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 $2k^{\mathit{\text{th}}}$ 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 ${(\hspace{0.17em}\text{log}\hspace{0.17em}T)}^{\theta },\hspace{0.17em}\theta >0$. 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.

Keywords:

• Riemann zeta function
• extreme value statistics
• random matrix theory

2000 Mathematics Subject Classification::

• Primary: 11M06
• Secondary: 11M50, 60G70, 60B20

Acknowledgments

We thank to M. Radziwiłł for insightful discussions and the referees for their careful reading of the first version of this article. We are also grateful to P. Bourgade and A. Nikeghbali for pointing out to us that Theorem 1.1 first appeared in [16].

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

L.-P. A. gratefully acknowledges the support from the grant NSF CAREER DMS-1653602. K. H. and R. R. were also financially supported in part by this grant. E. B. thanks the Heilbronn Institute for Mathematical Research for support.