\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm,graphicx,booktabs,hyperref} \usepackage[margin=1.1in]{geometry} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem*{ctheorem}{Conditional Theorem} \theoremstyle{definition} \newtheorem{hypothesis}{Hypothesis} \renewcommand{\thehypothesis}{H\arabic{hypothesis}} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\E}{\mathbb{E}} \newcommand{\PV}{\operatorname{PV}} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Cov}{Cov} \DeclareMathOperator{\Corr}{Corr} \title[Derivatives at close pairs of zeta zeros]{Derivatives of Hardy's function at close pairs of zeros and an exact unitary constant $\pi^2/15$} \author{Jaskirat Singh} \dedicatory{\upshape HIX Lab\\ \texttt{founder@hixlab.si}} \address{HIX Lab} \email{founder@hixlab.si} \subjclass[2020]{11M06, 11M26, 15B52, 60B20} \keywords{Riemann zeta function, Hardy $Z$ function, zero spacing, random matrix theory, circular unitary ensemble, Hua ensemble} \date{November 3, 2026} \begin{document} \begin{abstract} Let $\gamma_n<\gamma_{n+1}$ be consecutive ordinates of zeros of $\zeta(\tfrac12+it)$ with normalized gap $s_n$. We study the derivatives $Z'(\gamma_n)$, $Z'(\gamma_{n+1})$ of Hardy's function at close pairs. For the circular unitary ensemble we prove that the conditional variance of $\log|\Lambda'(\theta_1)|-\log|\Lambda'(\theta_2)|$ at adjacent eigenangles with normalized gap $s$ satisfies $\Var=c_N s^2(1+o(1))$ as $s\to0$, with the exact value $c_N=\frac{\pi^2}{15}\bigl(1-\frac{4}{N^2}\bigr)$, so that $c_\infty=\pi^2/15$, which we call the Singh constant. The proof reduces the problem through a Cayley transform to the Hua ensemble and closes it with two loop equations. We state a conditional theorem transferring the constant to the zeta function under the Riemann Hypothesis, a local GUE hypothesis and a tail bound, and we formulate a conjecture on the size of $Z'$ at close pairs. Numerical evidence from Odlyzko's tables up to zero number $10^{22}$ agrees with the predictions. This paper is a revised and expanded version of the Zenodo record doi:10.5281/zenodo.23899228. \end{abstract} \maketitle \section{Introduction} Montgomery's pair correlation conjecture \cite{Montgomery} and Odlyzko's computations \cite{Odlyzko87,Odlyzko92,Odlyzko01} show that the zeros of the Riemann zeta function repel like the eigenvalues of large random unitary matrices. Turing's method \cite{Turing} makes the verification of zeros on the critical line rigorous, and Keating and Snaith \cite{KeatingSnaith} together with Hughes, Keating and O'Connell \cite{HKO} identified the characteristic polynomial of a random unitary matrix as a model for $\zeta$ and its derivative at zeros. This paper is a revised and expanded version of \cite{SinghZenodo}. Write $\theta(t)$ for the Riemann Siegel theta function and $Z(t)=e^{i\theta(t)}\zeta(\tfrac12+it)$. For consecutive ordinates $\gamma_n<\gamma_{n+1}$ put \[ L=\log\frac{\gamma_n}{2\pi},\qquad s_n=\frac{(\gamma_{n+1}-\gamma_n)L}{2\pi},\qquad x_n=\log\frac{|Z'(\gamma_n)|}{L},\qquad y_n=\log\frac{|Z'(\gamma_{n+1})|}{L}. \] We show that, for small $s_n$, $|Z'(\gamma_n)|\asymp s_nL$, that $x_n-y_n$ has size of exact order $s_n$, and that $\Var(x_n-y_n\mid s_n=s)/s^2\to\pi^2/15$ in the unitary