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

Selberg’s central limit theorem states that the values of log |𝜁⁢(1/2+i⁢𝜏)|, where τ is a uniform random variable on [𝑇,2⁢𝑇], are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation √12 ⁢log  log 𝑇. It was conjectured by Radziwiłł that this distribution breaks down for values of order log  log 𝑇, where a multiplicative correction Ck would be present at level 𝑘 log  log 𝑇, k > 0. This constant should be the same as the one conjectured by Keating and Snaith for the leading asymptotic of the 2⁢𝑘𝑡ℎ 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 log |𝜁| in intervals of size ( log 𝑇)𝜃, 𝜃>0. The precision of the prediction enables the numerical detection of Ck even for low T’s of order 𝑇=108. 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.