-line zero. The criterion is therefore local: each off-line pair can make \(S\) negative only inside its own disc. Near an off-line zero, \(S\to-\infty\) as \(s\to\rho\) from the left, so no positive contribution from other zeros can compensate there. A proof of \(S>0\) inside a disc is equivalent to the absence of the zero. \textbf{Proposition 4.3} (concavity in the square of the distance to the line). For real \(t\) and \(v\ge0\) put \(\ell_t(v)=\log|\xi(\frac12+\sqrt v+it)|\). RH holds if and only if \(\ell_t\) is concave on \((0,\infty)\) for every \(t\) at which it is finite. Moreover \(\ell_t'(v)=S(\sigma,t)/(2(\sigma-\frac12))\) with \(\sigma=\frac12+\sqrt v\), and under RH \[\lim_{v\to0^+}\ell_t'(v)=\frac12\sum_\rho\frac1{(t-\gamma)^2},\qquad \ell_t'(v)\downarrow0\quad(v\to\infty).\] \emph{Proof.} By the functional equation \(|\xi(\frac12+a+it)|=|\xi(\frac12-a+it)|\), so \(\ell_t\) is a function of \(v=a^2\). Write \(\xi(\frac12+w)=\xi(\frac12)\prod(1+w^2/z^2)\) over the zeros \(\rho=\frac12+iz\) with \(\operatorname{Re}z>0\), the product converging absolutely since \(\sum|z|^{-2}<\infty\). For \(w=a+it\) and a real \(z=\gamma\), \(|\gamma^2+w^2|^2=\big(v+(t-\gamma)^2\big)\big(v+(t+\gamma)^2\big)\), so under RH \[2\ell_t(v)=\text{const}+\sum_{\gamma>0}\Big[\log\big(v+(t-\gamma)^2\big)+\log\big(v+(t+\gamma)^2\big)-4\log\gamma\Big],\] a convergent sum of concave functions of \(v\); the derivative at \(v=0\) is \(\frac12\sum_\rho(t-\gamma)^{-2}\) and every term of \(\ell_t'\) decreases to \(0\). Conversely, let \(z=\gamma-ib\) with \(00\) for \(\sigma>1\); and the derivative at the line is half the energy of the Euler helix (Remark 6.5). Under RH the normalized slope \(S/(2(\sigma-\frac12))\) starts at this energy on the line and decreases to \(0\) without changing sign. Numerically, on \(10\le t\le100\) (step \(0.05\)) and \(01\).} By the Euler product, \(\operatorname{Re}\frac{\zeta'}{\zeta}(\sigma+it)=-\sum_n\Lambda(n)n^{-\sigma}\cos(t\log n)\ge\frac{\zeta'}{\zeta}(\sigma)\). Hence \(S(\sigma,t)\ge G(\sigma,t)+\frac{\zeta'}{\zeta}(\sigma)\), and \(S>0\) whenever \(G(\sigma,t)>-\frac{\zeta'}{\zeta}(\sigma)=\frac1{\sigma-1}-\gamma_0+O(\sigma-1)\). We checked numerically that \(-\frac{\zeta'}{\zeta}(\sigma)<\frac1{\sigma-1}\) on \(1<\sigma\le12\) (the difference is at most \(-0.09\)), so \(S>0\) for \(\sigma\ge1+1/G(\sigma,t)\) in that range. In the same region the Euler product bounds the spiral radius from below, \(|\zeta(s)|\ge\zeta(2\sigma)/\zeta(\sigma)\), so that \(\sum_{n\ge2}C_2(n,s)\ge(\zeta(2\sigma)/\zeta(\sigma))^2-\zeta(2\sigma)\). For \(\sigma=1.2\) and \(0.50\) for \(\sigma\ge1-c(\log t)^{-2/3}(\log\log t)^{-1/3}\) and large \(t\). Section 6 explains why this type of argument does not extend deeper into the strip: the pole at \(s=1\) dominates any linear inequality with nonnegative coefficients. \item \emph{Bounded height.} All zeros with \(0<\gamma\le3\cdot10^{12}\) lie on the critical line \cite{PT}. By Corollary 4.2, \(S(\sigma,t)>0\) for all \(\sigma>\frac12\) and \(0a_0>0\). For \(\sigma>1\) and real \(t\), \[\sum_{k=0}^Ka_k\big[G(\sigma,kt)-S(\sigma,kt)\big]=\sum_{n\ge2}\frac{\Lambda(n)}{n^\sigma}P(t\log n)\ge0.\qquad\text{(4.1)}\] Consequently there is \(t_0(P)\) such that \(\zeta(\beta+i\gamma)\ne0\) for \[\beta>1-\frac{c(P)-\varepsilon}{\log|\gamma|},\qquad c(P)=\frac{2\big(\sqrt{a_1}-\sqrt{a_0}\big)^2}{\sum_{k\ge1}a_k},\] for every \(\varepsilon>0\) and \(|\gamma|\ge t_0(P,\varepsilon)\), and \(S(\sigma,t)>0\) for \(\sigma\ge1-(c(P)-\varepsilon)/\log(|t|+1)\) and \(|t|\ge t_0+1\). \emph{Proof.} By definition \(-\operatorname{Re}\frac{\zeta'}{\zeta}=G-S\), and for \(\sigma>1\), \(-\operatorname{Re}\frac{\zeta'}{\zeta}(\sigma+ikt)=\sum\Lambda(n)n^{-\sigma}\cos(kt\log n)\), which gives (4.1). For \(\sigma>1\) every term of \(S(\sigma,t)=\sum_\rho\operatorname{Re}\frac1{s-\rho}\) is positive, so \(S(\sigma,kt)\ge0\) for all \(k\), and if \(\rho=\beta+it\) is a zero then \(S(\sigma,t)\ge\frac1{\sigma-\beta}\). The term \(k=0\) is \(a_0[G(\sigma,0)-S(\sigma,0)]\le a_0G(\sigma,0)=\frac{a_0}{\sigma-1}+O(1)\), the pole. For \(k\ge1\), \(G(\sigma,kt)=\frac12\log|t|+O_K(1)\) by Stirling's formula. Hence \[\frac{a_1}{\sigma-\beta}\le\frac{a_0}{\sigma-1}+\frac{\log|t|}2\sum_{k\ge1}a_k+O_P(1).\] Put \(\sigma=1+\frac\eta{\log|t|}\) and \(L=\sum_{k\ge1}a_k\). Then \((1-\beta)\log|t|\ge\frac{a_1\eta}{a_0+\eta L/2}-\eta+o(1)\), and the maximum over \(\eta>0\) of the right side is \(2(\sqrt{a_1}-\sqrt{a_0})^2/L\), attained at \(\eta=2(\sqrt{a_0a_1}-a_0)/L\). The statement on \(S\) follows from Corollary 4.2, since every disc \(D_\rho\) meeting the line of abscissa \(\sigma\) at height \(t\) has \(|\gamma-t|<\frac12\) and \(\beta>\sigma\). \(\square\) The classical choice \(P=3+4\cos\theta+\cos2\theta\) gives \(c=0.02872=1/34.8\). Optimizing over nonnegative cosine polynomials of degree \(K\) (numerically, by sequential quadratic programming with random starts) gives \(1/c=26.6\), \(18.5\), \(17.45\), \(17.27\) and \(17.25\) for \(K=2,3,4,8,16\). These are the constants of the elementary method; refinements of the same inequality give \(1/c=9.65\) \cite{Ste}, \(5.70\) \cite{Ka} and \(5.573\) \cite{MoT}. Equation (4.1) is the form of the argument in the language of this section: a nonnegative combination of energy slopes at the heights \(0,t,2t,\dots\) is bounded by the gamma pressures and the pole, and the pole term \(a_0>0\), which cannot be avoided, limits the method to regions of width \(c/\log t\) (Section 6). Status: proved; the optimal constants for each \(K\) are numerical. In the remaining region only a statement in measure is available. \textbf{Theorem D} (exceptional set). Let \(\frac12<\sigma<1\) and \(T\ge2\), and let \(N(\sigma,T)\) be the number of zeros with \(\beta>\sigma\) and \(0<\gamma\le T\). Then \[\operatorname{meas}\{t\in[0,T]:S(\sigma,t)\le0\}\le\sum_{\substack{\beta>\sigma\\0<\gamma\le T+1/2}}2\sqrt{(\beta-\tfrac12)^2-(\sigma-\tfrac12)^2}\le2\sqrt{\sigma(1-\sigma)}\;N(\sigma,T+1).\] \emph{Proof.} Since \(\zeta\) has zeros on the critical line, the zero sum for \(S\) contains positive terms, and by Corollary 4.2 the set \(\{t:S(\sigma,t)\le0\}\) is contained in the union of the sections of the discs \(D_\rho\) at abscissa \(\sigma\). The section of \(D_\rho\) is an interval of length \(2\sqrt{(\beta-\frac12)^2-(\sigma-\frac12)^2}\) if \(\beta>\sigma\) and is empty otherwise. Since \(\beta<1\), this length is less than \(2\sqrt{\sigma(1-\sigma)}\). Only zeros with \(\gamma0\). Selberg's bound \(N(\sigma,T)\ll T^{1-(\sigma-1/2)/4}\log T\) \cite{Sel} gives this for every \(\sigma>\frac12\). The bounds of Huxley, \(N(\sigma,T)\ll T^{12(1-\sigma)/5+\varepsilon}\) \cite{Hu}, and of Guth and Maynard, \(N(\sigma,T)\ll T^{30(1-\sigma)/13+\varepsilon}\) \cite{GM}, give the stronger rates \(\theta(\sigma)=1-\frac{12}5(1-\sigma)\) and \(\theta(\sigma)=1-\frac{30}{13}(1-\sigma)\) where these are positive. By contrast, functions of Davenport--Heilbronn type have \(\gg T\) zeros in strips \(\sigma_1<\sigma<\sigma_2\) inside \(\frac12<\sigma<1\) \cite{KK}, so their exceptional sets have positive proportion. The Euler product therefore makes the failures of the pointwise inequality rare, and RH is the statement that they do not occur. \textbf{A density estimate from the square of the cross-energy.} The energy \(|X_N(s)|^2=\sum_{n\le N}n^{-2\sigma}+\sum_{2\le n\le N}C_2(n,s)\) is a sum of squares, and its mean over \(t\) keeps only the diagonal: each \(C_2(n,s)=2n^{-\sigma}\sum_{j\sigma\) and \(T<\gamma\le2T\). Then \[N(\sigma;T,2T)\le\frac{\log\zeta(\sigma+\frac12)}{2\pi(\sigma-\frac12)}\,T+o(T),\] so that the proportion of zeros with \(\beta>\sigma\) among those with \(T<\gamma\le2T\) is \(O\big(1/((\sigma-\frac12)\log T)\big)\). \emph{Proof.} Let \(\sigma_0=\frac12(\sigma+\frac12)\). Littlewood's lemma applied to \(\zeta\) on the rectangle \([\sigma_0,2]\times[T,2T]\) gives \[2\pi\int_{\sigma_0}^2N(u;T,2T)\,du=\int_T^{2T}\log|\zeta(\sigma_0+it)|\,dt-\int_T^{2T}\log|\zeta(2+it)|\,dt+O(\log T),\] the error coming from \(\arg\zeta\) on the horizontal sides, which is \(O(\log T)\) \cite[\S 9.4]{T}. The second integral is \(O(1)\), since \(\log\zeta(2+it)=\sum\Lambda(n)n^{-2-it}/\log n\). By the concavity of the logarithm, \[\int_T^{2T}\log|\zeta(\sigma_0+it)|\,dt\le\frac T2\log\Big(\frac1T\int_T^{2T}|\zeta(\sigma_0+it)|^2dt\Big)=\frac T2\log\zeta(2\sigma_0)+o(T),\] by the mean value \(\frac1T\int_T^{2T}|\zeta(\sigma_0+it)|^2dt\to\zeta(2\sigma_0)\) for \(\sigma_0>\frac12\) \cite[Thm 7.2]{T}, which is the vanishing of the mean of the cross terms \(C_2\). Since \(N(u;T,2T)\) is nonincreasing in \(u\), the left side is at least \(2\pi(\sigma-\sigma_0)N(\sigma;T,2T)\), and \(2\sigma_0=\sigma+\frac12\), \(\sigma-\sigma_0=\frac12(\sigma-\frac12)\). The proportion follows from \(N(2T)-N(T)\sim\frac T{2\pi}\log T\). \(\square\) This is the classical theorem of Bohr and Landau, written with the energy: the zeros off the line have density zero among all zeros because the mean energy at \(\sigma_0>\frac12\) is finite, while the energy at \(\sigma=\frac12\), where \(\zeta(2\sigma)\) has its pole, grows like \(\log T\). Numerically, on \(1000\le t\le2000\) the mean of \(\sum_{2\le n\le200}C_2(n,\sigma+it)\) is \(0.0012\), \(0.0005\), \(-0.0004\) for \(\sigma=0.6\), \(0.75\), \(0.9\), against \(\sum_{n\le200}n^{-2\sigma}=3.86\), \(2.47\), \(1.86\), and \(\frac1T\int_T^{2T}|\zeta|^2dt=4.10\), \(2.52\), \(1.87\) approaches \(\zeta(2\sigma)=5.59\), \(2.61\), \(1.88\) slowly, the deficit at \(\sigma=0.6\) being the tail \(\sum_{n>t/2\pi}n^{-2\sigma}\). At \(T=1000\) the bound of Proposition 4.5 is \(3755\), \(971\) and \(451\) for these \(\sigma\), against \(868\) zeros in \([T,2T]\); it becomes nontrivial only for large \(T\), as the factor \(1/\log T\) shows. The stronger estimates cited after Theorem D use higher moments and mollifiers, but the mechanism is the same quadratic positivity, which can show that zeros off the line are rare and cannot show that there are none: a single off-line zero changes the mean energy over \([T,2T]\) by an amount that the average does not see. Status: classical, proved; the computations are numerical. \textbf{Remark 4.6} (mollified energy and the Jensen gap). The bound of Proposition 4.5 is \(T\log(\text{mean energy})/(2\pi(\sigma-\sigma_0))\), and it improves when the energy is lowered by a mollifier, as in Selberg's method. Let \(M_y(s)=\sum_{d\le y}\mu(d)d^{-s}(1-\frac{\log d}{\log y})\), a smoothed truncation of \(1/\zeta\); in the language of Section 6 this sieves the wave by the primes up to \(y\) (Remark 6.10). The zeros of \(\zeta M_y\) include those of \(\zeta\), so the proof of Proposition 4.5 applied to \(\zeta M_y\) gives \[N(\sigma;T,2T)\le\frac T{2\pi(\sigma-\sigma_0)}\log\Big(\frac1T\int_T^{2T}|\zeta M_y(\sigma_0+it)|^2dt\Big)+o(T).\] On \(1000\le t\le2000\), which contains \(868\) zeros, the mean of \(|\zeta M_y|^2\) at \(\sigma_0=0.625\) is \(3.72\), \(1.80\), \(1.34\), \(1.19\) for \(y=1\), \(10\), \(100\), \(1000\), and the bound for \(\sigma=0.75\) is \(1672\), \(745\), \(376\), \(224\); at \(\sigma_0=0.55\) the means are \(5.12\), \(2.33\), \(1.66\), \(1.42\) and the bound for \(\sigma=0.6\) is still above \(868\). What limits the method is the gap in Jensen's inequality. The mean of \(\log|\zeta M_y(\sigma_0+it)|\) is \(0\) to within \(10^{-3}\), since there are no zeros to the right of \(\sigma_0\) in this range, and the log of the mean of \(|\zeta M_y|^2\) is close to twice the variance of \(\log|\zeta M_y|\): the variances are \(0.70\), \(0.34\), \(0.18\), \(0.11\), and \(e^{2\,\mathrm{var}}=4.05\), \(1.99\), \(1.43\), \(1.24\). To prove that there is no zero in \([T,2T]\) the variance would have to be below \(\pi(\sigma-\sigma_0)/T\), about \(10^{-4}\) here. Heuristically, \(\log|\zeta M_y|\) behaves like \(\sum_{p>y}p^{-\sigma_0}\cos(t\log p+\dots)\), whose variance is about \(\frac12\sum_{p>y}p^{-2\sigma_0}\asymp y^{1-2\sigma_0}/\log y\) (the measured values are larger by a factor between \(1\) and \(3\), the smoothing of \(M_y\) being imperfect). A variance of order \(1/T\) needs \(y\gtrsim T^{1/(2\sigma_0-1)}\), longer than \(T\), while the mean value theorems on which the method rests control mollifiers only up to length \(T\). With \(y=T\) the heuristic gives \(N(\sigma,T)\ll T^{2-2\sigma+\varepsilon}\), the form of the density hypothesis, which is known only in part of the strip. Quadratic means of the cross-energy, however mollified, can therefore show that off-line zeros are rare but not that they are absent; the obstruction is the variance of \(\log|\zeta M_y|\), which a mollifier of admissible length cannot make smaller than about \(T^{1-2\sigma_0}\). Status: the inequality is proved (Littlewood's lemma and Jensen's inequality); the variance law and the resulting exponent are heuristic; the computations are numerical. \textbf{Numerical results} (Figure \ref{fig:slope}). For \(\zeta\) we computed \(S(\sigma,t)\) by numerical differentiation of \(\log|\xi|\) on \(5\le t\le600\) (step \(0.01\)) and checked it against the zero sum (\(3.4526\) against \(3.4520\) at \(\sigma=0.7\), \(t=14\)). For \(t\ge20\) the minimum over \(t\) is attained at \(t\approx27.7\), in the gap between the zeros \(25.011\) and \(30.425\), for every \(\sigma\) tested: \begin{longtable}[]{@{}lrrrrrrr@{}} \toprule\noalign{} \(\sigma\) & \(0.51\) & \(0.55\) & \(0.6\) & \(0.7\) & \(0.8\) & \(0.9\) & \(1.0\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(\min_tS(\sigma,t)\) & \(0.0039\) & \(0.0195\) & \(0.0389\) & \(0.0776\) & \(0.1159\) & \(0.1534\) & \(0.1901\) \\ \(\max_t\big(-\operatorname{Re}\frac{\zeta'}{\zeta}\big)/G\) & \(0.995\) & \(0.974\) & \(0.948\) & \(0.895\) & \(0.844\) & \(0.793\) & \(0.744\) \\ \end{longtable} The margin is linear near the line, \(\min_tS\approx0.39(\sigma-\frac12)\), because \(S(\frac12,t)=0\) by the functional equation. The second row shows that the Euler-product term uses almost all of the gamma pressure near \(\sigma=\frac12\). For the Davenport--Heilbronn function \(f\) of \cite{C5} we used the completed function \((5/\pi)^{s/2}\Gamma(\frac{s+1}2)f(s)\) and its eight off-line zeros with \(\gamma<400\). The minimum of its slope over \(10\le t\le400\) is negative for every \(\sigma\le0.85\), with values between \(-30\) and \(-273\), and positive for \(\sigma\ge0.9\), beyond its largest \(\beta=0.8695\). Near \(0.8085+85.699i\), \(99.6\%\) of the grid points with negative slope lie in the disc of Theorem C, and \(91\%\) of the disc is negative. The failure set at abscissa \(\sigma\) consists of exactly one interval for each off-line zero with \(\beta>\sigma\): \begin{longtable}[]{@{}lrrrr@{}} \toprule\noalign{} \(\sigma\) & \(0.55\) & \(0.65\) & \(0.75\) & \(0.85\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot measure of \(\{t\le400:S<0\}\) & \(3.220\) & \(2.295\) & \(1.315\) & \(0.205\) \\ bound of Theorem D (sum of chords) & \(3.568\) & \(2.589\) & \(1.498\) & \(0.237\) \\ number of intervals \(=\) zeros with \(\beta>\sigma\) & \(8\) & \(6\) & \(4\) & \(1\) \\ \end{longtable} The bound of Theorem D is attained up to a factor of about \(0.9\) in this example. \textbf{An Euler-product twin.} The Davenport--Heilbronn function is \(f=c_1L(s,\chi)+c_2L(s,\bar\chi)\) with \(\chi\) the character mod \(5\) of Section 2, so \(L(s,\chi)\) has the same conductor, the same gamma factor and a functional equation of the same type, and differs from \(f\) only by its Euler product. Since the zeros of \(L(s,\chi)\) are symmetric under \(\rho\mapsto1-\bar\rho\), the proofs of Theorems 4.1, C and D apply to \(S_\chi=\partial_\sigma\log|(5/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi)|\) without change. On \(10\le|t|\le400\) we find \(\min_tS_\chi=0.0068\), \(0.034\), \(0.068\), \(0.135\) at \(\sigma=0.51\), \(0.55\), \(0.6\), \(0.7\), that is, about \(0.68(\sigma-\frac12)\), and no point with \(S_\chi<0\). On the same range the slope of \(f\) is negative on sets of measure \(6.7\), \(6.5\), \(5.6\) and \(3.8\). The Euler-product term of \(L(s,\chi)\) uses a fraction \(\max_t(-\operatorname{Re}\frac{L'}L)/G=0.994\) of the gamma pressure at \(\sigma=0.51\) and \(0.968\) at \(\sigma=0.55\), as for \(\zeta\) (\(0.995\) and \(0.974\)). The absence of a pole does not create slack: the tightness near the line is set by the density of the zeros. