The Rate of Convergence for Selberg's Central Limit Theorem under 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.