Linear statistics for zeros of Riemann's zeta function

TL;DR

This paper investigates the statistical distribution of the zeros of the Riemann zeta function, demonstrating that their scaled counts exhibit Gaussian behavior in the limit for certain moments.

Contribution

It provides a rigorous analysis of the moments of the zeros' counting function, showing convergence to Gaussian moments for the first few moments.

Findings

First few moments tend to Gaussian moments

Distribution of zeros exhibits Gaussian behavior

Results depend on the specific statistic considered

Abstract

We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered.