Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem

Abstract

Selberg’s central limit theorem states that the values of , where τ is a uniform random variable on , are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation . It was conjectured by Radziwiłł that this distribution breaks down for values of order , where a multiplicative correction Ck would be present at level , k > 0. This constant should be the same as the one conjectured by Keating and Snaith for the leading asymptotic of the moment of ζ. In this paper, we provide numerical and theoretical evidence for this conjecture. We propose that this correction has a significant effect on the distribution of the maximum of in intervals of size . The precision of the prediction enables the numerical detection of Ck even for low T’s of order . A similar correction appears in the large deviations of the Keating–Snaith central limit theorem for the logarithm of the characteristic polynomial of a random unitary matrix, as first proved by Féray, Méliot and Nikeghbali.