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{C2_energy_slope.png} \caption{The energy slope $S(\sigma,t)$. Left: Davenport--Heilbronn function near its zero $0.8085+85.699i$; $S<0$ (red) inside the disc of Theorem C (dashed), zero level solid. Centre: $\zeta$ near $t=27.7$, where the minimum over $t$ is attained; $S>0$ throughout. Right: $\min_tS(\sigma,t)$ for $\zeta$ ($20\le t\le600$) and for the Davenport--Heilbronn function ($10\le t\le400$), symmetric logarithmic scale.}\label{fig:slope}\end{figure} \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{PartVIII_gue.png} \caption{Left: empirical distribution of the unfolded margin $M_j$ for $\zeta$ ($300$ gaps), GUE ($6360$ gaps) and a Poisson sequence. Right: $M_j$ against the unfolded gap $\Delta_j$; the dashed line is $8/\Delta^2$.}\label{fig:gue}\end{figure} \textbf{The margin near the critical line and random matrices.} Write \(\Xi(t)=\xi(\frac12+it)\), which is real. Expanding in \(\sigma-\frac12\) gives \(S(\sigma,t)=(\sigma-\frac12)\,L(t)+O((\sigma-\frac12)^2)\) with \(L=(\Xi'^2-\Xi\Xi'')/\Xi^2=-(\Xi'/\Xi)'\). Under RH, \(L(t)=\sum_\gamma(t-\gamma)^{-2}\). The inequality \(\Xi'^2-\Xi\Xi''\ge0\) is the first Laguerre inequality; it is implied by RH, and the full family of generalized Laguerre inequalities is equivalent to RH \cite{CV}. The minimum of \(L\) over a gap between consecutive zeros measures how close \(S\) comes to \(0\) there. We unfolded the \(341\) zeros with \(0<\gamma<600\) by the Riemann--von Mangoldt formula, computed \(M_j=\min_{u}\sum_{|k-j|\le20}(u-x_k)^{-2}\) over each gap \((x_j,x_{j+1})\), and compared the \(300\) values with \(6360\) values from \(40\) bulk spectra of \(400\times400\) GUE matrices and with a Poisson sequence (Figure \ref{fig:gue}). The empirical quantiles at \(1\%\), \(5\%\), \(25\%\) and \(50\%\) are \(3.86\), \(4.53\), \(7.11\), \(10.68\) for \(\zeta\) and \(3.39\), \(4.28\), \(6.88\), \(10.75\) for GUE. The Kolmogorov--Smirnov test gives \(p=0.35\) against GUE and \(p=3\cdot10^{-34}\) against Poisson. The margin is determined by the gap: \(M_j\) has correlation \(0.9998\) with \(\Delta_j^{-2}\), and an isolated gap of length \(\Delta\) gives \(8/\Delta^2\). The tightest points of the inequality are therefore the largest gaps, and their distribution is the large-gap distribution predicted by random matrix theory \cite{Mo,Od}. \textbf{Remark 4.7} (statistical mechanics). For real \(\beta>1\), \(\zeta(\beta)=\sum_ne^{-\beta\log n}\) is the partition function of a system with energy levels \(\log n\) \cite{Ju}; Bost and Connes realize it as a quantum statistical system with a phase transition at \(\beta=1\) \cite{BC}. In this language \(-\frac{\zeta'}{\zeta}(\beta)\) is the mean energy, the bound in the case \(\sigma>1\) above compares the mean energy with the gamma pressure, and the zeros of \(\zeta\) are the zeros of the continued partition function at complex temperature. The Lee--Yang theorem places the zeros of ferromagnetic Ising partition functions on a circle by means of a positivity property that is preserved under products and limits \cite{LY}. The analogous property for \(\zeta\) would have to hold in \(\frac12<\sigma<1\), which lies beyond the transition at \(\beta=1\), where the partition sum diverges; this is the pole barrier of Section 6 in another form. The heat-flow deformation \(H_\lambda\) of \(\Xi\) studied by de Bruijn and Newman \cite{Ne} has only real zeros for \(\lambda\ge\Lambda\), RH is equivalent to \(\Lambda\le0\), and \(0\le\Lambda\le0.2\) \cite{RT,PT}. Under the backward flow, zeros leave the real line first at pairs of unusually close zeros (Lehmer pairs). These are the points where the margin \(M_j\) is largest, not smallest, so the heat-flow picture and the energy-slope margin measure different aspects of the zero distribution. \section{The energy slope as a Weil functional} \subsection{The explicit formula and Weil's criterion} Let \(h(r)=\int g(u)e^{iru}\,du\) with \(g\) smooth and compactly supported. For a zero \(\rho\) put \(\gamma_\rho=-i(\rho-\frac12)\), so that \(\gamma_\rho\) is real exactly when \(\rho\) lies on the critical line. The Riemann--Weil explicit formula \cite{We2}, \cite[Thm. 5.12]{IK} reads \[\begin{aligned}\sum_\rho h(\gamma_\rho)={}&h\big(\tfrac i2\big)+h\big(-\tfrac i2\big)-g(0)\log\pi+\frac1{2\pi}\int h(r)\operatorname{Re}\psi\Big(\frac14+\frac{ir}2\Big)dr\\&-\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}\big(g(\log n)+g(-\log n)\big).\end{aligned}\qquad\text{(5.1)}\] For the Davenport--Heilbronn function the completed function \((5/\pi)^{s/2}\Gamma(\frac{s+1}2)f(s)\) is entire and symmetric under \(s\mapsto1-s\); its explicit formula has no pole terms, the constant is \(+g(0)\log(5/\pi)\), the gamma term is \(\operatorname{Re}\psi(\frac34+\frac{ir}2)\), and \(\Lambda(n)\) is replaced by the coefficients \(b(n)\) of \(-f'/f\) \cite{C5}. With \(h(r)=e^{-(r/a)^2}\), \(2\le a\le5\), the two sides of each formula agree to \(10^{-14}\) for \(\zeta\) (682 zeros) and to \(10^{-16}\) for \(f\) \cite{C7}. Take \(g=\varphi*\tilde\varphi\), \(\tilde\varphi(u)=\overline{\varphi(-u)}\), so that \(h=|\hat\varphi|^2\) on the real line, and write \(W(\varphi)\) for the right side of (5.1). Under RH every \(\gamma_\rho\) is real and \(W(\varphi)=\sum_\rho|\hat\varphi(\gamma_\rho)|^2\ge0\). Weil proved the converse: RH holds if and only if \(W(\varphi)\ge0\) for every such \(\varphi\) \cite{We2,Bo}. If \(\varphi\) is supported in an interval of length \(L\), only \(n\le e^L\) enters, and by Parseval the prime part of \(W\) is the cross-energy of \(\varphi\) with its translates by \(\log p^k\), weighted by \(\Lambda(p^k)p^{-k/2}\) \cite{C7}: \[W(\varphi)=2\operatorname{Re}\Big[\hat\varphi\big(\tfrac i2\big)\overline{\hat\varphi\big(-\tfrac i2\big)}\Big]+\frac1{2\pi}\int|\hat\varphi(r)|^2\Big(\operatorname{Re}\psi\Big(\frac14+\frac{ir}2\Big)-\log\pi-2\operatorname{Re}\sum_{n\le e^L}\frac{\Lambda(n)}{n^{1/2+ir}}\Big)dr.\] \subsection{The Poisson kernel and its truncation} \textbf{Theorem E.} Let \(\sigma>\frac12\), \(\delta=\sigma-\frac12\) and \(h_{\delta,t}(r)=\delta/(\delta^2+(r-t)^2)\). \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item \(S(\sigma,t)=\sum_\rho h_{\delta,t}(\gamma_\rho)\), with the zeros summed in symmetric order. The function \(h_{\delta,t}\) is the Fourier transform of the positive-definite function \(g_{\delta,t}(u)=\frac12e^{-\delta|u|}e^{-itu}\). For an off-line pair, \(h_{\delta,t}(\gamma_\rho)+h_{\delta,t}(\overline{\gamma_\rho})=2\operatorname{Re}h_{\delta,t}(\gamma_\rho)\), which is negative exactly in the disc of Theorem C. \item For \(L>0\) let \(\varphi_L(u)=\sqrt{2\delta}\,e^{-\delta u}e^{-itu}\mathbf 1_{[0,L]}(u)\) and \(g_L=\varphi_L*\tilde\varphi_L\). Then \(g_L(u)=e^{-itu}\big(e^{-\delta|u|}-e^{-\delta(2L-|u|)}\big)\) for \(|u|\le L\) and \(g_L=0\) otherwise, and its Fourier transform is the entire function \[h_L(r)=\frac{2\delta\,\big(1-e^{-\delta L}e^{i(r-t)L}\big)\big(1-e^{-\delta L}e^{-i(r-t)L}\big)}{\delta^2+(r-t)^2}.\] Let \(W_L(\sigma,t)=\sum_\rho h_L(\gamma_\rho)=W(\varphi_L)\). Then: \end{enumerate} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \(W_L(\sigma,t)\) is given exactly by (5.1) with the prime powers \(n\le e^L\): \[\begin{aligned}W_L={}&h_L\big(\tfrac i2\big)+h_L\big(-\tfrac i2\big)-g_L(0)\log\pi+\frac1{2\pi}\int h_L(r)\operatorname{Re}\psi\Big(\frac14+\frac{ir}2\Big)dr\\&-2\sum_{n\le e^L}\frac{\Lambda(n)}{\sqrt n}\operatorname{Re}g_L(\log n).\end{aligned}\] \item If all zeros lie on the critical line, then \(h_L(\gamma)=2h_{\delta,t}(\gamma)\,|1-e^{-\delta L}e^{i(\gamma-t)L}|^2\) for every zero, and therefore \[\big(1-e^{-\delta L}\big)^2\le\frac{W_L(\sigma,t)}{2S(\sigma,t)}\le\big(1+e^{-\delta L}\big)^2.\] \item A zero with \(\beta>\sigma\) contributes to \(W_L(\sigma,t)\) a term of size \(e^{(\beta-\sigma)L}\), at every \(t\), with a phase that rotates with \(L\). \end{enumerate} \emph{Proof.} (a) By the Hadamard product, \(S=\sum_\rho\operatorname{Re}\frac1{s-\rho}\) in symmetric order \cite{La}, and \(s-\rho=\delta+i(t-\gamma_\rho)\). The maps \(\rho\mapsto1-\bar\rho\) and \(\rho\mapsto1-\rho\) act on \(\gamma_\rho\) as \(\gamma\mapsto\bar\gamma\) and \(\gamma\mapsto-\gamma\), so the multiset \(\{\gamma_\rho\}\) is invariant under conjugation. Hence \[\sum_\rho\operatorname{Re}\frac1{\delta+i(t-\gamma_\rho)}=\frac12\sum_\rho\Big[\frac1{\delta+i(t-\gamma_\rho)}+\frac1{\delta-i(t-\gamma_\rho)}\Big]=\sum_\rho h_{\delta,t}(\gamma_\rho).\] The Fourier transform of \(g_{\delta,t}\) is elementary, and the last assertion is Theorem C written in the variable \(\gamma_\rho\). \begin{enumerate} \def\labelenumi{(\alph{enumi})} \setcounter{enumi}{1} \tightlist \item The formula for \(g_L\) follows by direct integration, and \(h_L=|\hat\varphi_L|^2\) on the real line, which gives the displayed expression and its continuation. Since \(g_L\) is compactly supported and \(h_L\) is entire, (5.1) applies, with \(g_L(\log n)+g_L(-\log n)=2\operatorname{Re}g_L(\log n)\); this is (1). For real \(\gamma\) the factor \(|1-e^{-\delta L}e^{i(\gamma-t)L}|^2\) lies between \((1-e^{-\delta L})^2\) and \((1+e^{-\delta L})^2\), which with (a) gives (2). For \(\gamma_\rho=\gamma-i(\beta-\frac12)\) the factor \(e^{-\delta L}e^{i(\gamma_\rho-t)L}\) has modulus \(e^{(\beta-\sigma)L}\), which gives (3). \(\square\) \end{enumerate} Part (a) identifies the pointwise inequality \(S>0\) of Theorem 4.1 with Weil positivity along the family \(h_{\delta,t}\), and the disc of Theorem C with the region where \(h_{\delta,t}\) is negative at complex arguments. The test function \(g_{\delta,t}\) is not admissible in (5.1): \(h_{\delta,t}\) has poles at \(r=t\pm i\delta\), inside the strip \(|\operatorname{Im}r|<\frac12\), and the prime sum \(\sum\Lambda(n)n^{-1/2}g_{\delta,t}(\log n)\) converges only for \(\delta>\frac12\), that is, for \(\sigma>1\). This is the pole barrier again. Part (b) is the admissible version. It contains no remainder term: \(W_L\) is an exact finite expression in the primes up to \(e^L\). Under RH it is positive for every \(L\) and determines \(S\) to relative precision about \(2e^{-\delta L}\), so recovering the sign of \(S\) at \(\sigma=\frac12+\delta\) with a fixed margin requires the primes up to \(e^{C/\delta}\). \textbf{Corollary 5.1} (prime-resolution scale). Assume RH. Let \(0<\varepsilon<1\), \(\sigma=\frac12+\delta\) and \(P\ge2\). If \[\delta\,\log P\ge\log\frac{3}{\varepsilon},\] then the finite quantity \(W_{\log P}(\sigma,t)\), which involves only the prime powers \(n\le P\), satisfies, for every \(t\), \[\Big|\frac{W_{\log P}(\sigma,t)}{2S(\sigma,t)}-1\Big|\le\varepsilon.\] Equivalently, primes up to \(P\) determine \(S\) to relative precision \(\varepsilon\) at every distance \(\delta\) from the critical line with \(\delta\ge\delta_{\mathrm{res}}\), where \(\delta_{\mathrm{res}}=\delta_{\mathrm{res}}(P,\varepsilon)=\log(3/\varepsilon)/\log P\). \emph{Proof.} By Theorem E(b2), \(|W_L/2S-1|\le\max\{(1+a)^2-1,\,1-(1-a)^2\}=2a+a^2\le3a\) with \(a=e^{-\delta L}\le1\). Take \(L=\log P\). \(\square\) For example, primes up to \(10^8\) give \(\delta_{\mathrm{res}}=0.18\) at \(\varepsilon=0.1\) and \(0.31\) at \(\varepsilon=0.01\); resolving \(\delta=0.01\) to \(1\%\) requires \(P\ge10^{248}\). The corollary is conditional, and this cannot be removed by a better estimate. An unconditional bound for \(W_L-2S\) at abscissa \(\sigma\) must control the terms \(e^{(\beta-\sigma)L}\) of Theorem E(b3), that is, it requires knowing that no zero lies to the right of \(\sigma\) in the relevant range of heights. Such knowledge is available only inside the classical zero-free region and below the height \(3\cdot10^{12}\) to which RH has been verified \cite{PT}, where \(S>0\) already follows from Corollary 4.2. A finite certificate of the form ``\(W_L>B_L\Rightarrow S>0\)'' therefore certifies nothing beyond what is known. \subsection{Numerical results} \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{PartVIII_weil_poisson.png} \caption{The truncated Poisson functional $W_L(\sigma,t)$ of Theorem E(b), computed from the primes (or coefficients) up to $e^L$. Left: $\zeta$; the ratio $W_L/2S$ and the band $[(1-e^{-\delta L})^2,(1+e^{-\delta L})^2]$ implied by RH. Right: Davenport--Heilbronn function at $\sigma=0.6$, inside ($t=85.7$) and outside ($t=86.1$) the disc of its zero $0.8085+85.699i$; dashed lines show $2S$.}\label{fig:wp}\end{figure} \textbf{The truncated functional.} We computed \(W_L\) from the primes and, independently, from the zeros. The two agree to within \(3\cdot10^{-3}\), both for \(\zeta\) and for the Davenport--Heilbronn function with \(b(n)\) in place of \(\Lambda(n)\); the difference is the contribution of the zeros beyond the computed range. For \(\zeta\) at \((\sigma,t)=(0.6,27.7)\) and \((0.55,50)\), \(W_L/2S\) stays inside the band of (b2) for \(1\le L\le14\) (Figure \ref{fig:wp}, left). For the Davenport--Heilbronn function at \(\sigma=0.6\) (Figure \ref{fig:wp}, right) the behaviour is the opposite of the disc picture. At \(t=85.7\), inside the disc, where \(S<0\), \(W_L\) is positive and grows exponentially, reaching \(82\) at \(L=14\). At \(t=86.1\), outside the disc, where \(S>0\), \(W_L\) becomes negative at \(L=9\) and reaches \(-23\) at \(L=13.75\). The truncated functional therefore detects the off-line zero from the coefficients, but not locally: by (b3) every zero with \(\beta>\sigma\) affects \(W_L(\sigma,t)\) at every \(t\) once \(L\) is large. The locality of Theorem C holds only in the limit \(L\to\infty\), and that limit is what the Euler product does not control in \(\frac12<\sigma<1\). \textbf{Detection length.} The same mechanism, read from the side of an off-line zero, can be measured on \(f\), whose zeros are known. For each off-line zero \(\rho\) of \(f\) with \(\gamma<400\) we computed, at \(\sigma=0.55\), the smallest \(L\) (step \(0.25\)) at which the contribution of the pair \(\rho,1-\bar\rho\) to \(W_L(\sigma,t)\) exceeds in modulus the contribution of all zeros on the critical line, for some \(t\) with \(8\le|t-\gamma|\le12\), that is, outside the disc of Theorem C: \begin{longtable}[]{@{}rrrrr@{}} \toprule\noalign{} \(\beta\) & \(\gamma\) & \(\beta-\sigma\) & \(L_{\det}\) & \(L_{\det}(\beta-\sigma)\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(0.8695\) & \(240.40\) & \(0.320\) & \(16.5\) & \(5.3\) \\ \(0.8195\) & \(320.88\) & \(0.270\) & \(16.0\) & \(4.3\) \\ \(0.8085\) & \(85.70\) & \(0.259\) & \(19.3\) & \(5.0\) \\ \(0.7682\) & \(331.05\) & \(0.218\) & \(18.5\) & \(4.0\) \\ \(0.7243\) & \(176.70\) & \(0.174\) & \(22.8\) & \(4.0\) \\ \(0.6508\) & \(114.16\) & \(0.101\) & \(49.3\) & \(5.0\) \\ \(0.6285\) & \(366.64\) & \(0.079\) & \(69.5\) & \(5.5\) \\ \(0.5744\) & \(166.48\) & \(0.024\) & \(167.0\) & \(4.1\) \\ \end{longtable} A least-squares fit gives \(L_{\det}\approx4.05/(\beta-\sigma)+4.5\), with correlation \(0.993\) against \(1/(\beta-\sigma)\) (\(0.986\) against \(1/(\beta-\frac12)\)). The product \(L_{\det}(\beta-\sigma)\) stays between \(4.0\) and \(5.5\), its variation coming from the logarithm of the kernel factor \(2\delta/|\delta^2+(\gamma_\rho-t)^2|\) in (b3). The detection is not local in \(t\): at finite \(L\) the zero is seen at heights well outside its disc. The zero at \(\beta-\sigma=0.024\) requires \(L\approx167\), that is, the coefficients up to \(e^{167}\approx10^{72}\). \textbf{Weil matrices on windows.