Evidence of random matrix corrections for the large deviations of Selberg's central limit theorem

Selberg's central limit theorem for the Riemann zeta function gives that the logarithm of zeta on its half line is asymptotically Gaussian with a variance growing as log log T, for T the height up the critical line. One can ask about large deviations to this result, for example looking into the tail on the order of the variance. Radziwill states that one should still see the Gaussian density, but accompanied by a random matrix correction. We will introduce the broader area of modeling number-theoretic functions using random matrix theory, and hence present theoretical and numerical evidence in favor of this conjecture, and examine connections to the maximum of the logarithm of zeta, and to related branching processes.

This is joint work with L-P. Arguin, E. Amzallag, K. Hui, and R. Rao.