The Rate of Convergence for Selberg's Central Limit Theorem under the Riemann Hypothesis

TL;DR

Under the Riemann hypothesis, this paper improves the rate of convergence in Selberg's central limit theorem for the logarithm of the zeta function, refining previous bounds with a novel approach.

Contribution

The paper introduces a new, sharper rate of convergence for Selberg's CLT under RH, utilizing adapted techniques and a Selberg lemma with mollifiers near the critical line.

Findings

Achieves a convergence rate of rac{rac{\

Provides a refined bound in the Dudley distance for rac{rac{\

Improves upon previous convergence rates assuming the Riemann hypothesis.

Abstract

We assume the Riemann hypothesis to improve upon the rate of convergence of $(\log\log\log T)^2/\sqrt{\log\log T}$ in Selberg's central limit theorem for $\log|\zeta(1/2+it)|$ given by the author. We achieve a rate of convergence of $\sqrt{\log\log\log\log T}/\sqrt{\log\log T}$ in the Dudley distance. The proof is an adaptation of the techniques used by the author, based on the work of Radziwill and Soundararajan and Arguin et al., combined with a lemma of Selberg that provides for a mollifier close to the critical line $\operatorname{Re}(s)=1/2$ under the Riemann hypothesis.