A central limit theorem for the zeroes of the zeta function

TL;DR

Under the Riemann hypothesis, this paper extends a central limit theorem to the distribution of zeta zeros in mesoscopic intervals, linking number theory with random matrix theory results.

Contribution

It generalizes Fujii's central limit theorem for zeta zeros to a broader mesoscopic setting under the Riemann hypothesis.

Findings

Distribution of zeta zeros in mesoscopic intervals follows a normal distribution

Connects number theory results with random matrix theory

Provides new theorems on zeta zero distribution in the mesoscopic regime

Abstract

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.