On the mod-Gaussian convergence of a sum over primes

TL;DR

This paper establishes mod-Gaussian convergence for a Dirichlet polynomial approximating the imaginary part of the logarithm of the Riemann zeta function, leading to new insights into its distribution, moments, and large deviations on the critical line.

Contribution

It proves mod-Gaussian convergence for a specific Dirichlet polynomial related to the zeta function, extending results under the Riemann hypothesis and deriving explicit error bounds.

Findings

Proves mod-Gaussian convergence for the polynomial approximating Im log zeta.

Derives Selberg's central limit theorem with explicit error term.

Establishes large deviation principles for Im log zeta on the critical line.

Abstract

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $\operatorname{Im}\log\zeta(1/2+it)$. This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that $\operatorname{Im}\log\zeta(1/2+it)$ satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.