Connection between the Riemann zeta-function and random matrices via hyperfunctions

Abstract

Bohr pioneered the study of the statistical behavior of the Riemann zeta-function. A classical result by Bohr and Jessen revealed that the values of the Riemann zeta-function to the right of the critical line behave like a random variable. We now propose to extend Bohr's theory to the stage of hyperfunctions. In this paper, we introduce two random hyperfunctions: one is associated with the values of the Riemann zeta-function on the critical line, and the other is associated with the characteristic polynomial of a random matrix from the circular unitary ensemble. We then derive a relationship between these random hyperfunctions which is consistent with the Keating-Snaith conjecture on the moments of the Riemann zeta-function.