The multivariate rate of convergence for Selberg’s central limit theorem

Abstract

In this paper we quantify the rate of convergence in Selberg’s central limit theorem for $log|\mathit{\zeta }(1/2\mathbf{+}\mathit{i}\mathit{t})|$ based on the method of proof given by Radziwiłł and Soundararajan in (Enseign. Math. 63 (2017) 1–19). We achieve the same rate of convergence of ${(logloglog\mathit{T})}^2/\sqrt{loglog\mathit{T}}$ as Selberg in (In Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ (1992) 367–385) in the Kolmogorov distance by using the Dudley distance instead. We also prove the theorem for the multivariate case given by Bourgade in (Probab. Theory Related Fields 148 (2010) 479–500) with the same rate of convergence as in the single variable case.