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

We assume the Riemann hypothesis to improve upon the rate of convergence of $(\log \log \log T{)}^2\mathrm{/}\sqrt{\log \log T}$ in Selberg’s central limit theorem for $\log \mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$ given by the author. We achieve a rate of convergence of $\sqrt{\log \log \log \log T}\mathrm{/}\sqrt{\log \log T}$ 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 $\mathrm{Re}(s)=1\mathrm{/}2$ under the Riemann hypothesis.