Mesoscopic fluctuations of the zeta zeros

TL;DR

This paper extends Selberg's central limit theorem to multiple dimensions, revealing correlations in the mesoscopic fluctuations of the Riemann zeta zeros and their analogy with random matrix eigenvalues.

Contribution

It provides a multidimensional extension of Selberg's CLT for log zeta, addressing mesoscopic fluctuations and correlations, and draws parallels with random matrix theory.

Findings

Established a multidimensional CLT for log zeta zeros.

Identified correlations in mesoscopic fluctuations.

Demonstrated the n  log t correspondence in local scales.

Abstract

We prove a multidimensional extension of Selberg's central limit theorem for $\log\zeta$, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence $n \leftrightarrow \log t$ not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.