Mod-Gaussian convergence and the value distribution of $\zeta(1/2+it)$ and related quantities

TL;DR

This paper explores the distribution of the Riemann zeta function on the critical line using mod-Gaussian convergence, providing lower bounds for probabilities and evidence supporting conjectures about its value density.

Contribution

It introduces new lower bounds for local probabilities of approximately Gaussian vectors and applies these results to the zeta function and related L-functions.

Findings

Lower bounds for local probabilities of random vectors with increasing covariance

Evidence supporting the density conjecture of zeta function values on the critical line

Unconditional results for random matrices in classical groups and L-functions over finite fields

Abstract

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance. This is motivated by the conjecture concerning the density of the set of values of the Riemann zeta function on the critical line. We obtain evidence for this fact, and derive unconditional results for random matrices in compact classical groups, as well as for certain families of L-functions over finite fields.