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 $\log|\zeta(1/2+i \tau)|$, where $\tau$ is a uniform random variable on $[T,2T]$, is distributed like a Gaussian random variable of mean $0$ and standard deviation $\sqrt{\frac{1}{2}\log \log T}$. It was conjectured by Radziwi{\l}{\l} that this breaks down for values of order $\log\log T$, where a multiplicative correction $C_k$ would be present at level $k\log\log T$, $k>0$. This constant should be equal to the leading asymptotic for the $2k^{th}$ moment of $\zeta$, as first conjectured by Keating and Snaith using random matrix theory. 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 $\log|\zeta|$ in intervals of size $(\log T)^\theta$, $\theta>0$. The precision of the prediction enables the numerical detection of $C_k$ even for low $T$'s of order $T=10^8$. 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 F\'eray, M\'eliot and Nikeghbali.