Large deviations of Selberg's central limit theorem

Abstract: Selberg's celebrated central limit theorem shows that $\log\zeta(1/2+\rm{i} t)$ at a typical point $t$ at height $T$ behaves like a complex, centered Gaussian random variable with variance $\log\log T$. This talk will present recent results showing that the Gaussian decay persists in the large deviation regime, at a level on the order of the variance, improving on the best known bounds in that range. Time permitting, we will also present various applications, including on the maximum of the zeta function in short intervals.

This work is joint with Louis-Pierre Arguin.

number theoryprobability

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)