} The infimum of \(W\) over test functions on a window, \[\lambda_{\min}(L)=\min_{\operatorname{supp}\varphi\subset[-L/2,L/2]}\frac{W(\varphi)}{\|\varphi\|_2^2},\] involves only \(n\le e^L\). We computed it with the basis \(\varphi_m(u)=\sin(m\pi(u+L/2)/L)\), \(1\le m\le\lfloor10+12L\rfloor\), in two independent ways: from the explicit side of (5.1), and from the zero side, as the smallest singular value squared of \((\hat\varphi_m(\gamma_\rho))\) over the zeros with \(|\gamma|<600\), scaled by the Gram matrix \cite{C7}. \begin{longtable}[]{@{}lrr@{}} \toprule\noalign{} \(L\) & from the primes & from the zeros \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(0.25\) & \(3.305\cdot10^{-1}\) & \(3.305\cdot10^{-1}\) \\ \(0.50\) & \(3.719\cdot10^{-2}\) & \(3.718\cdot10^{-2}\) \\ \(0.75\) & \(5.798\cdot10^{-4}\) & \(5.798\cdot10^{-4}\) \\ \(1.00\) & \(1.079\cdot10^{-6}\) & \(1.079\cdot10^{-6}\) \\ \(1.25\) & \(3.47\cdot10^{-10}\) & \(3.40\cdot10^{-10}\) \\ \(1.50\) & --- & \(6.3\cdot10^{-15}\) \\ \(2.00\) & --- & \(1.7\cdot10^{-26}\) \\ \(2.50\) & --- & \(1.4\cdot10^{-36}\) \\ \end{longtable} The dashes mark values below the double-precision floor of the explicit side. The data fit \(\log_{10}\lambda_{\min}(L)\approx-6.3L^2\) (Figure \ref{fig:weil}, left). The transform of the minimizer vanishes near the zeros of \(\zeta\): at the first \(2\) for \(L=1\) and the first \(11\) for \(L=2\). The primes create the positivity. At \(L=1.25\) only \(p=2,3\) enter; without them (pole and archimedean terms only) \(\lambda_{\min}=-0.571\) and the minimizer vanishes at \(8.50\), \(13.49\), \(18.52\), \(23.56\); with them \(\lambda_{\min}>0\) and the minimizer vanishes at \(14.134\), \(21.022\), \(25.012\), against the zeros \(14.135\), \(21.022\), \(25.011\) (Figure \ref{fig:weil}, centre). Positivity is therefore decided by a cancellation whose depth grows like \(6.3L^2\) decimal digits, which explains why certified lower bounds are so small \cite{Liu}; earlier results establish positivity on small windows \cite{Yo,CC}. \textbf{Why the margin is so small.} The collapse of \(\lambda_{\min}(L)\) is not caused by the primes. It is already present for \(L<\log2\), before any prime enters, and it is a property of the density of the zeros, as a comparison with curves over finite fields shows. Let \(E\) be an elliptic curve over \(\mathbb F_p\) with \(\#E(\mathbb F_p)=p+1-a\) and \(a=2\sqrt p\cos\varphi\). Its zeta function depends on \(s\) only through \(p^{-s}\), and its zeros, all on the line by Hasse's theorem, are \(\frac12+i\gamma\) with \(\gamma\in\pm\varphi/\log p+\Omega\mathbb Z\), \(\Omega=2\pi/\log p\). We use the same test functions on \([-\frac L2,\frac L2]\) as for \(\zeta\). \textbf{Proposition 5.2.} If \(f\in L^2\) is supported in an interval of length \(L<\log p\) and \(F(r)=\int f(u)e^{iru}\,du\), then \[\sum_\gamma|F(\gamma)|^2=2\log p\,\|f\|_2^2,\] the sum running over the zeros of the zeta function of \(E\). The smallest value of Weil's form on such windows is therefore \(2\log p\), independent of \(L\) and of \(a\). \emph{Proof.} Let \(g=f*\tilde f\), which is continuous, supported in \((-L,L)\), and satisfies \(\hat g=|F|^2\). By Poisson summation, for each \(\theta\), \[\sum_{k\in\mathbb Z}\hat g(\theta+k\Omega)=\frac{2\pi}\Omega\sum_{n\in\mathbb Z}g\Big(\frac{2\pi n}\Omega\Big)e^{-2\pi in\theta/\Omega}=\log p\sum_ng(n\log p)\,e^{-in\theta\log p}.\] Since \(L<\log p\), only \(n=0\) contributes, and the sum is \(\log p\,g(0)=\log p\,\|f\|_2^2\). The zeros form two such progressions, \(\theta=\pm\varphi/\log p\). \(\square\) The terms \(n\ne0\), which enter when \(L\ge\log p\), are the closed points of \(E\) at \(u=n\log p\); they play the role of the primes. Numerically, the smallest value is \(9.23=2\log101\) for \(p=101\) up to \(L=4\), \(2.20=2\log3\) and \(1.39=2\log2\) for \(p=3\) and \(p=2\) up to \(L\approx\log p\) (Figure \ref{fig:margincurve}). The zeros of \(E\) have constant density \(\log p/\pi\); those of \(\zeta\) have density \(\frac1{2\pi}\log\frac t{2\pi}\), which comes from the factor \(\Gamma(\frac s2)\), and none below height \(14.13\). A function supported in a window of length \(L\) can be made small on a set of points only where the points are sparser than \(L/2\pi\), and it hides its mass there. Two experiments confirm that this is what happens. Removing the zeros of \(E\) (for \(p=101\)) with \(|t|<14\) makes its smallest value collapse like that of \(\zeta\): \(1.2\cdot10^{-4}\) at \(L=1\), \(5.9\cdot10^{-11}\) at \(L=2\). Adding these zeros to those of \(\zeta\) raises the value of \(\zeta\) from \(1.7\cdot10^{-26}\) to \(2.0\cdot10^{-3}\) at \(L=2\). The positions of the zeros of \(\zeta\) play no role beyond their density. Replacing them by the points defined by \(N_0(\gamma_k)=k-\frac12\), where \(N_0(T)=\frac T{2\pi}\log\frac T{2\pi e}+\frac78\) is the smooth counting function, gives \(\lambda_{\min}=1.3\cdot10^{-6}\), \(6.2\cdot10^{-15}\) and \(1.4\cdot10^{-26}\) at \(L=1\), \(1.5\), \(2\), against \(1.1\cdot10^{-6}\), \(6.3\cdot10^{-15}\) and \(1.7\cdot10^{-26}\) for the zeros of \(\zeta\); random displacements of \(25\%\) of the spacing change nothing, and a Poisson process with the same density gives smaller values, because of its larger gaps. Scaling the density by \(c\) gives \(\log_{10}\lambda_{\min}\approx-11.9L^2\), \(-6.6L^2\), \(-3.2L^2\) for \(c=\frac12,1,2\), that is, \[\log_{10}\lambda_{\min}(L)\approx-6.5\,L^2/c\qquad(L\le2)\] (Figure \ref{fig:marginlaw}). A heuristic for the form of this law: a function of exponential type \(L/2\) has at most about \(LR/\pi\) zeros in \([-R,R]\), and the zeros of \(\zeta\) in \([-R,R]\) number about \(c\frac R\pi\log\frac R{2\pi e}\), so the minimizer can follow them up to the height \(R\) with \(\log R\approx L/c\); if its residual beyond that height decays like a power \(R^{-\alpha L}\), then \(\log\lambda_{\min}\approx-\alpha L^2/c\), and the data give \(\alpha\approx6.5\log10\approx15\). We have not proved this. Its consequence is that the size of the margin carries no arithmetic information: it is the same for points with the density of the zeros and no relation to the primes, and no argument based on it can distinguish the zeros of \(\zeta\) from such points. \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Weil_margin_curve_vs_zeta.png} \caption{(a) Smallest value of Weil's form on windows of length $L$, from the zeros: $\zeta$, curves over $\mathbb F_p$ (constant value $2\log p$, Proposition 5.2), a curve with its zeros below height $14$ removed, and $\zeta$ with that gap filled. (b) Density of the zeros: constant for the curves, $\frac1{2\pi}\log\frac t{2\pi}$ for $\zeta$, with no zeros below $14.13$.}\label{fig:margincurve}\end{figure} \begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{Weil_margin_density_law.png} \caption{(a) The smallest value for the zeros of $\zeta$, for regular points with the same density, for these points displaced at random by $25\%$ of the spacing, and for a Poisson process with the same density. (b) Regular point sets with $c$ times the density of the zeros of $\zeta$, against $L^2/c$.}\label{fig:marginlaw}\end{figure} For the Davenport--Heilbronn function the same construction gives a form that is positive up to \(L^*\approx3.6\) and negative beyond: \(\lambda_{\min}=-1.8\cdot10^{-9}\), \(-0.017\), \(-0.226\), \(-0.730\) at \(L=3.6\), \(3.7\), \(3.8\), \(4.0\). The minimizer at \(L=4\) concentrates at \(r\approx84.5\), next to the off-line zero \(\rho_f=0.8085+85.6993i\), whose pair contributes \(-0.486\) (Figure \ref{fig:weil}, right). An independent discretization by cubic B-splines of mesh \(0.03\), for which the Weil matrix is Toeplitz, gives the same threshold: the smallest eigenvalue is \(-5\cdot10^{-7}\) at \(L=3.66\), \(-0.006\) at \(3.75\) and \(-1.2\) at \(3.9\), with the eigenvector peaked at \(r=84.5\). In this sense Weil positivity separates \(\zeta\) from \(f\) once the coefficients up to \(n\approx e^{3.6}\approx37\) are included. The same discretization for \(\zeta\) gives positive values on the spline space up to \(L\approx3.3\), where double precision is exhausted. \textbf{The prime powers one at a time.} On a window of length \(L\) only the \(n\le e^L\) enter (5.1), and the prime powers enter one at a time, at \(L=\log n\). Let \(\lambda_N(L)\) be the smallest eigenvalue of the form in which only the prime powers \(n\le N\) are kept; for \(N\ge e^L\) it is \(\lambda_{\min}(L)\). We computed \(\lambda_N(L)\) in the sine basis in double precision on a grid of step \(0.02\) in \(L\), and on cubic splines of mesh \(0.02\) in mpmath with \(36\) to \(79\) digits for \(1\le L\le2.8\). With the pole and archimedean terms only, the form is positive up to \(L=0.755\), consistent with \cite{CC}, which covers \(L\le\log2\). On the subspace \(\hat\varphi(\pm\frac i2)=0\), where the pole term vanishes, it is positive up to \(L=1.207\), beyond \(\log3\), and adding the prime \(2\) alone lowers this to \(1.146\). In every window the form becomes negative when the most recent prime power is removed, however short the time since it entered: at \(L=2.0\), removing \(7\) (which entered at \(1.946\)) gives \(-0.0031\); at \(L=2.6\), removing \(13\) (\(2.565\)) gives \(-2.5\cdot10^{-4}\); at \(L=2.8\), removing \(16\) (\(2.773\)) gives \(-1.4\cdot10^{-8}\) (Figure \ref{fig:pbp}a). The deficit grows roughly like \(\delta^6\) in \(\delta=L-\log q\) (fitted exponents between \(4.8\) and \(7.2\), Figure \ref{fig:pbp}b). Since \(\lambda_{\min}(L)\approx10^{-6.3L^2}\), a truncated form survives only for a length of about \(10^{-L^2}\) after the next prime power enters: positivity is maintained prime power by prime power, with no reserve. The spline values below \(10^{-14}\) are a discretization floor of order \(h^8\), not the true \(\lambda_{\min}\), which the table above gives. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Weil_prime_by_prime_mp.png} \caption{Weil positivity prime power by prime power. (a) $\lambda_N(L)$ for the form truncated at $n\le N$, on cubic splines of mesh $0.02$ in high precision; dotted lines at $L=\log q$; grey band: discretization floor. (b) The deficit $-\lambda_{q^-}(L)$ caused by omitting the most recent prime power $q$, against $\delta=L-\log q$ (sine basis); dotted lines: $\lambda_{\min}$ at $L=1$, $1.5$, $2$.}\label{fig:pbp}\end{figure} \textbf{Dirichlet polynomials as test functions.} The joint question of Remark 2.4 leads to the test functions \(\varphi=\sum_{n\le X}a_n\delta_{\log n}\), smoothed by a Gaussian. With \(w(r)=e^{-(r/T)^2}\) put \[A_{mn}=\sum_\rho w(\gamma_\rho)\Big(\frac mn\Big)^{i\gamma_\rho},\qquad1\le m,n\le X.\] Then \(\sum_{m,n}a_m\bar a_nA_{mn}=\sum_\rho w(\gamma_\rho)\big|\sum_na_nn^{i\gamma_\rho}\big|^2\) under RH, and (5.1) with \(h(r)=w(r)(m/n)^{ir}\) expresses each entry through the primes \(k\) with \(|\log k-\log\frac mn|\lesssim1/T\), the pole and the archimedean term. For the Davenport--Heilbronn function the same holds with its explicit formula. With \(T=60\) the entries computed from the primes (or the coefficients \(b(n)\)) and from the zeros agree to six digits or better. The smallest eigenvalue is: \begin{longtable}[]{@{}lrrrrrrr@{}} \toprule\noalign{} \(X\) & \(16\) & \(32\) & \(64\) & \(128\) & \(256\) & \(384\) & \(512\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(\zeta\) & \(0.28\) & \(1.9\cdot10^{-3}\) & \(2.6\cdot10^{-7}\) & \(7\cdot10^{-15}\) & \(0^*\) & \(0^*\) & \(0^*\) \\ Davenport--Heilbronn & \(18.9\) & \(8.0\) & \(0.11\) & \(3\cdot10^{-11}\) & \(-3.8\cdot10^{-4}\) & \(-1.6\) & \(-5.1\) \\ \end{longtable} (\(0^*\): below \(10^{-13}\) in modulus, the rounding level.) For \(f\) the form becomes negative between \(X=192\) and \(X=256\); at \(X=512\) the off-line zeros contribute \(-9.1\) to the minimal value and the zeros on the line \(+4.0\). For \(\zeta\) the margin collapses as on windows. At \(X=128\) the computed minimizer, normalized as \(c_n=a_n\sqrt n/a_1\), alternates between \(1\) at odd \(n\) and \(-1.14\) at even \(n\): the polynomial \(\sum c_nn^{-s}\) is of the type of the alternating series \(\eta(s)=(1-2^{1-s})\zeta(s)\), has modulus below \(10^{-4}\) at the first three zeros, and lies between \(0.2\) and \(2.3\) at the other points tested. This vector is not typical, however. The eigenvalues below \(10^{-9}\lambda_{\max}\) span a space of dimension \(11\) for \(\zeta\) and \(6\) for \(f\) at \(X=128\), and \(87\) and \(74\) at \(X=256\). On average \(78\%\) (\(X=128\)) and \(88\%\) (\(X=256\)) of the mass of these vectors lies on \(X/21\), \(E_1((s-1)\log x)=\int_x^\infty u^{-s}\,du/\log u\). Put \[E^*_x(s)=E_x(s)\exp\big(E_1((s-1)\log x)\big).\] \textbf{Proposition 6.1} (explicit formula in product form). Let \(t\ne0\) and let \(s\) not be a zero of \(\zeta\). Then \[\log E^*_x(s)=\log\zeta(s)+\sum_\rho E_1\big((s-\rho)\log x\big)+R_x(s)+O\big(x^{-2-\sigma}\big),\qquad R_x(s)=\sum_{k\ge2}\ \sum_{x^{1/k}1\). \emph{Proof.} \(\log E_x(s)=\sum_{n\le x}\Lambda(n)n^{-s}/\log n+R_x(s)\). Write the sum as \(\int_{2^-}^xu^{-s}\,d\psi(u)/\log u\) and insert \(\psi(u)=u-\sum_\rho u^\rho/\rho-\log2\pi-\frac12\log(1-u^{-2})\) \cite[\S 3.8]{T}, \cite{In}. For \(\sigma>1\), letting \(x\to\infty\) gives \(\log\zeta(s)\); the tail of the main term is \(E_1((s-1)\log x)\), the tail of each zero is \(-E_1((s-\rho)\log x)\), and the trivial zeros give \(O(x^{-2-\sigma})\). Both sides are analytic away from the zeros, the pole and the branch cuts of \(E_1\), and the sum over \(\rho\) converges locally uniformly in the stated order by the truncated explicit formula, so the identity extends by analytic continuation. \(\square\) The term of a zero is \(E_1((s-\rho)\log x)=x^{\rho-s}/((s-\rho)\log x)\,(1+O(1/(|s-\rho|\log x)))\). It decays like \(x^{\beta-\sigma}/\log x\) if \(\beta<\sigma\) and grows if \(\beta>\sigma\): on the line \(\operatorname{Re}s=\sigma\) the corrected product sees exactly the zeros to its right. The identity is a reorganization of the classical explicit formula; a smoothed form with the zeros truncated is the hybrid Euler--Hadamard product of Gonek, Hughes and Keating \cite{GHK}. \textbf{Corollary 6.2.} The following are equivalent: (i) RH; (ii) \(\log E^*_x(s)\to\log\zeta(s)\) locally uniformly on \(\{\frac12<\operatorname{Re}s<1,\ t\ne0\}\); (iii) for \(\frac12<\sigma<1\), \(t\ne0\), \(\log E^*_x(s)-\log\zeta(s)=O_s(x^{1/2-\sigma}\log^2x)\) locally uniformly. \emph{Proof.} (i)\(\Rightarrow\)(iii): under RH each mode is \(O(x^{1/2-\sigma}/(|s-\rho|\log x))\), the truncated explicit formula bounds their sum by \(O_s(x^{1/2-\sigma}\log^2x)\) \cite[Ch. XIV]{T}, and \(R_x(s)=O(x^{1/2-\sigma})\). (iii)\(\Rightarrow\)(ii) is immediate. (ii)\(\Rightarrow\)(i): each \(\log E^*_x\) is analytic on the region, so a locally uniform limit is an analytic logarithm of \(\zeta\) there, and \(\zeta\) has no zeros with \(\frac12<\beta<1\). \(\square\) This is the product form of the classical equivalence of RH with the analyticity of \(\log\zeta\) in \(\operatorname{Re}s>\frac12\). We computed \(E^*_x\) from all primes \(p\le2\cdot10^6\) and the right side of Proposition 6.1 from the \(682\) zeros with \(|\gamma|<600\): \textbf{The Euler product as a \(C_2\) energy.} The partial product and the cross-energy are related by an exact identity, valid inside the strip. For \(P\ge2\) let \(1=n_10\) and \(P\ge2\), \[|E_P(s)|^2=\prod_{p\le P}\big(1-p^{-2\sigma}\big)^{-1}+\sum_{k\ge2}C_2^{(P)}(n_k,s),\] the series converging absolutely. For \(\sigma>1\), letting \(P\to\infty\) gives \(|\zeta(s)|^2=\zeta(2\sigma)+\sum_{n\ge2}C_2(n,s)\). \emph{Proof.} For \(\sigma>0\) the product \(E_P(s)=\prod_{p\le P}\sum_{m\ge0}p^{-ms}\) of finitely many absolutely convergent geometric series equals \(\sum_kn_k^{-s}\), with absolute convergence. Telescoping \(|X^{(P)}_k|^2-|X^{(P)}_{k-1}|^2=n_k^{-2\sigma}+C_2^{(P)}(n_k,s)\) and letting \(k\to\infty\) gives the identity, since \(\sum_kn_k^{-2\sigma}=\prod_{p\le P}(1-p^{-2\sigma})^{-1}\). For \(\sigma>1\) the \(P\)-smooth integers exhaust all integers as \(P\to\infty\), with dominated convergence. \(\square\) With \(P=7\) and the \(14672\) seven-smooth integers up to \(10^{12}\), the right side equals \(0.886590\) at \(s=1.5+20i\), as does \(|E_7|^2\), and \(0.761394\) at \(s=0.6+20i\), against \(|E_7|^2=0.761404\); the difference is the contribution of the smooth integers above \(10^{12}\). Each partial Euler product is therefore the cross-energy of the partial sums of \(\zeta\) restricted to the \(P\)-smooth integers. Inside the strip the limit \(P\to\infty\) does not exist without the correction of Proposition 6.1: the energy over the smooth integers inherits the drift of size \(x^{1-\sigma}/\log x\) coming from the pole, which in the ordinary \(C_2\) is the term \(n^{1-2\sigma}/(1-s)\) of Theorem A. \begin{longtable}[]{@{}llll@{}} \toprule\noalign{} \(s\) & \(x\) & measured \(\log E^*_x-\log\zeta\) & predicted \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(0.8+20i\) & \(2\cdot10^6\) & \(-0.00068-0.00019i\) & \(-0.00070-0.00015i\) \\ \(0.7+20i\) & \(2\cdot10^6\) & \(-0.00285-0.00113i\) & \(-0.00292-0.00095i\) \\ \(0.6+20i\) & \(10^4\) & \(-0.01512-0.04052i\) & \(-0.01427-0.03802i\) \\ \(0.6+20i\) & \(2\cdot10^6\) & \(-0.01161-0.00629i\) & \(-0.01194-0.00552i\) \\ \(0.7+50i\) & \(2\cdot10^6\) & \(-0.00422+0.01193i\) & \(-0.00403+0.01198i\) \\ \end{longtable} The remaining discrepancy is the contribution of the zeros above height \(600\). The error of the Euler product is, point by point, the superposition of the modes of the zeros (Figure \ref{fig:euler}, left). Conversely, removing the computed modes from \(E^*_x\) reproduces \(\zeta\) to about \(1\%\) at \(0.6+20i\), \(0.75+50i\), and at \(\frac12+20i\) on the critical line. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{PartVI_euler_product.png} \caption{Left: the error $\operatorname{Re}[\log E^*_x(s)-\log\zeta(s)]$ at $s=\sigma+20i$ for $\sigma=0.6$ and $0.8$ (lines), against the sum over the zeros with $|\gamma|<600$ plus $R_x$ (circles). Right: $|E^*_x(\rho)|\log x/(|\zeta'(\rho)|e^{-\gamma_0})$ at the first five zeros.}\label{fig:euler}\end{figure} \textbf{At a zero.} Let \(\rho_*=\frac12+i\gamma_*\) be a simple zero. Isolating its own term and letting \(s\to\rho_*\), with \(E_1(w)=-\gamma_0-\log w+O(w)\), gives \[|E^*_x(\rho_*)|\log x=|\zeta'(\rho_*)|\,e^{-\gamma_0}\,\Big|\exp\Big(\sum_{\rho\ne\rho_*}E_1\big((\rho_*-\rho)\log x\big)+R_x(\rho_*)\Big)\Big|.\] The partial product decays like \(|\zeta'(\rho_*)|e^{-\gamma_0}/\log x\) provided the sum over the other zeros tends to \(0\), which is the Deep Riemann Hypothesis at \(\rho_*\); the constant agrees with Conrad \cite[Thm. 5.11]{Co}, see also \cite{Sh}. For the first zero, \(\zeta'(\rho_1)=0.7833+0.1247i\) and \(|\zeta'(\rho_1)|e^{-\gamma_0}=0.4453\). The ratio \(|E^*_x(\rho)|\log x/(|\zeta'(\rho)|e^{-\gamma_0})\) at the first five zeros stays within \(1\pm0.1\) for \(10^3\le x\le2\cdot10^6\), and its fluctuations are reproduced by the other \(681\) zeros to about \(0.005\) (Figure \ref{fig:euler}, right). For \(\rho_1\): \begin{longtable}[]{@{}lrrrrr@{}} \toprule\noalign{} \(x\) & \(10^3\) & \(10^4\) & \(10^5\) & \(10^6\) & \(2\cdot10^6\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot ratio & \(1.021\) & \(1.001\) & \(1.017\) & \(0.975\) & \(1.005\) \\ predicted & \(1.035\) & \(1.002\) & \(1.013\) & \(0.979\) & \(1.007\) \\ \end{longtable} The uncorrected product behaves differently: \(|E_x(\rho_1)|\) is \(0.023\), \(0.013\), \(0.417\) and \(1.4\cdot10^{-4}\) at \(x=10^4,10^5,10^6,2\cdot10^6\), because the pole term oscillates with amplitude \(x^{1/2}/(|1-\rho_1|\log x)\). The logarithmic increment of \(E_x\) at a prime is \(-2\log|1-p^{-s}|=2\operatorname{Re}p^{-s}+O(p^{-2\sigma})\), the prime analogue of \(C_2\) \cite{C2}; at a zero these increments add up to \(-2\log\log x+2\log|\zeta'(\rho)|-2\gamma_0\) plus the other modes. The constant \(|\zeta'(\rho)|\) is the one that governs the second-order behaviour of \(C_2\) at a zero \cite{C3}. In logarithmic form the same statement concerns the prime sum alone. Let \(D_x(s)=\sum_{p\le x}p^{-s}-\int_2^xu^{-s}\,du/\log u\). Removing the prime powers \(Q(s)=\sum_p\sum_{k\ge2}p^{-ks}/k\), which converge at \(\rho_*\), and using \(\int_2^xu^{-s}du/\log u=E_1((s-1)\log2)-E_1((s-1)\log x)\), the formula above gives \[\operatorname{Re}D_x(\rho_*)=-\log\log x+C_{\rho_*}+o(1),\qquad C_{\rho_*}=\log|\zeta'(\rho_*)|-\gamma_0-\operatorname{Re}Q(\rho_*)-\operatorname{Re}E_1\big((\rho_*-1)\log2\big),\] under the same proviso on the other zeros. The term \(-\log\log x\) is the resonance of the zero with itself, \(x^{\rho_*-s}=1\) at \(s=\rho_*\). For the first three zeros \(C_\rho=-1.142\), \(-0.406\), \(-0.076\), against \(\operatorname{Re}D_x(\rho)+\log\log x=-1.131\), \(-0.436\), \(-0.095\) at \(x=10^7\); the differences oscillate with the other zeros and decrease like \(1/\log x\). At the point \(\frac12+i\gamma\) below a hypothetical zero \(\frac12+\delta+i\gamma\) the same term becomes \(-\int_2^xu^{\delta-1}du/\log u\sim-x^\delta/(\delta\log x)\), the amplification of Remark 6.9. \textbf{The Davenport--Heilbronn control.} The analogue of Proposition 6.1 holds for \(f\) with \(\log f=\sum b(n)n^{-s}/\log n\), without a pole term. At height \(85.6993\) and \(x=2\cdot10^5\) the relative error \[\Big|\exp\Big(\sum_{n\le x}\frac{b(n)}{n^s\log n}\Big)\Big/f(s)-1\Big|\] is \(0.075\) and \(0.200\) for \(\sigma=0.95\) and \(0.90\), to the right of \(\rho_f=0.8085+85.6993i\), and \(1.334\) and \(1.062\) for \(\sigma=0.75\) and \(0.70\), to its left, with no decrease from \(x=10^4\). The equivalence of convergence with the absence of zeros to the right is a property of logarithmic Dirichlet series in general. What distinguishes \(\zeta\) is that its coefficients \(\Lambda(n)/\log n\) are nonnegative and supported on prime powers; for \(f\), \(72\%\) of \(\sum_{n\le2\cdot10^5}|b(n)|/n\) lies on integers that are not prime powers \cite{C5}. \textbf{Positivity and the pole.} For every \(x\), \(\sigma>0\) and real \(t\), the inequality of Mertens and de la Vallée Poussin \cite[\S 3.3]{T}, \[3\log|E_x(\sigma)|+4\log|E_x(\sigma+it)|+\log|E_x(\sigma+2it)|=\sum_{p\le x}\sum_k\frac{p^{-k\sigma}}k\big(3+4\cos\theta+\cos2\theta\big)\ge0,\] with \(\theta=kt\log p\), holds exactly. By Proposition 6.1 the corrected products satisfy the same identity with the pole corrections subtracted. At \((\sigma,t)=(0.7,14.13)\) and \(x=2\cdot10^6\) the positive sum is \(70.77\), the pole correction \(-74.45\), and the total \(-3.69\), against the limit \(3\log|\zeta(\sigma)|+4\log|\zeta(\sigma+it)|+\log|\zeta(\sigma+2it)|=-3.77\). Inside the strip the positive sum grows like \(3x^{1-\sigma}/((1-\sigma)\log x)\) and the correction from the real point \(\sigma\) cancels it. This cannot be avoided: a nonnegative trigonometric polynomial \(\sum a_j\cos j\theta\) has \(a_0>0\), and the \(a_0\) term sits at \(t=0\), at the pole. Every positivity argument of this type pays a correction of size \(x^{1-\sigma}\), which is why such arguments give zero-free regions only near \(\sigma=1\). Counting only the leading terms, a zero at \(\beta+it\) lowers the left side by about \(4x^{\beta-\sigma}/((\beta-\sigma)\log x)\), while the margin left by the pole is about \(3x^{1-\sigma}/((1-\sigma)\log x)\). Optimizing over \(\sigma\), the inequality can exclude exactly the zeros with \(\beta>1-0.427/\log x\), with the same constant for \(x\) from \(10^3\) to \(10^{100}\); with \(x\) a power of \(t\) this is the shape of the zero-free region of de la Vallée Poussin \cite[Ch. III]{T}. Twisting by characters does not remove this term. \textbf{Proposition 6.4.} Let \(F(\theta)=\sum_{|j|\le J}c_je^{ij\theta}\) be a nonnegative trigonometric polynomial, not identically zero, let \(\psi\) be a Dirichlet character mod \(q\), and put \[\Phi_x(\sigma,t)=\sum_{|j|\le J}c_j\log E_x\big(\sigma+ijt,\psi^j\big),\qquad\log E_x(s,\psi)=\sum_{p\le x}\sum_{k\ge1}\frac{\psi(p)^kp^{-ks}}k.\] Then \(\operatorname{Re}\Phi_x(\sigma,t)\ge0\) for all \(x\), \(\sigma>0\) and \(t\), and its \(j=0\) term is \(c_0\log E_x(\sigma,\psi^0)\) with \(c_0>0\) and \(\psi^0\) the principal character mod \(q\). For \(\sigma<1\) this term is \(c_0\,x^{1-\sigma}/((1-\sigma)\log x)\,(1+o(1))\). \emph{Proof.} The prime power \(p^k\), \(p\nmid q\), contributes \(p^{-k\sigma}k^{-1}F\big(k(\arg\psi(p)-t\log p)\big)\ge0\). The constant coefficient \(c_0\) is the mean of \(F\), which is positive. The asymptotic is the prime number theorem in the form \(\sum_{p\le x}p^{-\sigma}\sim x^{1-\sigma}/((1-\sigma)\log x)\). \(\square\) The same argument applies to several characters at once, with \(F\) a nonnegative trigonometric polynomial on a torus. A twist rotates the phase of each prime by \(\arg\psi(p)\) but leaves the mean of \(F\) attached to the principal character, which carries the pole. For a single nonprincipal character there is no pole: the uncorrected product \(E_x(s,\chi)\) for the character mod \(5\) of Section 2 approaches \(L(s,\chi)\) in the strip, with \(|E_x/L-1|=0.070\), \(0.025\), \(0.017\), \(0.009\), \(0.003\) at \(s=0.75+85.7i\) for \(x=10^3,10^4,10^5,10^6,2\cdot10^6\), where the logarithmic series of \(f\) stays above \(1.3\). But no linear inequality with nonnegative coefficients can be built from it alone. The quadratic analogue, averaging \(|\sum_p\chi(p)a_p|^2\) over a family of characters, is the large sieve, which is how the density estimates used in Theorem D are proved. \textbf{The Euler helix.} Proposition 6.1 has a geometric form. For \(s=\sigma+it\) with \(t\ne0\), not a zero, and \(u=\log x\), put \[H_s(u)=u\Big(\sum_{n\le e^u}\frac{\Lambda(n)}{n^s\log n}+E_1\big((s-1)u\big)-\log\zeta(s)\Big),\] the corrected partial product without the term \(R_x\), minus its limit, multiplied by \(u\). Since \(E_1(w)=e^{-w}w^{-1}\big(1+O(|w|^{-1})\big)\), Proposition 6.1 gives \[H_s(u)=\sum_\rho\frac{e^{(\rho-s)u}}{s-\rho}\Big(1+O\Big(\frac1{|s-\rho|\,u}\Big)\Big).\] Each zero contributes a vector of length \(e^{(\beta-\sigma)u}/|s-\rho|\) that turns with angular velocity \(\gamma-t\). With \(u\) as a third coordinate this is a helix of constant radius if \(\beta=\sigma\), an expanding spiral if \(\beta>\sigma\) and a contracting one if \(\beta<\sigma\). On the critical line, in the frame that turns with \(e^{itu}\), the pair \(\rho,\bar\rho\) with \(\rho=\frac12+i\gamma\) contributes \[-i\Big(\frac{e^{i\gamma u}}{t-\gamma}+\frac{e^{-i\gamma u}}{t+\gamma}\Big),\] two circles turning in opposite directions, that is, an ellipse with semi-axes \(|t-\gamma|^{-1}\pm|t+\gamma|^{-1}\). The pairs of zeros on the line are ellipses, as for the prime signal of Section 3, now centred at the point \(s\). \textbf{Remark 6.5} (energy of the Euler helix). Assume RH, and let \(s=\frac12+it\) with \(t\) not an ordinate. If the mean square of \(H_s\) is computed term by term, one obtains \[\lim_{U\to\infty}\frac1U\int_0^U|H_s(u)|^2\,du=\sum_\gamma\frac{m_\gamma^2}{(t-\gamma)^2}\ \ge\ \sum_\gamma\frac{m_\gamma}{(t-\gamma)^2}=-\frac{d^2}{dt^2}\log\big|\xi\big(\tfrac12+it\big)\big|,\] where the sums run over the distinct ordinates and \(m_\gamma\) is the multiplicity. The first equality is Parseval's identity for the almost periodic sum above; the second follows from the Hadamard product, in which each zero contributes \(m_\gamma\log|t-\gamma|\) to \(\log|\xi(\frac12+it)|\). Equality holds if and only if the zeros are simple, as in Theorem B. If a zero has \(\beta>\frac12\), its spiral grows like \(e^{(\beta-\frac12)u}\) and the mean square is infinite. We do not justify the interchange of the sum over the zeros with the mean; the computation below supports it. At \(t=20\), \(H_s\) computed from the prime powers up to \(10^8\) agrees with the sum over the \(682\) zeros with \(|\gamma|<600\) to a root mean square of \(0.056\) on \(8600\), and \(-(\log|\xi|)''(20)=1.077\) computed directly from \(\xi\). At \(\sigma=0.3\) and \(0.7\), \(|H_s|\) grows and decays like \(e^{\pm0.2u}\) (Figure \ref{fig:eulerhelix}). Seen from \(\sigma=0.3\) every zero lies to the right and produces an expanding spiral, as an off-line zero would on the critical line. The construction is the explicit formula and Parseval's identity in product form. It shows the Euler product inside the strip as a superposition of circles, one per zero, but it does not exclude spirals. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Euler_helix.png} \caption{The Euler helix (Remark 6.5). (a) $H_s(u)$ at $s=\frac12+20i$ from the prime powers up to $10^8$ (solid) and from the $682$ zeros with $|\gamma|<600$ (dashed), both smoothed in $u$. (b) In the frame turning with $e^{itu}$, the curve $e^{itu}H_s(u)$ for $12600\), which are not in the sum, contribute oscillations of every frequency; sampled at isolated points they alias onto the grid and produce the root mean square \(0.056\). We therefore average \(H_s\) exactly over bins of width \(h=5\cdot10^{-4}\), using for the pole term and for each zero the primitive \[\int u\,E_1(cu)\,du=\frac{u^2}2E_1(cu)-\frac{e^{-cu}}2\Big(\frac uc+\frac1{c^2}\Big),\] and convolve with a Gaussian of width \(0.02\) in \(u\), which removes the contribution of the zeros with \(|\gamma\mp t|>300\). For \(45\) heights \(16.5\le t\le117.7\) and the prime powers up to \(10^8\), the smoothed \(H_s\) then agrees with the smoothed sum over the zeros to a relative root mean square between \(2\cdot10^{-7}\) and \(8\cdot10^{-6}\) (median \(2.3\cdot10^{-6}\)), with no growth on \(10\le u\le18.3\) (Figure \ref{fig:rigidity}a). The remaining error is numerical. Replace a zero \(\rho=\frac12+i\gamma\) and its conjugate by the pairs \(\frac12\pm\delta+i\gamma\) and \(\frac12\pm\delta-i\gamma\), each of multiplicity \(\frac12\), which respects the functional equation. The first-order terms in \(\delta\) cancel, and the sum over the zeros changes by \[\frac{\delta^2}2\,\partial_\beta^2\big[uE_1((s-\rho)u)+uE_1((s-\bar\rho)u)\big]+O(\delta^4),\qquad \partial_\beta^2\,uE_1((s-\rho)u)=u^3E_1''((s-\rho)u).\] A symmetric pair is therefore seen by the helix only at second order. To measure it we fit the residual \(H_s-\sum_\rho\) by least squares on the second derivatives in \(\beta\) of the three zeros nearest to \(t\), together with their first derivatives in \(\gamma\), after the same smoothing; the coefficient of the second derivative estimates \(\delta^2/2\). On the actual data these coefficients are noise, and twice their standard deviation over the \(45\) heights defines a threshold \(\delta_{\min}\). With the primes up to \(x\) we obtain \[\delta_{\min}=1.3\cdot10^{-4},\ 0.9\cdot10^{-4},\ 0.5\cdot10^{-4},\ 0.6\cdot10^{-4},\ 0.4\cdot10^{-4}\qquad\text{for }\log x=10,\ 12,\ 14,\ 16,\ 18.3,\] a decrease close to \((\log x)^{-1.9}\) (Figure \ref{fig:rigidity}b). When the zero at \(21.022\) is split by \(\delta=0.003\), \(0.01\), \(0.03\) in synthetic data that keep the true residual, the fit returns \(0.0030\), \(0.0100\) and \(0.0291\) for every \(x\) between \(e^{10}\) and \(10^8\); the last value is low by the term of order \(\delta^4\) (Figure \ref{fig:rigidity}c). Without smoothing, a direct comparison of root mean squares could not separate the split zeros from the true ones below \(\delta\approx0.03\). That threshold came from the sampling and not from the second-order nature of the effect. The Euler product with primes up to \(10^8\) thus fixes the real parts of the zeros near a given height to within a few units of \(10^{-5}\), and this precision improves as \(x\) grows. The computation does not approach RH, for three reasons. (i) A measurement bounds \(|\beta-\frac12|\) but never gives \(\beta=\frac12\). Symmetry closes this gap locally: if a disc centred on the critical line contains exactly one zero counted with multiplicity, then \(\rho\) and \(1-\bar\rho\) both lie in it, so \(\rho=1-\bar\rho\) and \(\beta=\frac12\). This is the logic of Turing's method (Section 2). With interval arithmetic and rigorous bounds for the zeros outside the model, the computation would verify from the primes alone that the zeros near height \(t\) lie on the line. (ii) This verification is local: the number of primes needed grows with \(t\), and no finite computation covers all heights. (iii) The fit compares the helix with a list of known zeros. A zero outside the list appears only as a residual, and excluding it requires control of all zeros at once, which is RH itself or a density estimate. The smoothed \(H_s\) is Weil's explicit formula evaluated on Gaussian test functions; using it to locate zeros is standard in numerical work, and RH is equivalent to the positivity of Weil's functional on all test functions (Section 5), which no finite family reaches. Status: numerical. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Euler_helix_rigidity.png} \caption{Rigidity of the Euler helix (Remark 6.6), with the prime powers up to $x\le10^8$. (a) Relative root mean square of the difference between the smoothed $H_s$ computed from the primes and from the zeros, at $45$ heights $16.5\le t\le117.7$ (grey) and their median (black). (b) Threshold $\delta_{\min}$ of the second-order detector for symmetric pairs $\frac12\pm\delta$ (squares), with a fitted power law, and the threshold of a direct root-mean-square comparison when every zero is split, at $t=21.1$ (circles). (c) The zero at $\gamma=21.022$ split by $\delta=0.003$, $0.01$, $0.03$ in synthetic data, and the values of $\delta$ recovered by the detector.}\label{fig:rigidity}\end{figure} \textbf{Remark 6.7} (amplification and parity). For curves over finite fields the analogue of RH is proved by amplification. If \(C\) has genus \(g\) over \(\mathbb F_q\), then \(N_k-q^k-1=-\sum_{i\le2g}\alpha_i^k\), and a bound \(|N_k-q^k-1|\le Cq^{k/2}\) valid for all \(k\) forces \(|\alpha_i|=\sqrt q\), because a root with \(|\alpha|>\sqrt q\) produces a deviation that grows geometrically in \(k\). Figure \ref{fig:amplification}a shows this for \(q=101\): the numerator \(1-20T+101T^2\) of an elliptic curve stays within the Weil bound, while \(1-25T+101T^2\), whose counts \(N_k\) and numbers of closed points are nonnegative integers but which is not the numerator of a curve, leaves it by a factor \(4.5\cdot10^5\) at \(k=20\). Stepanov's method \cite{St,Bo73} proves the upper bound \(N_k\le q^k+O(q^{k/2})\) with an auxiliary polynomial; the lower bound needs an auxiliary family. In the example the upper bound holds for every \(k\) and the analogue of RH fails: the large root is real, the analogue of a real zero. Deligne \cite{De} extended the amplification to all varieties through tensor powers. Over \(\mathbb Q\) the powers \(\alpha^k\) become \(x^\rho\), and \(N_k-q^k-1\) becomes \(x-\psi(x)\). Here one side suffices. If \(\psi(x)\le x+Cx^\theta\) for all large \(x\), the Mellin transform of the nonnegative function \(x+Cx^\theta-\psi(x)\) is \(C/(s-\theta)+1/(s-1)+\zeta'(s)/(s\zeta(s))\) up to an entire function, which is regular on the real axis for \(s>\theta\) since \(\zeta\) has no zeros in \((0,1)\). By Landau's theorem on transforms of nonnegative functions its abscissa of convergence is at most \(\theta\), and \(\zeta\) has no zeros with \(\beta>\theta\) \cite[Ch. V]{In}. RH is therefore equivalent to an upper bound for the primes, \(\psi(x)\le x+O(x^{1/2}\log^2x)\), which is the form the amplification would take over \(\mathbb Q\). Upper bounds for primes are what sieves provide, and Selberg's \(\Lambda^2\) sieve is a positivity method: it minimizes the quadratic form \(\sum_n\big(\sum_{d\mid n,\,d\le R}\lambda_d\big)^2\) with \(\lambda_1=1\), as Stepanov's method minimizes the size of an auxiliary polynomial. On the intervals \((10^8,10^8+y]\), with \(R=\sqrt y\) and the optimal weights, the bound exceeds the number of primes by the factors \(3.07\), \(2.61\), \(2.24\) for \(y=10^4\), \(10^5\), \(10^6\) (Figure \ref{fig:amplification}b). Splitting the sum according to Liouville's function gives \(859+832\), \(7159+6966\) and \(62341+59012\) for \(\lambda(n)=-1\) and \(\lambda(n)=+1\). The sieve data \(\#\{n\equiv0\bmod d:\lambda(n)=\pm1\}\) agree within the statistical fluctuation for every \(d\le R\), so the sieve cannot distinguish the half that contains the primes from the half that contains none. A factor close to \(2\) is the parity barrier \cite{FI10}; the remaining factor \(1.15\) at \(y=10^6\) is the ratio \(\log x/\log y\) of the level of distribution, and \(2y/\log y\) is the Brun--Titchmarsh bound \cite{MoV}. The barrier is not a numerical gap. With no zeros at all, the estimate \(\psi(x+y)-\psi(x)\approx y\) is correct to \(0.1\%\), \(0.3\%\) and \(1.5\%\) for the three intervals, and the \(341\) zeros below \(600\) do not change this materially (Figure \ref{fig:amplification}c). What is missing is an inequality. The information that separates the primes from the products of two primes in the sifted set is the single sum \(\sum\lambda(n)\) over the sifted \(n\), and bounding it is the cancellation of Liouville's function in short intervals. Matomäki and Radziwiłł \cite{MR} proved this cancellation for almost all intervals using factorizations \(n=pm\), that is, bilinear information of type II, the same kind of input that gave primes of the forms \(x^2+y^4\) and \(x^3+2y^3\) \cite{FI98,HB}. The cross term \(C_2(n,s)=2\operatorname{Re}\big(X_{n-1}(s)\overline{n^{-s}}\big)\) is bilinear in the additive sense: its coefficients are all \(1\) and it does not split \(n\) into factors, so its arithmetic content comes only through the zeros. It does not supply the type II input. Status: the reductions are classical; the computations are ours. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Amplification_parity.png} \caption{Amplification and parity (Remark 6.7). (a) Normalized deviation $|N_k-q^k-1|/(2q^{k/2})$ for $q=101$ and the numerators $1-20T+101T^2$ (an elliptic curve) and $1-25T+101T^2$ (not a curve). (b) Selberg's $\Lambda^2$ upper bound for the primes in $(10^8,10^8+y]$ with $R=\sqrt y$, split according to $\lambda(n)=-1$ and $\lambda(n)=+1$, relative to the number of primes. (c) Error of the explicit formula for $\psi(x+y)-\psi(x)$ at $x=10^8$ when the zeros with $|\gamma|\le T$ are used; dotted lines mark $y$, the size of the parity gap in this normalization.}\label{fig:amplification}\end{figure} \textbf{Remark 6.8} (heat flow and the Laguerre hierarchy). The properties of the cross-energy that could force the zeros onto the line can be tested on the heat-flow deformation \[H_t(z)=\int_{-\infty}^{\infty}e^{tu^2}\Phi(u)\cos(zu)\,du,\qquad H_0(z)=\xi(\tfrac12+iz).\] There is a constant \(\Lambda\), the de Bruijn--Newman constant, such that \(H_t\) has only real zeros if and only if \(t\ge\Lambda\) \cite{Ne}. Rodgers and Tao proved \(\Lambda\ge0\) \cite{RT}, the Polymath project \(\Lambda\le0.22\) \cite{Pm}, and RH is the statement \(\Lambda\le0\). Hence \(H_t\) has non-real zeros for every \(t<0\), and a property that implies RH must hold at \(t=0\) and fail for every \(t<0\). We computed \(H_t\) and its derivatives exactly by the trapezoidal rule (step \(\frac1{150}\), \(60\) digits) on \(2t_c\) and changes sign at the collision point linearly in \(t-t_c\) (Figure \ref{fig:heat}b). (ii) The energy slope of Theorems C and D, in the form \(\partial_y|H_t(x+iy)|^2\ge0\) for \(y>0\), fails exactly where non-real zeros exist. (iii) The Jensen polynomials \(J^{d,0}\) of \(H_t\) with \(d\le120\) remain hyperbolic at \(t=-0.6\), \(-1\) and \(-1.5\), although \(H_t\) has non-real zeros at height about \(50\). The local energy properties detect the collision exactly, and the low-degree global ones do not see it. Inside any finite window \(\xi\) has a margin, here \(|t_c|=0.47\). Since \(\Lambda\ge0\), for every \(t<0\) some pair collides at a greater height, so collisions occur arbitrarily close to \(t=0\), and no finite computation can decide the question. The first Laguerre expression alone is not sufficient. The generating identity \(|f(x+iy)|^2=\sum_{n\ge0}L_n(f)(x)\,y^{2n}\) with \(L_n(f)=\sum_{k=0}^{2n}(-1)^{k+n}f^{(k)}f^{(2n-k)}/(k!\,(2n-k)!)\) shows that \(f\) has only real zeros if and only if \(L_n(f)\ge0\) for all \(n\) and \(x\) \cite{CV}. A pair \(x_0\pm iy_0\) contributes \(-2/y_0^2\) to \(-(\log|f|)''\) at \(x_0\), while real zeros with mean spacing \(\Delta\) contribute about \(\pi^2/\Delta^2\), so pairs with \(y_0>(\sqrt2/\pi)\Delta\) pass the test \(L_1\ge0\). For \(\zeta\) this would apply to off-line zeros above height about \(2000\), where \(\Delta<1\). On the deformation, the non-real zeros \(47.07+4.10i\) and \(55.66+5.01i\) at \(t=-4\) are first detected by \(L_2\), and \(49.89+6.53i\) at \(t=-6\) by \(L_3\); all others are detected by \(L_1\) (Figure \ref{fig:heat}c). On the side of the primes the hierarchy is a hierarchy of moments of the Euler helix. If all zeros are on the line, then \(L_n/\xi^2=e_n(w)\), the elementary symmetric function of the numbers \(w=(t\mp\gamma)^{-2}\), by the Hadamard product. If moreover the ordinates are linearly independent over \(\mathbb Q\), the moments \(M_{2m}\) of \(|H_s|\) over \(u\) determine the power sums of \(w\), and \(L_n/\xi^2\) is a polynomial in \(M_2,\dots,M_{2n}\). In particular \[\frac{L_1}{\xi^2}=M_2,\qquad\frac{L_2}{\xi^2}=\frac12\big(M_4-M_2^2\big)=\frac12\operatorname{Var}|H_s|^2.\] At \(s=\frac12+20i\) the exact value \(L_2/\xi^2=0.12026\) agrees with \(e_2(w)=0.12025\) from the zeros and with \(\frac12(M_4-M_2^2)=0.115\) to \(0.120\) computed from the prime powers up to \(10^8\) on windows \([u_0,18.4]\), \(8\le u_0\le12\). These moments are automatically nonnegative: the helix of a function with all zeros on the line is a sum of circles, and a variance is not negative. An off-line zero does not make \(L_2\) negative on the helix side; it turns a circle into a growing spiral and makes the moments infinite. In the language of the primes the whole Laguerre hierarchy therefore reduces to the finiteness of the energy of the Euler helix, which is Remark 6.5 and Theorem B, and the higher orders add no constraint. Status: the identities are classical or follow from the Hadamard product (the moment identity uses linear independence); the computations are numerical. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Heat_flow_Laguerre.png} \caption{Heat flow and the Laguerre hierarchy (Remark 6.8). (a) Real zeros of $H_t$ in $100\). With the prime powers up to \(10^8\), \(E(\varepsilon)\) agrees with the double sum over the \(682\) zeros with \(|\gamma|<600\), the constant term and the diagonal contribution of the higher zeros to within \(0.3\) to \(0.6\%\) for \(0\le\varepsilon\le1\) (Figure \ref{fig:damped}a). The price is the size of the primes needed. We added to the zeros a pair \(\frac12\pm\delta+iT\) (with conjugates) and compared the mean energy on \([2,U]\) with that of the same pair placed on the line. The difference reaches \(10\%\) at \(U=13\), \(39\) and \(115\) for \(T=27.5\) and \(\delta=0.1\), \(0.03\), \(0.01\); at \(U=25.5\), \(81\) and \(240\) for \(T=100.3\); and at \(U=43.5\) and \(145\) for \(T=500.7\) and \(\delta=0.1\), \(0.03\) (Figure \ref{fig:damped}b,c). This follows \(x^*=e^{U^*}\approx(cT^2)^{1/(2\delta)}\), roughly \(T^{1/\delta}\): an energy detects an off-line zero at any height, but only with the primes up to about \(T^{1/\delta}\). Since RH is verified up to \(T=3\cdot10^{12}\) \cite{PT}, detecting there a zero with \(\delta=0.01\) would need \(x\) of order \(10^{1250}\). That an off-line zero influences \(\psi(x)\) only for \(x\) beyond a power \(1/\delta\) of its height is the classical heuristic; here it is measured on the energy. Status: the inequality and the location of the singularity are proved; the computations are numerical. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Damped_energy.png} \caption{The damped energy (Remark 6.9). (a) $\varepsilon E(\varepsilon)$ on $2\le u\le\log10^8$ from the prime powers (solid) and from the zeros with the constant term and the tail (dashed); the dotted line is $2+\gamma_0-\log4\pi$, the limit as $\varepsilon\to0$ when the range of $u$ is infinite. (b) Mean energy on $[2,U]$ with a pair at height $100.3$ on the line (black) and at $\beta=\frac12+\delta$. (c) The size $x^*=e^{U^*}$ of the primes at which the mean energy exceeds its value under RH by $10\%$, against $1/\delta$.}\label{fig:damped}\end{figure} \textbf{Remark 6.10} (the sieved wave is the Euler helix). The wave \(W(n,s)=2X_{n-1}(s)\overline{n^{-s}}\) of Section 2 and the Euler helix of Remark 6.5 are one object. Remove from the partial sums the multiples of the primes \(p\le y\). By multiplicativity, \[X^{(y)}_{n-1}(s)=\sum_{\substack{ky\) with \(p\le y\). With the primes up to \(10^6\) the two sides agree to about \(1\%\), the accuracy of the grid on which \(H_s\) was stored. At \(t=20\), \(|A^*_y-1|\) is \(0.035\) and \(0.005\) for \(y=10^3\) and \(10^6\) at \(\sigma=0.7\), decreases like \(1/\log y\) at \(\sigma=\frac12\), and grows at \(\sigma=0.3\), where every zero produces an expanding spiral (Figure \ref{fig:sieved}c). Hence RH holds if and only if \(A^*_y(s)\to1\) for every \(s\) with \(\sigma>\frac12\), which is the statement on corrected partial products of this section. Sieving the wave by the Euler product turns its free amplitude into the Euler helix and adds no new information; the obstruction is again the pole correction, which has to be removed by hand. Status: the identities are proved; the computations are numerical. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Sieved_wave.png} \caption{The sieved wave (Remark 6.10). (a) $W_y$ at $s=\frac12+20i$ with no sieve and with the primes $\le3$ and $\le7$ removed; crosses: the centres $2c_y/(1-s)$. (b) Distance of $W_y$ to its centre (maximum over blocks of $210$ consecutive $n$) at $\rho_1$ (solid) and at $\frac12+20i$ (dotted). (c) $|A^*_y(s)-1|=|e^{-H_s(\log y)/\log y}-1|$ at $t=20$ for $\sigma=0.3$, $0.5$, $0.7$, from the prime powers up to $10^8$.}\label{fig:sieved}\end{figure} \textbf{Remark 6.11} (prime operators and the functional equation). Extend the wave to real arguments by \(W(x,s)=2X_{\lceil x\rceil-1}(s)\overline{x^{-s}}\). Removing the multiples of a prime \(p\) from the partial sums acts on the sieved wave by \[W_{y\cup\{p\}}(n,s)=W_y(n,s)-p^{-2\sigma}\,W_y(n/p,s),\] because the multiples of \(p\) below \(n\) contribute \(p^{-s}X^{(y)}_{\lceil n/p\rceil-1}(s)\) and \(\overline{n^{-s}}=\overline{p^{-s}}\;\overline{(n/p)^{-s}}\). The identity holds to \(10^{-15}\) at \(s=\frac12+20i\), \(0.7+33i\) and \(\rho_1\) for \(p=2,3,5,7\). The phase \(t\) cancels: on the critical line each prime acts by the real operator \(1-p^{-1}T_p\), \(T_pW(n)=W(n/p)\), the same at every height, so the primes act on the waves of all zeros at once by the same operators. On a mode \(n^\lambda\) the operator multiplies by \(1-p^{-2\sigma-\lambda}\), and all primes together by \(1/\zeta(2\sigma+\lambda)\). For the three modes of \(W\) this gives \(1/\zeta(1)=0\) on the centre (\(\lambda=1-2\sigma\)), \(1/\zeta(s)\) on the rotating term (\(\lambda=-\sigma+it\)), and \(0\) on the drift (\(\lambda=-2\sigma\)); equivalently \(W=\sum_mm^{-2\sigma}T_m[2n^{-\bar s}]\) is a positive combination of dilates of the elementary wave \(2n^{-\bar s}\). At a zero, the rotating amplitude \(0\) meets the eigenvalue \(1/\zeta(\rho)=\infty\). For the Davenport--Heilbronn function no such operator exists: the best fit of the sieved wave by \(W_f-cT_pW_f\) leaves a relative residual \(1.00\) for \(p=2,3,7\). For \(L(s,\chi)\) the operator is \(1-\chi(p)p^{-2\sigma}T_p\); the structure is that of every Euler product, so an argument for RH resting on it alone would apply to the whole Selberg class. Combined with the functional equation \(\zeta(s)=\chi(s)\zeta(1-s)\), the sieve gives the sieved factor \[\chi_y(s)=\frac{\zeta_y(s)}{\zeta_y(1-s)}=\chi(s)\prod_{p\le y}\frac{1-p^{-s}}{1-p^{s-1}}.\] On the critical line \(|\chi_y(s)|=1\) for every \(y\), since \(1-s=\bar s\); at \(s=\frac12+20i\) the computed values are \(1\) for \(1\le y\le10^6\). Off the line the factor is erratic: at \(s=0.7+20i\), \(|\chi_y|=0.79\), \(0.70\), \(1.53\), \(0.72\), \(19.5\), \(<10^{-3}\), \(3\cdot10^{-4}\) for \(y=1,10,\dots,10^6\). The reason is that the reflected point \(1-\sigma=0.3\) lies to the left of the critical line, where the Euler product diverges whether or not RH holds, because the zeros on the line lie to its right. The helix measures this: \(|H_s(u)/u|\) decreases from \(0.147\) to \(0.0025\) as \(y=e^u\) goes from \(10^2\) to \(10^8\) at \(s=0.7+20i\), and grows from \(0.79\) to \(2.94\), like \(y^{0.2}/\log y\), at \(s=0.3+20i\). The functional equation therefore pairs the half of the strip where the Euler product converges, provided there are no zeros to the right, with the half where it always diverges. The compatibility of the prime operators with the reflection on \(\sigma=\frac12\) is a symmetry and not a constraint, and at an off-line pair the relation reads \(0=\chi_y\cdot0\). Combining the two structures would require control of the Euler product on the divergent side. Status: the identities are proved; the computations are numerical. \textbf{Proposition 6.12} (no linear polarization without the pole). Let \(F(x)=\sum_{j=1}^Kc_j\cos(t_jx+\phi_j)\) with real \(c_j,\phi_j\) and distinct \(t_j>0\), so that \(F\) has no constant term. If \(F(\log p)\ge0\) for all but finitely many primes \(p\), then \(F\equiv0\). \emph{Proof.