On the number variance of zeta zeros and a conjecture of Berry

TL;DR

This paper investigates the variance of the Riemann zeta-function's zeros and related quantities, proving new estimates and conjectures under the Riemann hypothesis and other assumptions, revealing non-universal behaviors.

Contribution

It provides new estimates for the variance of zeta zeros, proves Berry's conjecture on number variance in a non-universal regime, and calculates lower-order terms in the zeta function's second moment.

Findings

Proved estimates for the variance of zeta zeros assuming RH.

Confirmed Berry's conjecture on number variance under certain conditions.

Calculated lower-order terms in the second moment of the zeta function.

Abstract

Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta-function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed nonzero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, GUE statistics do not describe the distribution of the zeros. We also calculate lower-order terms in the second moment of the logarithm of the modulus of the Riemann zeta-function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).