The Multivariate Rate of Convergence for Selberg's Central Limit Theorem

TL;DR

This paper quantifies the rate at which the distribution of the logarithm of the Riemann zeta function converges to a normal distribution, extending the results to multivariate cases with the same convergence rate.

Contribution

It provides a multivariate rate of convergence for Selberg's central limit theorem using Dudley distance, matching the known single-variable rate.

Findings

Achieves the same convergence rate as Selberg in the multivariate case.

Extends the rate of convergence results to the multivariate setting.

Uses Dudley distance to measure convergence, improving upon previous methods.

Abstract

In this paper we quantify the rate of convergence in Selberg's central limit theorem for $\log|\zeta(1/2+it)|$ based on the method of proof given by Radziwill and Soundararajan. We achieve the same rate of convergence of $(\log\log\log T)^2/\sqrt{\log\log T}$ as Selberg in the Kolmogorov distance by using the Dudley distance instead. We also prove the theorem for the multivariate case given by Bourgade with the same rate of convergence as in the single variable case.