} For real \(u\ne0\) the partial sums \(\sum_{p\le x}p^{-1-iu}\) are bounded as \(x\to\infty\); this is the prime number theorem in the form \(\zeta(1+iu)\ne0\) \cite[Ch. III]{T}, \cite{IK}. Since \(\sum_{p\le x}p^{-1}\to\infty\), the weighted mean \(\langle G\rangle_x=\sum_{p\le x}G(\log p)p^{-1}\big/\sum_{p\le x}p^{-1}\) tends to \(0\) for every \(G(x)=\cos(ux+\phi)\) with \(u\ne0\), and to \(1\) for \(G=1\). Hence \(\langle F\rangle_x\to0\). Expanding the square, \[F^2=\frac12\sum_jc_j^2+\frac12\sum_jc_j^2\cos(2t_jx+2\phi_j)+\sum_{j0\) in (4.1) and Proposition 6.4: a combination of the values \(\operatorname{Re}\frac{\zeta'}{\zeta}(\sigma+it_j)\) that is positive prime by prime must contain the point \(t=0\), that is, the pole. The obstruction is the uniform distribution of the angles \(t\log p\), which follows from the absence of zeros on \(\sigma=1\). Finitely many primes can be polarized: maximizing \(\min_pF(\log p)\) by linear programming over \(80\) frequencies in \([0.5,40]\) with \(|c_j|\le1\) gives a positive minimum \(12.6\), \(5.5\) and \(1.6\) on the primes up to \(30\), \(100\) and \(1000\), but the optimized \(F\) is then negative on \(57\%\), \(52\%\) and \(59\%\) of the primes from there to \(10^6\), and its mean over \(x\) is \(0\) to within \(10^{-3}\). A positivity without the pole term can only be quadratic, \(|\sum a_nn^{-s}|^2\ge0\); this is the route of Weil's functional (Section 5), where the margin decreases like \(10^{-6.3L^2}\), and of Selberg's sieve, where it meets the parity barrier (Remark 6.7). Status: the proposition is proved; the optimization is numerical. \textbf{Remark 6.13} (the wave over a curve and the degree form). The constructions of Section 2 can be carried out for an elliptic curve \(E/\mathbb F_q\), where the missing positivity is known. With \(T=q^{-s}\) one has \(\zeta_E(s)=Z(T)=(1-aT+qT^2)/((1-T)(1-qT))\), \(a=q+1-N_1\), and the coefficient of \(T^n\) is the number of effective divisors of degree \(n\), \(b_0=1\) and \(b_n=N_1(q^n-1)/(q-1)\) for \(n\ge1\). Define the partial sums \(X_{N-1}(s)=\sum_{n2\sqrt2=2.828\). The dumbbell has a pair of real poles off the circle, at \(\beta=0.861\) and \(0.139\) (Figure \ref{fig:graphsFq}a). The numbers \(N_m\) of closed non-backtracking cycles, computed from the Hashimoto matrix, agree with \(q^m+1+2(r-1)[m\text{ even}]+\sum(\alpha^m+\bar\alpha^m)\), and the normalized deviation stays bounded for the Ramanujan graph and grows geometrically for the dumbbell (b). The wave built from the partial sums of \(\sum_n\operatorname{tr}(A_n)u^n=\operatorname{tr}\,(1-u^2)(I-Au+qu^2I)^{-1}\), where \(\operatorname{tr}A_n\) counts the closed non-backtracking walks of length \(n\), has the centre \(-2(1+q^{-1})q^N|u|^{2N}/(1-qu)\) coming from the trivial pole, of constant modulus on \(\sigma=\frac12\) as for \(\zeta\). In \(200\)-digit arithmetic, \(|W-\text{centre}|\) at two points of the critical circle decays with slopes \(-0.50\) and \(-0.47\) in \(\log_q\) per step for the Ramanujan graph, the second affected by the beating of the \(78\) nontrivial frequencies, and \(-0.139=\beta-1\) for the dumbbell (c). The Toeplitz matrix \([s_{j-k}]\) built from the primes, \(s_m=q^{-m/2}\big(N_m-q^m-1-2(r-1)[m\text{ even}]\big)\) and \(s_0=2(n-1)\), which is the analogue of the Weil matrices of Section 5, is positive definite up to size \(40\) for the Ramanujan graph and has a negative eigenvalue from size \(13\) for the dumbbell (d). The structural positivity of a graph is the symmetry of \(A\): the eigenvalues are real, so a nontrivial pole lies either on the circle or on the real axis, and a graph that is not Ramanujan uses the second possibility. Over \(\mathbb Q\) this alternative is closed, since \(\zeta(\sigma)<0\) for \(0<\sigma<1\); a self-adjoint operator in the sense of Hilbert and Pólya would therefore suffice for \(\zeta\), while for graphs it does not. The explicit Ramanujan graphs of \cite{LPS} obtain the bound \(|\lambda|\le2\sqrt q\) from the Ramanujan--Petersson conjecture, proved by Deligne, that is, from geometry. \item \emph{Dirichlet \(L\)-functions over \(\mathbb F_q[T]\).} For a nonprincipal character \(\chi\) modulo a monic irreducible \(M\) of degree \(d\), \(L(u,\chi)=\sum_f\chi(f)u^{\deg f}\) over monic \(f\) is a polynomial of degree at most \(d-1\), its zeros satisfy \(|u|=q^{-1/2}\) by Weil's theorem \cite{Ro}, and it satisfies \(L(u,\chi)=\epsilon(\chi)(\sqrt q\,u)^{d-1}L(1/(qu),\bar\chi)\). The combination \(F=L(\chi)+\epsilon(\chi)L(\bar\chi)\) satisfies \(F(u)=(\sqrt q\,u)^{d-1}F(1/(qu))\), the analogue of the Davenport--Heilbronn construction, but its coefficients are not multiplicative. For \(q=3\), \(M=T^7+2T^6+T^5+1\) and the character \(\chi(g)=e^{2\pi i/2186}\) on a generator \(g\), the six zeros of \(L\) lie on the circle, while \(F\) has a real pair off it with \(\beta=-0.120\) and \(1.120\), outside the strip, as is possible without an Euler product (Figure \ref{fig:graphsFq}e). The prime sums \(\psi_n=\sum_{\deg P^j=n}\deg P\,\chi(P)^j\), computed by enumerating the irreducibles for \(n\le8\), agree with \(-\sum\alpha^n\) to \(10^{-12}\) and satisfy \(|\psi_n|\le6q^{n/2}\), while \(\sum\alpha^n\) for \(F\) grows geometrically (f). Here the wave carries no information: the Dirichlet series is a polynomial, so \(W(N,\rho)=2L(\rho)\overline{\rho^N}=0\) for \(N\ge d\) at every zero, of \(L\) and of \(F\) alike (g). The Toeplitz matrix built from \(s_m=q^{-m/2}\sum\alpha^m\) is positive semidefinite for \(L\), with smallest eigenvalue exactly \(0\) (to \(10^{-13}\)) from size \(7\) on, the rank being \(d-1=6\), and is indefinite for \(F\) from size \(4\), with smallest eigenvalue \(-5\cdot10^8\) at size \(30\) (h). For \(L\) the positivity is that of a Gram matrix: \(\sqrt q^{-1}\alpha_j\) are the eigenvalues of a unitary operator, the normalized Frobenius on the first cohomology of the corresponding curve. \end{enumerate} In the three settings of Remarks 6.13 and 6.14 the wave measures \(\beta\) when the Dirichlet series is infinite and is blind when it is finite, and in none of them does it decide the analogue of RH. What decides it is in each case Weil's quadratic positivity, and where it holds it comes from a unitary or self-adjoint operator: the dual isogeny, the symmetric adjacency matrix together with a bound on its spectrum, or Frobenius on \(H^1\). Status: the identities are classical; the computations are exact or carried out to \(10^{-12}\). \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Graphs_and_FqT.png} \caption{Graphs and $\mathbb F_q[T]$ (Remark 6.14). Top, cubic graphs on $40$ vertices ($q=2$), a Ramanujan graph and a dumbbell with $\lambda_2=2.918$: (a) nontrivial poles $\sqrt q\,u$ of $Z_G$; (b) normalized deviation of the counts $N_m$ of closed non-backtracking cycles; (c) distance of the wave to its centre at $\sqrt q\,u=(3-4i)/5$; (d) smallest eigenvalue of the Toeplitz matrix built from the $N_m$. Bottom, $q=3$, $\deg M=7$: (e) zeros $\sqrt q\,u$ of $L(u,\chi)$ and of the Davenport--Heilbronn-type combination; (f) $|\psi_n|/q^{n/2}$; (g) $|W(N,\rho)|$ at the zeros, identically $0$ for $N\ge7$; (h) smallest eigenvalue of the Toeplitz matrix.}\label{fig:graphsFq}\end{figure} \textbf{Remark 6.15} (the wave as a quantum state). Sample \(W(n,s)\) at \(64\) points with \(\log n\) equally spaced in \([7,9]\), form the \(3\times62\) trajectory matrix \(H_{jk}=W(n_{j+k},s)\) and the density matrix \(\rho=HH^*/\operatorname{tr}HH^*\). We record the von Neumann entropy \(S(\rho)/\log3\in[0,1]\), the \(\ell_1\)-coherence, and the drift \(\kappa\) of the dominant eigenvector \(v\), defined by \(v_{j+1}/v_j=e^{\kappa h}\) with \(h\) the step in \(\log n\), so that a single mode \(W\propto n^{\kappa}\) gives a pure state with drift \(\kappa\). The recursion \(W(n+1,s)=(n/(n+1))^{\bar s}[W(n,s)+2n^{-2\sigma}]\) splits exactly into a transient and a steady part, \[W(n,s)=2\zeta(s)\,n^{-\bar s}-2\zeta(s,n)\,n^{-\bar s},\] with the Hurwitz function \(\zeta(s,n)\) (verified to \(10^{-15}\)). The steady part is \(-2n^{1-2\sigma}/(1-s)+O(n^{-2\sigma})\), the centre of Section 2, and \(\zeta(s)\) is the amplitude of the transient. At a zero the transient is absent and the state is pure with drift \(\kappa=1-2\beta\); elsewhere the second eigenvalue of \(\rho\) decays like \(n^{2\sigma-2}\) as the window moves, and at a zero like \(n^{-2}\). Measured slopes of \(\log\lambda_2\) are \(-0.962\) at \(\frac12+20i\) (predicted \(-1\)), \(-0.587\) at \(0.7+i\gamma_1\) (\(-0.6\)) and \(-2.000\) at \(\rho_1\) (Figure \ref{fig:berry}a). On \(\sigma=\frac12\) the entropy falls from \(0.1\)--\(0.5\) between zeros to \(10^{-5}\)--\(10^{-3}\) at every zero, with \(\kappa=10^{-4}\) (Figure \ref{fig:qstate}a, b). For the Davenport--Heilbronn function the zeros are instead maximally mixed, \(S=0.96\)--\(0.99\) at the five zeros on the line with \(80\frac12\), \(\psi_s=\zeta(2\sigma)^{-1/2}\sum_nn^{-s}|n\rangle\). Then \[\psi_s=e^{-itL}\psi_\sigma,\] and \(e^{-itL}\) is diagonal in the basis \(|n\rangle\) and a product of diagonal unitaries over the primes. Hence every coherence measure with respect to the basis \(|n\rangle\), every entropy of the populations and every entanglement between groups of primes is a function of \(\sigma\) alone; the zeros are invisible to all of them. For the truncation \(n\le4096\) on \(\sigma=\frac12\) the relative entropy of coherence \(6.066379389410\), the \(\ell_1\)-coherence \(1799.345\) and the entanglement between the prime \(2\) and the odd primes \(0.233336\) coincide to twelve digits at \(t=0\), \(14.13\), \(20\) and \(37.59\). The relative entropy of coherence of \(\psi_s\) is the entropy of the populations \(n^{-2\sigma}/\zeta(2\sigma)\), \[C(\sigma)=\log\zeta(2\sigma)-2\sigma\frac{\zeta'}{\zeta}(2\sigma)\sim\frac1{2\sigma-1}\qquad(\sigma\to\tfrac12^+),\] the thermal entropy of the gas of primes \cite{Ju,BC} at inverse temperature \(2\sigma\); \(\psi_s\) is normalizable exactly for \(\sigma>\frac12\), and the critical line is the boundary of the Hilbert space (Figure \ref{fig:cohEuler}a). The zeros appear in the dynamics. With the alternating state \(|a\rangle=N^{-1/2}\sum_{n\le N}(-1)^{n+1}|n\rangle\), the frequency \(\frac12\) of the additive Fourier basis, one has \(\langle a|e^{-itL}|\psi_{1/2}\rangle\propto(1-2^{1/2-it})\zeta(\frac12+it)\) up to the truncation, so the zeros on the line are the times at which the evolved state becomes orthogonal to \(|a\rangle\). For \(N=4096\) the fidelity drops to \(10^{-9}\)--\(10^{-10}\) at the ten zeros with \(t<50\), while at \(\sigma=0.6\) it stays above \(10^{-6}\) (Figure \ref{fig:cohEuler}b). For the other frequencies \(a/q\) the overlap is \(F(s,a/q)=\sum_ne^{2\pi ina/q}n^{-s}\), and every zero found with \(t<40\) lies off the line, with \(\max|\beta-\frac12|=0.66\), \(0.73\), \(0.79\) for \(q=3,4,6\) and \(0.77\), \(0.78\), \(0.81\) for \(\frac15\), \(\frac25\), \(\frac18\) (Figure \ref{fig:cohEuler}c), in accordance with \cite{DH}. The additive character \(e^{2\pi ina/q}\) is a superposition of \(\sum_{d\mid q}\#\{\text{primitive characters mod }d\}\) primitive \(L\)-functions, which is \(1\) only for \(q\le2\), \(2\) for \(q=3,4,6\) and \(4\) for \(q=5,8\) (Figure \ref{fig:cohEuler}d). Measured against a single multiplicative character the orthogonality times lie on the line; measured against a coherent superposition of characters they do not. The dependence on the amount of coherence can be measured. For the real character \(\chi_5\) modulo \(5\) let \[\Lambda(s,\chi_5)=\tfrac12s(s-1)(5/\pi)^{s/2}\Gamma(\tfrac s2)L(s,\chi_5),\] and \(G_\varepsilon=\xi+\varepsilon\Lambda(\cdot,\chi_5)\), which satisfies \(G_\varepsilon(s)=G_\varepsilon(1-s)\) and is real on the line for every real \(\varepsilon\). For \(t<100\), \(\xi\) has \(29\) zeros and \(\Lambda(\cdot,\chi_5)\) has \(54\), all on the line. For every \(\varepsilon>0\), \(G_\varepsilon\) has the larger number, and the \(13\) additional pairs with \(t<100\) lie where the two terms have equal size, at \[\beta\approx\frac{2\log(1/\varepsilon)}{\log5},\] spaced by \(4\pi/\log5=7.81\) in \(t\) (Proposition 6.17): the median \(\beta\) is \(8.585\), \(5.733\), \(3.795\) for \(\varepsilon=10^{-3}\), \(10^{-2}\), \(0.05\), against \(8.584\), \(5.723\), \(3.723\) (Figure \ref{fig:cohlaw}a, b). As \(\varepsilon\to0\) these zeros recede to infinity, while the zeros of \(\xi\) stay on the line apart from one pair that leaves it for \(0.1\le\varepsilon\le0.15\). As \(\varepsilon\) grows the distant zeros enter the strip and land on the line in pairs, and the number of zeros on the line increases from \(29\) to \(54\) (Figure \ref{fig:cohlaw}c). Each landing is a square-root bifurcation: the pair that lands at \(t=57.69\) for \(\varepsilon_c=0.6197\) satisfies \(\beta-\frac12\propto(\varepsilon_c-\varepsilon)^{0.495}\) over three decades (Figure \ref{fig:cohlaw}d), the same collision law as in the heat flow (Remark 6.8), proved in Proposition 6.18. The two pure functions have their zeros on the line, every superposition has zeros off it, and these come either from infinity, at distance of order \(\log(1/\varepsilon)\), or from collisions on the line. In this language RH says that \(\zeta\) contains no second component, however small. Status: the invariance and the formula for \(C(\sigma)\) (Proposition 6.19), the location of the distant zeros (Proposition 6.17) and the collision law (Proposition 6.18) are proved; the rest is numerical. \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Coherence_entropy_Euler_state.png} \caption{Coherence of the Euler state (Remark 6.16). (a) Relative entropy of coherence $C(\sigma)$ of $\psi_s$. (b) $|\langle a|e^{-itL}|\psi_\sigma\rangle|^2$ for the alternating state, $N=4096$; dotted: zeros. (c) Zeros of $\sum e^{2\pi ina/q}n^{-s}$ with $t<40$, $\beta\ge\frac12$ shown, except the zeros of the local factors on $\sigma=1$. (d) Number of primitive $L$-functions in the additive character against $\max|\beta-\frac12|$.}\label{fig:cohEuler}\end{figure} \begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{Coherence_law.png} \caption{Superposition of $\xi$ and $\Lambda(\cdot,\chi_5)$ (Remark 6.16). (a) Zeros of $G_\varepsilon$ with $\beta\ge\frac12$, $t<100$, for five values of $\varepsilon$. (b) Median $\beta$ of the distant zeros against $2\log(1/\varepsilon)/\log5$. (c) Number of zeros on the line and in $\frac12<\beta<1$. (d) Landing of one pair: $\beta-\frac12$ against $\varepsilon_c-\varepsilon$.}\label{fig:cohlaw}\end{figure} \textbf{Proposition 6.17} (distant zeros of a superposition). Let \(\chi_5\) be the real character modulo \(5\), \(G_\varepsilon=\xi+\varepsilon\Lambda(\cdot,\chi_5)\) as in Remark 6.16 with \(0<\varepsilon\le5^{-5/2}\), and put \[\sigma_\varepsilon=\frac{2\log(1/\varepsilon)}{\log5}\ \ (\ge5),\qquad s_k=\sigma_\varepsilon+\frac{2\pi i(2k+1)}{\log5},\] \[R_k=\Big\{|\sigma-\sigma_\varepsilon|\le1,\ |t-\operatorname{Im}s_k|\le\frac{2\pi}{\log5}\Big\}\quad(k\in\mathbb Z).\] Then: (i) \(G_\varepsilon\) has no zeros with \(\sigma\ge\frac52\) outside \(\bigcup_kR_k\); (ii) each \(R_k\) contains exactly one zero \(\rho_k\) of \(G_\varepsilon\), and it is simple; (iii) with \(\delta=\zeta(\sigma_\varepsilon-1)^2-1\), \[\rho_k=s_k-\frac2{\log5}\operatorname{Log}A(\rho_k),\qquad|\rho_k-s_k|\le\frac2{\log5}\,\frac{\delta}{1-\delta},\qquad\rho_k=s_k+\frac4{\log5}\big(2^{-s_k}+3^{-s_k}\big)+O(4^{-\sigma_\varepsilon}),\] where \(A(s)=L(s,\chi_5)/\zeta(s)\). By the functional equation and \(G_\varepsilon(\bar s)=\overline{G_\varepsilon(s)}\), the points \(1-\rho_k\) are zeros as well. \emph{Proof.} The factors \(\frac12s(s-1)\Gamma(\frac s2)\) are common to \(\xi\) and \(\Lambda(\cdot,\chi_5)\), so \(G_\varepsilon=\xi\,H\) with \(H(s)=1+\varepsilon5^{s/2}A(s)\), and \(\xi\ne0\) for \(\sigma>1\). The Dirichlet series \(A(s)=\sum a(n)n^{-s}\) has \(a=\chi_5*\mu\), so \(a(1)=1\), \(a(2)=a(3)=-2\) and \(|a(n)|\le2^{\omega(n)}\le d(n)\); hence \(|A(s)|\le\zeta(\sigma)^2\) and \(|A(s)-1|\le\zeta(\sigma)^2-1\) for \(\sigma>1\). Note \(|\varepsilon5^{s/2}|=5^{(\sigma-\sigma_\varepsilon)/2}\). \begin{enumerate} \def\labelenumi{(\roman{enumi})} \item If \(\frac52\le\sigma\le\sigma_\varepsilon-1\), then \(|\varepsilon5^{s/2}A(s)|\le5^{-1/2}\zeta(\frac52)^2=0.804\), so \(|H|\ge0.196\). If \(\sigma\ge\sigma_\varepsilon+1\ge6\), then \(|\varepsilon5^{s/2}A(s)|\ge5^{1/2}(2-\zeta(6)^2)>2\), so \(|H|>1\). \item Let \(H_0(s)=1+\varepsilon5^{s/2}\), whose zeros are the points \(s_k\), all simple, one in each \(R_k\). On the vertical sides of \(R_k\), \(|\varepsilon5^{s/2}|=5^{\mp1/2}\) and \(|H_0|\ge1-5^{-1/2}=0.553\). On the horizontal sides, \(\frac t2\log5\equiv0\pmod{2\pi}\), so \(\varepsilon5^{s/2}>0\) and \(H_0\ge1\). On \(R_k\), \(|H-H_0|=|\varepsilon5^{s/2}||A-1|\le5^{1/2}\delta\le5^{1/2}(\zeta(4)^2-1)=0.383<0.553\). By Rouché's theorem \(H\) and \(H_0\) have the same number of zeros in \(R_k\), namely one. \item At \(\rho_k\) we have \(\varepsilon5^{\rho_k/2}A(\rho_k)=-1=\varepsilon5^{s_k/2}\), so \(5^{(\rho_k-s_k)/2}A(\rho_k)=1\). Since \(|\operatorname{Im}(\rho_k-s_k)|\le2\pi/\log5\) and \(|A(\rho_k)-1|\le\delta<1\), the principal branch gives the first identity, and \(|\operatorname{Log}(1+w)|\le|w|/(1-|w|)\) gives the bound. Finally \(\operatorname{Log}A(\rho)=-2\cdot2^{-\rho}-2\cdot3^{-\rho}+O(4^{-\beta})\), and \(2^{-\rho_k}-2^{-s_k}=O(2^{-\sigma_\varepsilon}|\rho_k-s_k|)=O(4^{-\sigma_\varepsilon})\). \(\square\) \end{enumerate} The proof applies to any two primitive Dirichlet characters of the same parity and conductors \(q_10\) they lie on the line, at \(t=t_c\pm\sqrt{2(\varepsilon-\varepsilon_c)/r''(t_c)}+O(|\varepsilon-\varepsilon_c|)\); if \((\varepsilon-\varepsilon_c)/r''(t_c)<0\) they are \[\rho_\pm=\tfrac12\pm\sqrt{\frac{2|\varepsilon-\varepsilon_c|}{|r''(t_c)|}}+it_c+O(|\varepsilon-\varepsilon_c|).