Moments of the derivative of the characteristic polynomial of unitary matrices
E. Alvarez, J. Brian Conrey, MO Rubinstein, N. C. Snaith · Random Matrices Theory and Application · 2025
Let [Formula: see text] be the characteristic polynomial of a Haar distributed unitary matrix [Formula: see text]. It is believed that the distribution of values of [Formula: see text] model the distribution of values of the Riemann zeta-function [Formula: see text]. This principle motivates many avenues of study. Of particular interest is the behavior of [Formula: see text] and the distribution of its zeros (all of which lie inside or on the unit circle). In this paper, we present several identities for the moments of [Formula: see text] averaged over [Formula: see text], for [Formula: see text] as well as specialized to [Formula: see text]. Additionally, we prove, for positive integer [Formula: see text], that the polynomial [Formula: see text] of degree [Formula: see text] in [Formula: see text] divides the polynomial [Formula: see text] which is of degree [Formula: see text] in [Formula: see text] and that the ratio, [Formula: see text], of these moments factors into linear factors modulo [Formula: see text] if [Formula: see text] is prime. We also discuss the relationship of these moments to a solution of a second-order nonlinear Painléve differential equation. Finally we give some formulas in terms of the [Formula: see text] hypergeometric series for the moments in the simplest case when [Formula: see text], and also study the radial distribution of the zeros of [Formula: see text] in that case.