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 $\text{log}\hspace{0.17em}|\zeta (1/2+\mathrm{i}\tau )|$, where τ is a uniform random variable on $[T,2T]$, are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation $\sqrt{\frac{1}{2}\text{log}\hspace{0.17em}\hspace{0.17em}\text{log}\hspace{0.17em}T}$. It was conjectured by Radziwiłł that this distribution breaks down for values of order $\text{log}\hspace{0.17em}\hspace{0.17em}\text{log}\hspace{0.17em}T$, where a multiplicative correction C_k would be present at level $k\hspace{0.17em}\text{log}\hspace{0.17em}\hspace{0.17em}\text{log}\hspace{0.17em}T$, k > 0. This constant should be the same as the one conjectured by Keating and Snaith for the leading asymptotic of the $2k^{th}$ 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 $\text{log}\hspace{0.17em}\left|\zeta \right|$ in intervals of size ${(\hspace{0.17em}\text{log}\hspace{0.17em}T)}^{\theta },\theta >0$. 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 Féray, Méliot and Nikeghbali.