model. \begin{theorem}[Singh Theorem]\label{thm:main} Let $U$ be Haar distributed on $U(N)$, $N\ge3$, with eigenangles $\theta_1,\dots,\theta_N$, and let $\Lambda(\theta)=\prod_{k=1}^N(1-e^{i(\theta_k-\theta)})$. For adjacent eigenangles $\theta_1<\theta_2$ with normalized gap $s=\frac{N}{2\pi}(\theta_2-\theta_1)$ put $x=\log|\Lambda'(\theta_1)|$ and $y=\log|\Lambda'(\theta_2)|$. Then \[ \lim_{s\to0}\frac{\Var(x-y\mid s)}{s^2}=c_N:=\frac{\pi^2}{15}\Bigl(1-\frac{4}{N^2}\Bigr),\qquad \lim_{N\to\infty}c_N=c_\infty=\frac{\pi^2}{15}. \] We call $c_\infty=\pi^2/15$ the Singh constant. \end{theorem} The main ingredient is the following identity for the Hua ensemble \cite{Hua}. \begin{theorem}\label{thm:hua} For $n\ge1$ and $h>\frac12$ let $u_1,\dots,u_n$ have joint density proportional to \[ P_{n,h}(u)=\prod_{k=1}^n(1+u_k^2)^{-(n+h)}\prod_{j\frac12$, since the one variable marginal decays like $|u|^{-2-2h}$. Let $\phi$ be a polynomial of degree at most one and $F$ a polynomial of degree at most one. For each $k$, \[ \int_{\mathbb R^n}\partial_{u_k}\bigl[(1+u_k^2)\phi(u_k)F(u)P_{n,h}(u)\bigr]\,du=0, \] because for the choices of $\phi$ and $F$ used below the product $\phi(u_k)F(u)$ has degree at most one in $u_k$, so the bracket is $O(|u_k|^{2+1+2(n-1)-2(n+h)})=O(|u_k|^{1-2h})$ as $|u_k|\to\infty$, uniformly integrable in the remaining variables, and tends to zero for $h>\frac12$. Using \[ \frac{\partial_{u_k}P_{n,h}}{P_{n,h}}=-\frac{2(n+h)u_k}{1+u_k^2}+\sum_{j\ne k}\frac{2}{u_k-u_j}, \] we obtain the identity \begin{equation}\label{eq:loop} \E\Bigl[\sum_k\bigl((1+u_k^2)\phi\bigr)'(u_k)F+\sum_k(1+u_k^2)\phi(u_k)\partial_kF-2(n+h)\sum_k u_k\phi(u_k)F+\sum_k\sum_{j\ne k}\frac{2(1+u_k^2)\phi(u_k)}{u_k-u_j}F\Bigr]=0 . \end{equation} \emph{First equation.} Take $\phi=1$ and $F=\sum_k u_k$. Symmetrizing, \[ \sum_k\sum_{j\ne k}\frac{2(1+u_k^2)}{u_k-u_j}=\sum_{j2\delta$, Taylor's theorem gives \[ f(-\tfrac\delta2-\theta_k)-f(\tfrac\delta2-\theta_k)=\tfrac{\delta}{2}\cot(\theta_k/2)+R_k,\qquad |R_k|\le C\delta^3|\theta_k|^{-3}. \] Near $0$ the one point density of the conditional law is $\asymp(\theta^2-\delta^2/4)^2$ on $|\theta|\ge\delta/2$, so \[ \E\Bigl[\Bigl(\sum_{|\theta_k|>2\delta}R_k\Bigr)^2\Bigr]\ll\delta^6\int_{2\delta}^{\pi}\theta^{-2}\,d\theta\ll\delta^5,\qquad \E\Bigl[|S|\sum_{|\theta_k|>2\delta}|R_k|\Bigr]\ll\delta^3 . \] Angles with $\delta/2\le|\theta_k|\le2\delta$ occur with probability $O(\delta^5)$ and each contributes $O(\log^2\delta)$, so they add $O(\delta^5\log^2\delta)$ to the second moment. The reflection $\theta\mapsto-\theta$ preserves the conditional law and maps $S\mapsto-S$, hence $\E[S]=\E[x-y]=0$. Therefore \[ \Var(x-y\mid s)=\delta^2\,\E_{w_0}[S^2]\,(1+o(1)),\qquad c_N=\Bigl(\frac{2\pi}{N}\Bigr)^2\E_{w_0}[S^2], \] where $\E_{w_0}$ is the circular ensemble of $n$ angles with weight $w_0$. \emph{Step 4.} Put $u_k=\cot(\theta_k/2)$. Then $|1-e^{i\theta}|^2=\frac{4}{1+u^2}$, $d\theta=\frac{2\,du}{1+u^2}$ up to orientation, and $|e^{i\theta_j}-e^{i\theta_k}|^2=\frac{4(u_j-u_k)^2}{(1+u_j^2)(1+u_k^2)}$. Each $u_k$ receives the exponent $(n-1)+1+2=n+2$, so the law of $(u_k)$ is $P_{n,2}$, and $S=\frac12\sum_k u_k$. \emph{Step 5.