On the mod-Gaussian convergence of a sum over primes

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.