Gram's law for the zeta zeros and the eigenvalues of Gaussian unitary ensembles (Q1097917)

The main purpose of this note is to study the quantity \[ G_ M(k,\alpha)=| \{-1\leq m\leq M: N(g_{m+\alpha})-N(g_ m)=k\}| \] for \(\alpha >0\) and integers \(k\geq 0\). Here as usual N(t) is the number of nontrivial zeros \(\rho\) of \(\zeta(s)\) for which \(0<Im \rho <t\), so that \(N(t)=(1/\pi)\theta (t)+1+S(t)\), where \(S(t)=(1/\pi)\arg \zeta (1/2+it)\) \((=O(\log t))\) and \[ \theta(t)=Im \log \Gamma (1/4+(i/2)t)-t/2 \log \pi =t/2 \log (t/2\pi)-t/2-\pi /8+O(1/t). \] For \(x\geq -1\) the Gram points \(g_ x\) are defined by \(\theta (g_ x)=x\pi\) (so that \(g_ x\sim 2\pi x/\log x\) as \(x\to \infty)\). The Gram points are well-known from their rôle in the investigation of the zeros of \(\zeta(s)\) on the critical line [see \textit{H. M. Edwards}' book ``Riemann's zeta-function'' (Academic Press, London 1974; Zbl 0315.10035) and the numerical work of \textit{J. van de Lune}, \textit{D. T. Winter} and \textit{H. J. J. te Riele}, Math. Comput. 46, 667-681 (1986; Zbl 0585.10023)]. The author states without proof an asymptotic formula for the sum \[ \sum_{m\leq M}(S(g_{m+\alpha})-S(g_ m))^{\ell}\quad (\ell \geq 1\text{ an integer, }\alpha \ll \log M). \] He also conjectures that \[ (1)\quad \lim_{M\to \infty}(1/M)G_ M(k,\alpha)=E(k,\alpha) \] for each integer \(k\geq 0\) and \(0<\alpha <\alpha_ 0<\infty\). The constant E(k,\(\alpha)\) is precisely given and it is connected with the so-called pair correlation of the eigenvalues of the Gaussian Unitary Ensembles [see more on this in \textit{A. M. Odlyzko}, Math. Comput. 48, 273-308 (1987; Zbl 0615.10049)]. The author states that from (1) (or its continuous variant) he is able to deduce several corollaries. These seem interesting, one of them being \[ \int^{T}_{0}(S(t+\frac{2\pi \alpha}{\log T})-S(t))^ 2 dt\quad \sim \quad (\alpha -\alpha^ 2+O(\alpha^ 4))T, \] where \(T\to \infty\) and \(\alpha\to 0\).