} By Theorem \ref{thm:hua} with $h=2$, $\E_{w_0}[S^2]=\frac14\cdot\frac{n(n+4)}{15}=\frac{(N-2)(N+2)}{60}$, hence \[ c_N=\frac{4\pi^2}{N^2}\cdot\frac{N^2-4}{60}=\frac{\pi^2}{15}\Bigl(1-\frac4{N^2}\Bigr). \qquad\qed \] \begin{corollary} As $N\to\infty$ and then $s\to0$, $\Corr(x,y\mid s)\to1$ and $\Cov(x,y\mid s)=\Var(x\mid s)-\frac{\pi^2}{30}s^2+O(s^4)$. In the sine process limit, $\E_0^{(2)}\bigl[(\PV\sum_k\xi_k^{-1})^2\bigr]=\pi^2/15$, where $\E_0^{(2)}$ is the second order Palm measure at a double point at $0$. \end{corollary} \section{Transfer to the zeta function} For a close pair $\gamma_n<\gamma_{n+1}$ let $\gamma_m$ be the midpoint and let $\xi_k=(\gamma-\gamma_m)L/2\pi$ for the other ordinates $\gamma$, and $R_3=\sum_k|\xi_k|^{-3}$. \begin{lemma}\label{lem:A} Assume the Riemann Hypothesis. Then $\log|Z(t)|=\sum_{\gamma}^{*}\log|t-\gamma|+H(t)$, where $\sum^{*}$ pairs $\gamma$ with $-\gamma$ and $H'(t)=O(1)$. Consequently \[ x_n-y_n=s_n\,\PV\sum_k\frac1{\xi_k}+O(s_n^3R_3)+O\Bigl(\frac{s_n}{L}\Bigr). \] \end{lemma} \begin{proof} On the critical line $\xi(\tfrac12+it)=-\tfrac12(t^2+\tfrac14)\pi^{-1/4}|\Gamma(\tfrac14+\tfrac{it}2)|Z(t)$ and, under RH, $\xi(\tfrac12+it)=\xi(\tfrac12)\prod_{\gamma>0}(1-t^2/\gamma^2)$. Taking logarithms gives the decomposition with $H(t)=-\log(t^2+\frac14)-\log|\Gamma(\frac14+\frac{it}2)|+O(1)$ plus the regularizing constants $\sum_{\gamma>0}\log\gamma^2$. By Stirling's formula and the density $\frac1{2\pi}\log\frac{t}{2\pi}$ of ordinates, the derivative of the regularized sum has mean $\frac{1}{2\pi}\PV\int_0^\infty\frac{2t\log u}{t^2-u^2}\,du+O(1/t)=-\frac{\pi}{4}+O(1/t)$, which combines with $H'$ to give a bounded total. Since $Z'(\gamma_n)=\lim_{t\to\gamma_n}Z(t)/(t-\gamma_n)$, $\log|Z'(\gamma_n)|=\sum_{\gamma\ne\gamma_n}^{*}\log|\gamma_n-\gamma|+H(\gamma_n)$. Subtracting the same identity at $\gamma_{n+1}$ and expanding as in Step 3 of the proof of Theorem \ref{thm:main} gives the claim, the factor $L/2\pi$ converting $\gamma_{n+1}-\gamma_n$ into $s_n$. \end{proof} \begin{hypothesis}[local GUE law]\label{H1} For each fixed $K$, the law of $\{\xi_k:|\xi_k|\le K\}$ conditioned on $s_n<\varepsilon$ and $\gamma_n\in[T,2T]$ converges, as $T\to\infty$ and then $\varepsilon\to0$, to the law of the sine process under $\E_0^{(2)}$. \end{hypothesis} \begin{hypothesis}[tail bound]\label{H2} Uniformly for large $T$ and small $\varepsilon$, \[ \lim_{K\to\infty}\E\Bigl[\Bigl(\PV\sum_{|\xi_k|>K}\frac1{\xi_k}\Bigr)^2\Bigm| s_n<\varepsilon\Bigr]=0,\qquad \sup\E\bigl[R_3^2\mid s_n<\varepsilon\bigr]<\infty . \] \end{hypothesis} Hypothesis \ref{H1} follows from the GUE $m$ level correlation conjecture for all $m$ and all supports, which Rudnick and Sarnak \cite{RudnickSarnak} establish for restricted support. Hypothesis \ref{H2} is a conditional mean square bound for $\zeta'/\zeta$ near the critical line, of the type related to pair correlation by Goldston, Gonek and Montgomery \cite{GGM}. \begin{ctheorem} Assume the Riemann Hypothesis, \ref{H1} and \ref{H2}. Then \[ \lim_{\varepsilon\to0}\lim_{T\to\infty}\frac{\E\bigl[(x_n-y_n)^2\mid s_n<\varepsilon,\ \gamma_n\in[T,2T]\bigr]}{\E\bigl[s_n^2\mid s_n<\varepsilon,\ \gamma_n\in[T,2T]\bigr]}=\frac{\pi^2}{15}. \] \end{ctheorem} \begin{proof} By Lemma \ref{lem:A}, $x_n-y_n=s_n\Sigma+E$ with $\Sigma=\PV\sum_k\xi_k^{-1}$ and $\E[E^2\mid s_n<\varepsilon]=o(\E[s_n^2\mid s_n<\varepsilon])$, using \ref{H2} for $R_3$. Split $\Sigma=\Sigma_{\le K}+\Sigma_{>K}$. By \ref{H2} the second moment of $\Sigma_{>K}$ tends to zero as $K\to\infty$ uniformly, and by \ref{H1} the second moment of $\Sigma_{\le K}$ converges to its sine process value, which tends to $\pi^2/15$ as $K\to\infty$ by the Corollary. Since $s_n$ and the local configuration are asymptotically independent in the limit of \ref{H1}, the ratio converges to $\pi^2/15$. \end{proof} \section{Conjecture} \begin{conjecture}[Singh Conjecture]\label{conj} With the notation above and $W_n=\sqrt{|Z'(\gamma_n)Z'(\gamma_{n+1})|}/L$, as $T\to\infty$ and $\varepsilon\to0$: \begin{enumerate} \item[(i)] $\exp\E\bigl[\log(W_n/s_n)\mid s_n<\varepsilon,\ \gamma_n\approx T\bigr]=C_\infty+\dfrac{B'}{\log(T/2\pi)}+O\Bigl(\dfrac1{\log^2T}\Bigr)+O(\varepsilon^2)$ with $C_\infty=0.676\pm0.031$ and $B'=2.99\pm0.35$; \item[(ii)] $\Var\bigl[\log(W_n/s_n)\mid s_n<\varepsilon,\ \gamma_n\approx T\bigr]=\frac12\log\log T+O(1)$; \item[(iii)] $\Var\bigl[x_n-y_n\mid s_n=s\bigr]=\dfrac{\pi^2}{15}s^2\bigl(1+O(s^2)\bigr)$, so that $\Corr(x_n,y_n\mid s_n=s)\to1$ as $s\to0$. \end{enumerate} Together with $\mathbb P(s_n<\varepsilon)\sim\frac{\pi^2}{9}\varepsilon^3$, clause (i) gives $|Z'(\gamma_n)|\asymp s_n\log(\gamma_n/2\pi)$ for close pairs. \end{conjecture} Clause (iii) is the Conditional Theorem. Clauses (i) and (ii) are supported by the data of Section \ref{sec:num}. \section{Numerical evidence}\label{sec:num} \subsection{Data} We use Odlyzko's tables \cite{Odlyzko01}: the first $2{,}001{,}052$ zeros ($\gamma\le1{,}132{,}490.66$) and $10^4$ consecutive zeros starting at zero number $10^{12}+1$, $10^{21}+1$ and $10^{22}+1$. Values of $Z$ and $Z'$ were computed from the Riemann Siegel formula with the first two correction terms; for $t\approx10^{21}$ and $10^{22}$ the main sums of length $N=4.79\times10^9$ and $1.48\times10^{10}$ were evaluated at many points simultaneously by grouping frequencies $\log n$ into short intervals and expanding in Taylor series, in the spirit of Odlyzko and Sch\"onhage. Against an independent high precision evaluation the maximal errors were $3.7\times10^{-10}$ for $Z$ and $3.5\times10^{-11}$ for $Z'$. At the tabulated zeros near $10^{21}$ and $10^{22}$ the median of $|Z|$ is $6\times10^{-8}$ and $Z$ alternates in sign at all $199$ tested midpoints. \subsection{Small gaps} Maximum likelihood fits of $p(s)\propto s^\beta$ on $[0,s_0]$ for the first $2\times10^6$ zeros give $\beta=1.953\pm0.025$ ($s_0=0.2$, $14{,}279$ gaps) and $\beta=1.953\pm0.014$ ($s_0=0.3$); the pure power fit is biased below $2$ by the term $-\frac{2\pi^4}{45}s^4$ of the GUE density. The smallest normalized gap is $s=0.00545$, between zeros $1{,}115{,}578$ and $1{,}115{,}579$ at $\gamma\approx663{,}318.5083$, where $|Z'|=0.1751$ and $0.1756$. \begin{figure}[ht] \includegraphics[width=\textwidth]{gap_hist_gue.png} \caption{Normalized gap distribution against the Wigner surmise, with the small gap region.