\] \item Let \(K\) be a compact subset of the strip and \(I\) an interval such that no \(G_\varepsilon\), \(\varepsilon\in I\), vanishes on \(\partial K\). Then the number of zeros of \(G_\varepsilon\) in \(K\) off the line is constant on every subinterval of \(I\) that contains no critical value \(r(t_c)\) with \(\frac12+it_c\in K\), and it changes by \(2\) at each nondegenerate one. \end{enumerate} \emph{Proof.} Near the line, \(G_\varepsilon(\frac12+iz)=X(z)+\varepsilon Y(z)=Y(z)(\varepsilon-r(z))\), with \(X\), \(Y\) entire and real on the real axis, which gives (i). The zeros of \(G_\varepsilon\) are symmetric under \(s\mapsto1-\bar s\) by the functional equation and \(G_\varepsilon(\bar s)=\overline{G_\varepsilon(s)}\); in the variable \(z\) this is \(z\mapsto\bar z\). In (ii), \(r-\varepsilon\) has a single zero near \(t_0\) by the implicit function theorem; since the zero set is invariant under conjugation, that zero is real. In (iii), write \(r(t_c+w)-\varepsilon_c=\frac12r''(t_c)w^2u(w)\) with \(u\) analytic and \(u(0)=1\), and \(\phi(w)=w\sqrt{u(w)}\), which is analytic and invertible near \(0\) with \(\phi'(0)=1\). Then \(r(t_c+w)=\varepsilon\) if and only if \(\phi(w)=\pm c\) with \(c^2=2(\varepsilon-\varepsilon_c)/r''(t_c)\), so \(w=\pm c+O(|c|^2)\). If \(c\) is real the two zeros are real, that is, on the line. If \(c=i|c|\) they are \(t_c\pm i|c|+O(|c|^2)\), and \(s=\frac12+iz\) gives \(\rho_\pm\). (iv) By the argument principle the number of zeros in \(K\) is constant on \(I\). The number on the line changes only where a zero on the line ceases to be simple, which by (ii) and (iii) happens only at critical values, where it changes by \(2\). \(\square\) The proposition reduces the zeros on the line to the graph of the single real function \(r(t)=-\xi(\frac12+it)/\Lambda(\frac12+it,\chi_5)\): a horizontal line at height \(\varepsilon\) meets it at the zeros on the line, a local maximum of \(r\) above \(0\) is a value of \(\varepsilon\) at which a pair leaves the line, and a local minimum is a value at which a pair lands (Figure \ref{fig:levels}a). For \(0\frac12\) (respectively \(N<\infty\) and any \(\sigma\)) let \(\psi_s=\zeta(2\sigma)^{-1/2}\sum_nn^{-s}|n\rangle\) (respectively \(\psi_{s,N}=P_N\psi\) normalized). \begin{enumerate} \def\labelenumi{(\roman{enumi})} \item \(\sum_nn^{-s}|n\rangle\in\ell^2(\mathbb N)\) if and only if \(\sigma>\frac12\), and then \(\psi_s=e^{-itL}\psi_\sigma\); likewise \(\psi_{s,N}=e^{-itL}\psi_{\sigma,N}\). \item Let \(\ell^2(\mathbb N)=\bigotimes'_p\ell^2(\mathbb N_0)\) by \(|n\rangle=\bigotimes_p|v_p(n)\rangle\) (restricted product with respect to \(|0\rangle\)). Then \(e^{-itL}=\bigotimes_pe^{-itL_p}\) with \(L_p|k\rangle=k\log p\,|k\rangle\). Consequently every function of a state that is invariant under diagonal unitaries in the basis \(|n\rangle\) (populations, every coherence measure of the resource theory of coherence in this basis, the entropy of the populations) and every function invariant under local unitaries for a bipartition of the primes (entanglement entropies, Schmidt coefficients) takes the same value on \(\psi_s\) and \(\psi_\sigma\), and on \(\psi_{s,N}\) and \(\psi_{\sigma,N}\), for every \(t\). \item The relative entropy of coherence of \(\psi_s\) is the entropy of the populations \(n^{-2\sigma}/\zeta(2\sigma)\): \[C(\sigma)=\log\zeta(2\sigma)-2\sigma\frac{\zeta'}{\zeta}(2\sigma)=\frac1{2\sigma-1}-\log(2\sigma-1)+1-\gamma_0+O(2\sigma-1)\qquad(\sigma\to\tfrac12^+),\] and for the truncation on the line \(C_N(\frac12)=\frac12\log N+\log\log N-\frac{\gamma_0}2+O(1/\log N)\). \item For \(\sigma>0\) and the alternating vector \(a_N=N^{-1/2}\sum_{n\le N}(-1)^{n+1}|n\rangle\), \[\sqrt{N\textstyle\sum_{n\le N}n^{-2\sigma}}\ \langle a_N|e^{-itL}|\psi_{\sigma,N}\rangle=\sum_{n\le N}(-1)^{n+1}n^{-s}=(1-2^{1-s})\zeta(s)+O_s(N^{-\sigma}),\] uniformly for \(s\) in compact subsets of \(\sigma>0\). In particular the zeros of \(\zeta\) in \(0<\sigma<1\) are the points at which the rescaled overlap tends to \(0\). \end{enumerate} \emph{Proof.} (i) \(\|\sum n^{-s}|n\rangle\|^2=\sum n^{-2\sigma}\), and \(n^{-s}=e^{-it\log n}n^{-\sigma}\); \(P_N\) is diagonal and commutes with \(L\). (ii) By unique factorization \(\log n=\sum_pv_p(n)\log p\), so \(e^{-it\log n}=\prod_pe^{-itv_p(n)\log p}\), which is the stated product; for a bipartition \(S\cup S^c\) of the primes it is \(U_S\otimes U_{S^c}\) with diagonal factors, and since \(P_N\) commutes with \(e^{-itL}\), also \(\psi_{s,N}=(U_S\otimes U_{S^c})\psi_{\sigma,N}\), although \(\psi_{\sigma,N}\) is not a product state. (iii) With \(p_n=n^{-2\sigma}/\zeta(2\sigma)\), \(-\sum p_n\log p_n=\log\zeta(2\sigma)+2\sigma\sum p_n\log n\) and \(\sum n^{-b}\log n=-\zeta'(b)\). The expansion follows from \(\zeta(b)=\frac1{b-1}+\gamma_0+O(b-1)\) and \(\frac{\zeta'}{\zeta}(b)=-\frac1{b-1}+\gamma_0+O(b-1)\) with \(b=2\sigma\). For the truncation, \(\sum_{n\le N}n^{-1}=\log N+\gamma_0+O(N^{-1})\) and \(\sum_{n\le N}n^{-1}\log n=\frac12\log^2N+O(1)\). (iv) The left side equals \(\sum_{n\le N}(-1)^{n+1}n^{-s}\) by definition. Grouping consecutive terms, \(|(2k-1)^{-s}-(2k)^{-s}|\le|s|(2k-1)^{-\sigma-1}\), so the tail beyond \(N\) is \(O(|s|N^{-\sigma}/\sigma)+O(N^{-\sigma})\), and the full series is \((1-2^{1-s})\zeta(s)\) for \(\sigma>0\). \(\square\) Numerically, \(C(\sigma)\) computed from \(\zeta\) differs from the expansion by \(0.039\), \(0.019\), \(0.0038\), \(0.0004\) at \(\sigma=0.6\), \(0.55\), \(0.51\), \(0.501\), and \(C_{4096}(\frac12)=6.066\) against \(5.988\) from the first three terms. Part (ii) says that the arithmetic of the zeros cannot be read from any intrinsic property of the Euler state; only overlaps with vectors that are not diagonal in the multiplicative structure, such as \(a_N\), depend on \(t\). \textbf{Proposition 6.20} (convolution identities do not see \(s\)). Let \(f,g\) be arithmetic functions, \(h=f*g\), \(G_M(s)=\sum_{m\le M}g(m)m^{-s}\) and \(H_N(s)=\sum_{n\le N}h(n)n^{-s}\). Then for every \(N\) and every \(s\in\mathbb C\) \[\sum_{d\le N}f(d)\,d^{-s}\,G_{\lfloor N/d\rfloor}(s)=H_N(s).\] More generally the identity remains true when \(n^{-s}\) is replaced by \(x^{v(n)}=\prod_px_p^{v_p(n)}\) for an arbitrary sequence \((x_p)\) of complex numbers. \emph{Proof.} The left side is \(\sum_{dm\le N}f(d)g(m)(dm)^{-s}\), and grouping the terms with \(dm=n\) gives \(H_N(s)\). Only \((dm)^{-s}=d^{-s}m^{-s}\) was used, which also holds for \(x^{v(dm)}=x^{v(d)}x^{v(m)}\). \(\square\) With \(f=\mu\), \(g=1\) this gives \(\sum_{d\le N}\mu(d)d^{-s}X_{\lfloor N/d\rfloor}(s)=1\); with \(f=\Lambda\), \(g=1\) it gives \(-X'_N(s)=\sum_{d\le N}\Lambda(d)d^{-s}X_{\lfloor N/d\rfloor}(s)\); and for \(\sqrt N0\) \\ helix crossings to \(T=400\) (\S{}2) & \(202\) of \(202\) & \(287\) of \(303\) \\ spiral of a zero (Prop. 2.3) & rate \(n^{-1/2}\) & rates \(n^{-0.81}\), \(n^{-0.19}\) \\ zeros on prime circles (\S{}2) & bias toward \(\pi\) & spikes of both signs \\ prime-signal energy (Thm. B) & \(\approx C_B\) up to \(10^8\) & grows, rate \(\approx2\beta-1\) \\ local energy (Thm. 3.4) & inside \(C_B\pm B(H)\) & grows from \(0.22\) to \(2.06\) \\ energy slope (Thms. C, D) & \(\min_tS\approx0.39(\sigma-\frac12)\) & one failure interval per zero \\ \(W_L\) (Thm. E) & inside the RH band & negative outside the disc \\ Weil matrices (\S{}5) & margin \(10^{-6.3L^2}\) & negative for \(L>3.6\) \\ Euler product (\S{}6) & converges right of zeros & converges right of zeros \\ Euler helix, pairs \(\frac12\pm\delta\) (Rem. 6.6) & \(\delta_{\min}\approx4\cdot10^{-5}\) at \(x=10^8\) & --- \\ heat flow \(H_t\) (Rem. 6.8) & first collision at \(t_c=-0.47\) (height \(49.7\)) & --- \\ \end{longtable} The row of the Euler product is the only one in which \(f\) behaves like \(\zeta\): convergence of a logarithmic Dirichlet series is controlled by the zeros to its right, whether or not there is an Euler product. In every other row the difference comes from the coefficients. Those of \(\log\zeta\) are \(\Lambda(n)/\log n\ge0\) and live on prime powers; those of \(\log f\) change sign and live mostly on composite integers. Landau's formula (2.3) displays this difference most directly: summed over the zeros, \(x^\rho\) returns \(-\frac T{2\pi}\Lambda(x)\), a one-signed function supported on prime powers. The results also show where the Euler product is used and where it is not. \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item \emph{In linear form the pole cancels positivity.} Any inequality \(\sum a_j\cos j\theta\ge0\) has \(a_0>0\), so its use in the strip pays a correction of size \(x^{1-\sigma}\) from the point \(t=0\) (Section 6). More generally, no combination of cosines without a constant term is nonnegative at all \(\log p\), because the angles \(t\log p\) are uniformly distributed (Proposition 6.12). This is why the classical method gives zero-free regions only near \(\sigma=1\), and twisting by characters does not change it (Proposition 6.4). \item \emph{In quadratic form the primes create positivity.} The pole enters Weil's form as a rank-two term, and the prime terms move the zeros of the minimizer onto the zeros of \(\zeta\) (Section 5). The margin, however, decreases like \(10^{-6.3L^2}\), and positivity on all windows is equivalent to RH. The size of the margin is fixed by the density of the zeros alone: points with the same density and no arithmetic content give the same values, and for curves over finite fields, whose zeros have constant density, the margin is the constant \(2\log p\) (Proposition 5.2). In a basis of Dirichlet polynomials the collapse of the margin is a counting effect, the same for the Davenport--Heilbronn function. \item \emph{In energy form positivity is automatic and finiteness is the question.} Averaged over \(t\), the square of the cross-energy keeps only its diagonal; this quadratic positivity without a pole gives the theorem of Bohr and Landau that off-line zeros have density zero (Proposition 4.5) and, with higher moments, the density theorems, but an average cannot exclude a single zero. With a mollifier the bound improves until it meets the gap in Jensen's inequality, the variance of \(\log|\zeta M_y|\), which mollifiers of length at most \(T\) cannot reduce below about \(T^{1-2\sigma}\) (Remark 4.6). Theorem B expresses RH as the finiteness of \(\sum_\rho\lim_nC_2(n,\rho)\), that is, of the energy of the prime signal. \item \emph{Pointwise, the obstruction is local.} By Theorems C and D, a failure of the energy slope at abscissa \(\sigma\) occurs only inside the discs of off-line zeros, and the density theorems, which use the Euler product through mean values of Dirichlet polynomials, make the failure set sparse. For functions of Davenport--Heilbronn type it has positive proportion \cite{KK}. A proof along these lines would require a version of these mean-value arguments valid inside a single disc. By Theorem E this is Weil positivity along the Poisson family, and the truncated functionals \(W_L\) show where the difficulty lies: they are exact and computable for each \(L\), but the limit \(L\to\infty\) at fixed \(\sigma<1\) is equivalent to controlling all zeros with \(\beta>\sigma\). \item \emph{From finite prime data the obstruction is uniformity, not resolution.} The Euler helix fixes the real parts of the zeros near a given height to within \(4\cdot10^{-5}\) with the primes up to \(10^8\), although symmetric off-line pairs enter only at second order (Remark 6.6). Combined with the functional equation this can give exact local statements, as in Turing's method, but it requires a list of the zeros and a number of primes that grows with the height. \item \emph{Amplification needs an upper bound for the primes.} Over \(\mathbb Q\) the amplification that proves the analogue of RH for curves becomes a one-sided bound \(\psi(x)\le x+O(x^{1/2}\log^2x)\). Sieves give upper bounds by positivity, but they stop at the parity barrier, which we measure as a factor close to \(2\) in Selberg's sieve on short intervals; removing it needs bilinear information of the kind used by Matomäki and Radziwiłł, which the cross-energy does not provide (Remark 6.7). \item \emph{The Laguerre hierarchy is the finiteness of the helix energy.} The properties that force real zeros are the Laguerre inequalities \(L_n\ge0\) for all \(n\). On the heat-flow deformation, \(L_1\) and the energy slope change sign exactly at the first collision, while low-degree Jensen polynomials miss it, and wide pairs need higher \(n\). On the helix side, \(L_1\) and \(L_2\) are the mean and half the variance of \(|H_s|^2\), and all \(L_n\) are moments of a sum of circles: nonnegative when they exist, infinite when a zero leaves the line (Remark 6.8). The crossing law measures the spectral variable \(\lambda=s(1-s)\): \(n^*=\lambda+\frac1{12}+O(\lambda^{-1})\) (Section 2). \item \emph{What the energy slope proves.} The slope gives the zero-free region of de la Vallée Poussin with the constants of the elementary method (Proposition 4.4), and the cross-energy evaluated at its own crossing reduces to \(Z(t)\cos\varphi(t)\) on the line, with the zeros of \(\zeta\) as dislocations of its nodal pattern in the strip (Section 2). Both are proved; neither reaches inside the strip beyond the classical results. \item \emph{Holomorphic sums are blind, energies are expensive.} Sums over the zeros that are holomorphic in \(\rho\) come linearly from the primes and do not see \(\beta\); non-holomorphic ones, such as \(\sum|\rho|^{-2}\) or the crossing points, see \(\beta\) and are reached from the primes only through energies. The damped energy has its first singularity at \(2\sup\beta-1\), but detects an off-line zero at height \(T\) only with the primes up to about \(T^{1/\delta}\) (Remark 6.9). \item \emph{The wave and the helix are one object.} Sieving the partial sums by the primes up to \(y\) multiplies the free amplitude of the wave \(W\) by the partial Euler product; after the pole correction is removed, what remains is \(\exp(-H_s(\log y)/\log y)\), the Euler helix (Remark 6.10). The two halves of the paper, partial sums and partial products, meet there. \item \emph{The primes act on all waves by the same real operator, and the reflection pairs convergence with divergence.} On the critical line each prime acts on the wave by \(1-p^{-1}T_p\), independently of the height, with spectral form \(1/\zeta(2\sigma+\lambda)\); the Davenport--Heilbronn function has no such operator. The functional equation, sieved, keeps modulus \(1\) on the line by symmetry, but maps the half of the strip where the Euler product converges onto the half where it always diverges (Remark 6.11). This is where the two structures that RH requires meet, and where finite prime data stop. \item \emph{A single combination of the three structures.} The functional equation, the logarithm of the \(C_2\) energy and the derivative combine into one criterion: RH holds if and only if \(\log|\xi(\frac12+\sqrt v+it)|\) is concave in \(v=(\sigma-\frac12)^2\) for every \(t\) (Proposition 4.3). Its slope at the line is the energy of the Euler helix, the Euler product makes it positive for \(\sigma>1\), and one off-line pair breaks the concavity only near its own height, as the Davenport--Heilbronn function shows. \end{enumerate} We list the questions that the computations suggest and that we cannot answer. \begin{itemize} \tightlist \item Is there a positive quantity built from the partial Euler products that does not pay the pole correction of item 1, for instance an average over twists in which the \(t=0\) term cancels? \item Can the exceptional set of Theorem D be bounded inside one disc, using only the information in \(\sum_{n\le e^L}\Lambda(n)n^{-s}\) for \(L\) of size \(C/(\sigma-\frac12)\)? \item The one-signed spectrum of (2.3) is a property of the exact positions of the zeros: displacements of a small fraction of the mean spacing destroy it (Section 2). Is there a statement about \(\{\gamma\log p\bmod2\pi\}\) that is equivalent to the Euler product and implies a constraint on \(\beta\)? \end{itemize} \textbf{Two known structures and a missing one.