Large gaps between zeros of the zeta-function on the critical line and moment conjectures from random matrix theory
From the random matrix model for the Riemann zeta-function \(\zeta(s)\), \textit{J. P. Keating} and \textit{N. C. Snaith} [Commun. Math. Phys. 214, No. 1, 57--89 (200; Zbl 1051.11048)] conjectured that \[ \int_{0}^{T} \Biggl| \zeta \biggl({1\over 2}+it\biggr)\Biggr| ^{2k}\, dt \sim a(k){G^{2}(k+1)\over G(2k+1)}T(\log T)^{k^2} \quad (k\in \mathbb N), \] as \(T\to \infty\). Here \(G(z)\) is the Barnes double gamma-function, and \(a(k)\) is an explicitly given constant in terms of \(k\). The cases \(k=1, 2\) of this, being the only proved cases, are part of the classical literature about \(\zeta(s)\). For the discrete moments with a fixed \(k > -{1\over 2}\), \textit{C. P. Hughes} [J. Phys. A 36, No. 12, 2907--2917 (2003; Zbl 1074.11047)] conjectured upon assuming the Riemann Hypothesis that \[ \sum_{\gamma_n \leq T} \Biggl| \zeta\biggl({1\over 2}+i \biggl(\gamma_n +{2\pi\alpha\over \log {T\over 2\pi}}\biggr)\biggr)\Biggr| ^{2k} \sim F_{k}(2\pi\alpha) a(k){G^{2}(k+1)\over G(2k+1)}{T\over 2\pi}(\log T)^{k^2 +1} , \] uniformly for \(| \alpha| \leq \log {T\over 2\pi}\). Here the \(\gamma_n\) denote the positive ordinates of the nontrivial zeros of \(\zeta(s)\) in increasing order, and \(F_{k}\) is a function explicitly given in terms of the \(k\)-th spherical Bessel function of the first kind. Other than the trivial \(k=0\) case, the only known case of this conjecture is for \(k=1\) due to Gonek. In this paper, the authors apply the established techniques involving the comparison of the continuous global average with the local average obtained from the discrete average to the problem of gaps between zeta zeros, assuming the Riemann Hypothesis and the above conjectures (the cases with \(k\) and \(2k\) for the continuous moment, and the case \(k\) of the discrete moment). They then show that for any fixed positive integer \(r\), \[ (\gamma_{n+r}-\gamma_n){\log\gamma_{n}\over 2\pi r} \geq \theta \] holds for any \(\theta \leq {4k\over \pi e r}\) for more than \(c(\log T)^{-4k^{2}}\) proportion of the zeros \(\gamma_n \in [0,T]\), with a computable constant \(c=c(k,\theta,r)\).
moment conjectures for the zeta-function
scientific article; zbMATH DE number 5305525