} \end{figure} \subsection{Clause (i)} Table \ref{tab:C} lists fits of $W_n=Cs_n^\alpha$ on $s_n<0.3$ and the geometric mean $C_1$ of $W_n/s_n$. \begin{table}[ht] \centering \begin{tabular}{lrrcc} \toprule Height & $L$ & pairs & $\alpha$ & $C_1$ \\ \midrule $t<10^5$ & 9.1 & 345 & $1.116\pm0.063$ & $1.001\pm0.022$\\ $10^5\le t<5\cdot10^5$ & 10.8 & 2094 & $1.052\pm0.028$ & $0.964\pm0.010$\\ $5\cdot10^5\le t\le1.13\cdot10^6$ & 11.8 & 3726 & $0.994\pm0.024$ & $0.923\pm0.008$\\ zero $10^{12}$ & 24.5 & 262 & $0.871\pm0.151$ & $0.814\pm0.036$\\ zero $10^{21}$ & 44.6 & 257 & $0.940\pm0.157$ & $0.771\pm0.042$\\ zero $10^{22}$ & 46.8 & 266 & $0.771\pm0.162$ & $0.720\pm0.040$\\ \bottomrule \end{tabular} \caption{Slope exponent and constant by height.}\label{tab:C} \end{table} The exponents are consistent with $\alpha=1$ ($\chi^2=8.4$ on $5$ degrees of freedom, weighted mean $1.020$). The model $C_1=C_\infty+B'/L$ gives $C_\infty=0.676\pm0.031$, $B'=2.99\pm0.35$, $\chi^2=3.1/4$; a power law $AL^\kappa$ gives $\kappa=-0.187\pm0.024$, $\chi^2=6.5/4$. The variance of $\log(W_n/s_n)$ grows from $0.17$ at $t<10^5$ to $0.81$ near zero $10^{22}$, an increase of $0.64$, against the increase $\frac12(\log\log 1.4\cdot10^{21}-\log\log 10^5)\approx0.72$ predicted by clause (ii). \begin{figure}[ht] \includegraphics[width=\textwidth]{zprime_per_height.png} \caption{Fits of $W_n$ against $s_n$ by height, and $\alpha$, $C_1$ against $L$ with both models.} \end{figure} \subsection{Clause (iii)} For all $10^4$ consecutive zeros of each high set and a large sample of the first $2\times10^6$ zeros, we measured $\Cov(x_n,y_n\mid s)$, $\Corr(x_n,y_n\mid s)$ and $\Var(x_n-y_n\mid s)$ for $s<1$, and compared them with the unitary ensemble at $N=12,25,46$. The covariance has no logarithmic singularity at $s=0$: fitting $a+b\log s$ on $s<0.5$ gives $b=-0.026\pm0.006$, $-0.056\pm0.079$, $0.034\pm0.110$, $0.061\pm0.103$ for the four zeta sets and $b\approx-0.04$ for the unitary ensemble. The correlation tends to $1$ and $\Var(x-y\mid s)\propto s^g$ with $g=2.06$, $1.70$, $1.96$, $1.94$. \begin{figure}[ht] \includegraphics[width=\textwidth]{cov_vs_s.png} \caption{Conditional covariance, correlation and variance of the difference, for zeta zeros and the unitary ensemble.} \end{figure} The exact values of $\E_{w_0}[S^2]$ computed from Toeplitz determinants for $n=1,\dots,15$ are $\frac1{12},\frac15,\frac7{20},\frac8{15},\frac34,1,\frac{77}{60},\dots,\frac{19}4$, in agreement with $\frac{n(n+4)}{60}$. Figure \ref{fig:c} compares the exact finite $s$ curve $\E[(x-y)^2\mid s]/s^2$ with simulations ($\chi^2=19.0$, $17.7$, $18.2$ on $10$ bins) and with zeta. Near zero $10^{21}$ and $10^{22}$ the agreement is good ($\chi^2=8.8/9$ and $4.8/9$); near $10^{12}$ the data lie $5$ to $10$ percent low ($\chi^2=24.4/9$), and for $t\le1.13\times10^6$ the ratio is about $0.42$ against $0.63$, a low height effect consistent with the slow convergence described by Bogomolny and Keating \cite{BogomolnyKeating}. \begin{figure}[ht] \includegraphics[width=\textwidth]{c_check.png} \caption{$\E[(x-y)^2\mid s]/s^2$: exact unitary curves against simulations (left) and against zeta zeros with $N=\mathrm{round}(L)$ (right). The dotted line is $\pi^2/15$.}\label{fig:c} \end{figure} \section{Open problems} The Conditional Theorem reduces clause (iii) to the Riemann Hypothesis, the local law \ref{H1} for all supports and the tail bound \ref{H2}. 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