} The results can be summarized by the role of three ingredients. The functional equation alone does not force the zeros onto the line: the Davenport--Heilbronn function has one of the same type and off-line zeros, and every criterion of this paper separates it from \(\zeta\) only through its coefficients. The Euler product alone does not force them either: it controls the half of the strip to the right of the zeros and diverges on the other half, and products without a functional equation need not even continue into the strip. Together the two are expected to imply RH for every function of the Selberg class, but they do not meet directly: the functional equation maps the half where the partial Euler products converge onto the half where they diverge (Remark 6.11). What is missing is a third ingredient, a positivity that holds for structural reasons. Over function fields it is supplied by the geometry of the curve, through intersection theory and the cohomology used by Weil and Deligne; for an elliptic curve it is the positivity of the degree \(\deg(m-nF)=|m-n\alpha|^2\), which the wave of Section 2 reproduces in every other respect but does not see (Remark 6.13), and for graphs and for Dirichlet \(L\)-functions over \(\mathbb F_q[T]\) it is the positivity of a Toeplitz matrix of Weil type coming from a unitary or self-adjoint operator (Remark 6.14). Read as quantum states, the Euler product is a product state whose coherence and entropy do not depend on \(t\), its zeros are orthogonality times of the evolution \(e^{-it\log n}\), and zeros leave the line exactly when characters are superposed (Remarks 6.15, 6.16); over \(\mathbb Q\) it has been sought as a self-adjoint operator (Hilbert--Pólya), as the positivity of Weil's functional, or as the real-rootedness of \(\Psi\) (Pólya). In the language of this paper it appears as the finiteness of the energy of the prime signal and of the Euler helix (Theorem B, Remark 6.5), as the Laguerre inequalities (Remark 6.8), as the energy slope (Theorem 4.1), and as positivity of the Weil matrices (Section 5). Each of these is equivalent to RH, and none is supplied by the partial sums, the partial products or the explicit formula. Every construction made here from the cross-energy either reduces to one of them or detects zeros without constraining them: the crossing law measures the spectral variable, the wave \(W\) counts zeros wherever it is applied, the damped energy sees any off-line zero at a cost of primes up to about \(T^{1/\delta}\), and sieving by primes stops at the parity barrier. The contribution of the paper is a description of where the missing positivity would have to act, from these several directions, with numerical verification and with the Davenport--Heilbronn function as a control throughout. We do not prove the Riemann hypothesis. The material of Parts I--V that is not included here (the Fresnel transition law, the monotonicity windows, the Jensen formula for the \(C_2\) wave and the anatomy of individual zeros) is available in \cite{C1}--\cite{C5}. \section{Numerical methods and data} All computations use double precision in Python (NumPy, SciPy). The zeros of \(\zeta\) with \(0<\gamma<600\) (341 ordinates) were computed once and are reused throughout. \(\zeta\) is evaluated by Euler--Maclaurin summation, and the Davenport--Heilbronn function \(f\) through Hurwitz zeta functions as in \cite{C5}; its eight off-line zeros with \(\gamma<400\) are those of \cite{C5}. \(\Lambda(n)\) is computed by sieve up to \(10^8\), and the coefficients \(b(n)\) of \(-f'/f\) by Dirichlet convolution up to \(2\cdot10^7\). Every value in the tables was obtained by two independent routes where one was available, from the primes (or coefficients) and from the zeros, and the agreement is stated in the text. \begin{itemize} \tightlist \item \emph{Section 2.} \(Z_n\) for \(n\le2\cdot10^4\) against (2.1); the detector \(\max_n|D_n(t)|\) on a grid of step \(0.002\); the sums \(\sum_{0<\gamma<400}\cos(\gamma\log n)\), their analogues for \(f\), and the displacement and GUE controls (directories \texttt{spiral/}, \texttt{landau/}). \item \emph{Section 3.} \(E(u)\) from the prime powers up to \(10^8\); autocorrelations; \(\mathcal J(\delta)\) by quadrature of \(|\zeta'/\zeta+s/(s-1)|^2|s|^{-2}\) on \(|t|\le T\) with a tail estimate; the window energies of \(E_f\) up to \(x=2\cdot10^7\) (\texttt{signal/}, \texttt{c2law/}). \item \emph{Section 4.} \(S(\sigma,t)\) by central differences with step \(10^{-5}\) applied to \(\log|\xi|\) and to the completed \(f\), on grids of step \(0.01\) in \(t\) and \(0.005\) in \(\sigma\); GUE spectra of size \(400\), bulk half, unfolded by the semicircle law (\texttt{slope/}). \item \emph{Section 5.} \(W_L\) from (5.1) with prime powers or coefficients up to \(e^{14}\) and the archimedean integral over \(|r|\le3000\) with an analytic tail; the Weil matrices in the sine basis with numerical integration over \(|r|\le800\) (step \(0.004\)), and in the cubic B-spline basis of mesh \(0.03\) (Toeplitz, archimedean term by Gauss--Legendre quadrature); eigenvalues by \texttt{numpy.linalg.eigh}, and on the zero side by singular values, which resolve values below the floor of the explicit side (\texttt{weil/}). \item \emph{Sections 2, 4, 5 (twins and Dirichlet polynomials).} Zeros of \(L(s,\chi)\), \(\chi\) mod \(5\), from sign changes of \(W^{-1/2}\Lambda(\frac12+it,\chi)\) on a grid of step \(0.01\), refined by Brent's method; the matrices \(A_{mn}\) from (5.1) with prime powers up to \(4000\) and from the zeros (\texttt{twin/}, \texttt{joint/}). \item \emph{Section 6.} Partial products over \(p\le2\cdot10^6\); \(E_1\) from \texttt{scipy.special.exp1}; mode sums over the \(682\) zeros with \(|\gamma|<600\) (\texttt{euler/}). \item \emph{Remarks 6.6--6.8 and the spectral variable (Section 2).} Euler helix with exact bin averages and Gaussian smoothing (\texttt{rigidity/}); Selberg's sieve, Liouville's function and the explicit formula on \((10^8,10^8+y]\) (\texttt{parity/}); moments \(b_k\), the series \(\Psi\), Jensen polynomials, the heat flow \(H_t\) and the Laguerre expressions in Python's \texttt{decimal} arithmetic with \(60\) to \(500\) digits (\texttt{spectral/}, \texttt{heat/}). \end{itemize} \textbf{Data availability.} The scripts are included in the ancillary directory \texttt{anc/} of the source package. The preprints \cite{C1}--\cite{C8} on which this paper is based, with their own scripts and data, are deposited on Zenodo under the DOIs listed in the references. \appendix \section{Compendium of formulas} This appendix collects the formulas used in the paper and some related ones, grouped by structure. Each group states its status: an exact identity, an asymptotic result, a criterion equivalent to RH, or a statement that would imply RH and is not proved. Throughout, \(s=\sigma+it\), \(\rho=\beta+i\gamma\) is a nontrivial zero, \(\rho^*=1-\bar\rho\) is its reflection, and \(X_N(s)=\sum_{n\le N}n^{-s}\). \subsection{Partial sums} \emph{Status: exact identities and classical asymptotics.} \[X_N(s)-X_{N-1}(s)=N^{-s},\qquad -X_N'(s)=\sum_{n\le N}\frac{\log n}{n^s}.\qquad\text{(A.1)}\] For fixed \(s\ne1\) with \(0<\sigma<1\), Euler--Maclaurin summation gives \[X_N(s)=\frac{N^{1-s}}{1-s}+\zeta(s)+\frac12N^{-s}+O_s(N^{-\sigma-1}),\qquad\text{(A.2)}\] and at a zero \[X_N(\rho)=\frac{N^{1-\rho}}{1-\rho}+\frac12N^{-\rho}+O_\rho(N^{-\beta-1}),\] \[-X_N'(\rho)=N^{1-\rho}\Big[\frac{\log N}{1-\rho}-\frac1{(1-\rho)^2}\Big]-\zeta'(\rho)+O_\rho(N^{-\beta}\log N).\qquad\text{(A.3)}\] \textbf{The \(X-Y\) decomposition.} Put \(X^C_N(s)=X_N(s)-\frac12N^{-s}\) and \(Y^C_N(s)=N^{1-s}/(1-s)\). By (A.2), for \(0<\sigma<1\), \[\zeta(s)=\lim_{N\to\infty}\big[X^C_N(s)-Y^C_N(s)\big],\qquad D_N(s):=X^C_N(s)-Y^C_N(s)=\zeta(s)+O_s(N^{-\sigma-1}).\qquad\text{(A.4)}\] The two components diverge separately for \(\sigma<1\); only their difference converges. The continuous component has \[|Y^C_N(s)|^2=\frac{N^{2-2\sigma}}{(1-\sigma)^2+t^2},\qquad \frac{d}{dN}|Y^C_N(s)|^2=\frac{2(1-\sigma)N^{1-2\sigma}}{(1-\sigma)^2+t^2},\qquad\text{(A.5)}\] which is linear in \(N\) if and only if \(\sigma=\frac12\), with slope \(1/(t^2+\frac14)\). At a zero, \(D_N(\rho)=O_\rho(N^{-\beta-1})\), hence \[|X^C_N(\rho)|^2-|Y^C_N(\rho)|^2=O_\rho(N^{-2\beta}),\qquad \frac{|X^C_N(\rho)|^2}{|Y^C_N(\rho)|^2}=1+O_\rho(N^{-2}).\qquad\text{(A.6)}\] The two energies agree asymptotically at every zero, on or off the critical line; their comparison therefore does not locate the zeros. \subsection{The cross-energy} \emph{Status: exact identities; asymptotics as in Theorem A and Corollary 2.1.} \[C_2(N,s)=|X_N(s)|^2-|X_{N-1}(s)|^2-N^{-2\sigma}=2\operatorname{Re}\big[N^{-\bar s}X_{N-1}(s)\big],\] \[C_2(N,s)=2N^{-\sigma}\sum_{n\frac12,\end{cases}\qquad \lim_{N\to\infty}\frac{\log C_2(N,\rho)}{\log N}=1-2\beta,\qquad\text{(A.10)}\] the second for \(0<\beta<1\), and \[\lim_{N\to\infty}N^{2\beta-1}C_2(N,\rho)=A_\rho=N^{2\beta-1}\frac{d}{dN}|Y^C_N(\rho)|^2,\qquad\text{(A.11)}\] so that at a zero \(C_2\) is asymptotically the growth rate of the continuous energy (A.5). The leading terms of (A.9) on the critical line vanish at \(N\approx\gamma^2+\frac14\); the crossing point of Proposition 2.2 is \(n^*=\gamma^2+\frac13+\frac1{180\gamma^2}+O(\gamma^{-4})\). \textbf{Reflected zeros.} \(A_{\rho^*}=2\beta/(\beta^2+\gamma^2)\) and \(C_2(N,\rho^*)=A_{\rho^*}N^{2\beta-1}-N^{2\beta-2}+O(N^{2\beta-3})\). The product and ratio satisfy \[C_2(N,\rho)C_2(N,\rho^*)=A_\rho A_{\rho^*}-\frac{A_\rho+A_{\rho^*}}N+O_\rho(N^{-2})\ \longrightarrow\ \frac{4\beta(1-\beta)}{|\rho|^2|1-\rho|^2},\qquad\text{(A.12)}\] \[\frac{C_2(N,\rho)}{C_2(N,\rho^*)}\sim\frac{A_\rho}{A_{\rho^*}}N^{2-4\beta},\qquad \lim_{N\to\infty}\Big[\frac{\log\big(C_2(N,\rho)/C_2(N,\rho^*)\big)}{\log N}\Big]^2=4(1-2\beta)^2.\qquad\text{(A.13)}\] On the critical line the limit in (A.12) is \((\gamma^2+\frac14)^{-2}\) and the limit in (A.13) is \(0\). \emph{Criterion (Theorem A).} RH holds if and only if \(0<\lim_NC_2(N,\rho)<\infty\) for every \(\rho\). \emph{Sufficient condition, not proved:} \(C_2(N,\rho)=O_{\rho,\varepsilon}(N^\varepsilon)\) for every \(\rho\) and \(\varepsilon>0\); applied to \(\rho\) and \(\rho^*\) it forces \(\beta=\frac12\). \subsection{Functional equation and derivatives} \emph{Status: exact identities; they hold equally for hypothetical off-line zeros.} With \(h(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\) and \(\xi(s)=h(s)\zeta(s)\), \[\xi(s)=\xi(1-s),\qquad \xi^{(k)}(s)=(-1)^k\xi^{(k)}(1-s),\qquad \xi'(\rho^*)=-\overline{\xi'(\rho)},\qquad \xi''(\rho^*)=\overline{\xi''(\rho)}.\qquad\text{(A.14)}\] At a simple zero, \(\xi'(\rho)=h(\rho)\zeta'(\rho)\), \(\xi''(\rho)=2h'(\rho)\zeta'(\rho)+h(\rho)\zeta''(\rho)\), and \[\frac{\xi''(\rho)}{\xi'(\rho)}=2\frac{h'(\rho)}{h(\rho)}+\frac{\zeta''(\rho)}{\zeta'(\rho)},\qquad \frac{\xi''(\rho^*)}{\xi'(\rho^*)}=-\overline{\Big(\frac{\xi''(\rho)}{\xi'(\rho)}\Big)}.\qquad\text{(A.15)}\] \subsection{Primes in the partial sums} \emph{Status: exact identities, except (A.19), which is asymptotic.} \[\log n=\sum_{d\mid n}\Lambda(d),\qquad -\frac{\zeta'}{\zeta}(s)=\sum_{n\ge1}\frac{\Lambda(n)}{n^s}\ (\sigma>1),\qquad\text{(A.16)}\] \[-X_N'(s)=\sum_{d\le N}\frac{\Lambda(d)}{d^s}X_{\lfloor N/d\rfloor}(s),\qquad\text{(A.17)}\] \[\frac{\partial C_2(N,s)}{\partial\sigma}=-(\log N)\,C_2(N,s)-2\operatorname{Re}\Big[N^{-\bar s}\sum_{d1\) not a prime power, \[\psi(x)=x-\sum_\rho\frac{x^\rho}{\rho}-\log2\pi-\frac12\log(1-x^{-2}),\qquad -\frac{x^\rho}\rho-\frac{x^{\bar\rho}}{\bar\rho}=-\frac{2x^\beta}{|\rho|}\cos(\gamma\log x-\arg\rho),\qquad\text{(A.20)}\] the sum over zeros taken in symmetric order. In the variable \(u=\log x\) the normalized error of the paper is \(E(u)=e^{-u/2}(\psi(e^u)-e^u)\), and a pair of zeros contributes an oscillation of amplitude \(2e^{(\beta-1/2)u}/|\rho|\) and frequency \(\gamma\). The Mellin transform of \(\psi(x)-x\) is \[\int_1^\infty(\psi(x)-x)x^{-s-1}\,dx=-\frac{\zeta'(s)}{s\zeta(s)}-\frac1{s-1}\qquad(\sigma>1),\qquad\text{(A.21)}\] and RH holds if and only if this function continues holomorphically to \(\sigma>\frac12\), if and only if \(\psi(x)-x=O_\varepsilon(x^{1/2+\varepsilon})\) for every \(\varepsilon>0\). \textbf{Energy.} Put \[I(X)=\int_1^X\frac{(\psi(x)-x)^2}{x^2}\,dx=\int_0^{\log X}E(u)^2\,du,\qquad Q(X)=\frac{I(X)}{\log X}.\qquad\text{(A.22)}\] By Theorem 3.2 and Theorem B, RH holds if and only if \(\sup_{X\ge2}Q(X)<\infty\), and then \(Q(X)\to\sum_\rho m_\rho^2|\rho|^{-2}\), which equals \(C_B=2+\gamma_0-\log4\pi\) when the zeros are simple. If \(I(X)=O(\log X)\), then \(\int_1^\infty|\psi(x)-x|x^{-\sigma-1}dx<\infty\) for \(\sigma>\frac12\) by the Cauchy--Schwarz inequality on dyadic intervals, which excludes zeros with \(\beta>\frac12\). Expanding the square, \[\int_1^X\frac{\psi(x)^2}{x^2}\,dx=\sum_{m,n\le X}\Lambda(m)\Lambda(n)\Big(\frac1{\max(m,n)}-\frac1X\Big),\qquad\text{(A.23)}\] \[I(X)=\sum_{m,n\le X}\Lambda(m)\Lambda(n)\Big(\frac1{\max(m,n)}-\frac1X\Big)-2\sum_{n\le X}\Lambda(n)\log\frac Xn+X-1.\qquad\text{(A.24)}\] The double sum splits into the diagonal \(D(X)=\sum_{n\le X}\Lambda(n)^2(\frac1n-\frac1X)\sim\frac12(\log X)^2\) and the cross terms \(T(X)=2\sum_{n\le X}\Lambda(n)\psi(n^-)(\frac1n-\frac1X)\). Identity (A.24) expresses the energy through prime powers alone, without reference to \(\zeta\) or its zeros; the terms on the right are of size \(X\), and the statement \(I(X)=O(\log X)\) asks for their cancellation down to \(O(\log X)\). It is a classical reformulation of the boundedness in Theorem B, not a new constraint. \textbf{Numerical law.} From the primes up to \(10^7\), \begin{longtable}[]{@{}llllll@{}} \toprule\noalign{} \(X\) & \(10^3\) & \(10^4\) & \(10^5\) & \(10^6\) & \(10^7\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(Q(X)\) & \(0.40866\) & \(0.31875\) & \(0.26449\) & \(0.22795\) & \(0.20200\) \\ \((Q(X)-C_B)\log X\) & \(2.504\) & \(2.510\) & \(2.513\) & \(2.511\) & \(2.511\) \\ \end{longtable} so that, in this range, \[I(X)=C_B\log X+K+o(1),\qquad K\approx2.51.\qquad\text{(A.25)}\] The decrease of \(Q\) is the approach to \(C_B\) at the rate \(K/\log X\): subtracting \(K/\log X\) from \(Q(10^7)\) gives \(0.0462\), against \(C_B=0.0461914\). The constant \(K\) collects the transient part of the energy (the interval \(1\le x<2\), where \(\psi=0\), contributes exactly \(1\)). A finite computation does not establish boundedness. \subsection{Statements that would imply RH} \emph{Status: each is equivalent to RH or implies it; none is proved.} For every zero \(\rho\) and every \(\varepsilon>0\): \[C_2(N,\rho)=O_{\rho,\varepsilon}(N^\varepsilon),\qquad X_N(\rho)=O_{\rho,\varepsilon}(N^{1/2+\varepsilon}),\qquad\text{(A.26)}\] and, for the primes, \[\psi(x)-x=O_\varepsilon(x^{1/2+\varepsilon}),\qquad I(X)=O(\log X),\qquad \sum_\rho\frac1{|\rho|^2}=2+\gamma_0-\log4\pi.\qquad\text{(A.27)}\] The last is the vanishing of the non-holomorphic gap \(\mathcal Q\) of Section 3. The two energy structures of the paper, the cross-energy \(C_2(N,\rho)\) of the integers and the energy \(I(X)\) of the prime powers, are linked by Theorem B; what is missing is an inequality, independent of the location of the zeros, that excludes the growth \(N^{1-2\beta}\) of \(C_2(N,\rho)\) for \(\beta<\frac12\). \subsection{Statements not included as results} Two assertions of the author's earlier preprints are not valid in general. First, \(C_2(N,\sigma,t)=0\) does not imply \(\sigma=\frac12\): finite crossings of the cross-energy occur off the critical line. Second, the asymptotic vanishing of \(X^C_N-Y^C_N\) does not force \(|X^C_N|^2\) and \(|Y^C_N|^2\) to be linear in \(N\); by (A.6) the two energies agree at every zero whatever its abscissa. These points do not affect the exact identities above. \begin{thebibliography}{GHK}\small \bibitem[BBLS]{BBLS} L. B\'aez-Duarte, M. Balazard, B. Landreau, E. Saias, Notes sur la fonction $\zeta$ de Riemann, 3, \emph{Adv. 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