Moments of the derivative of the characteristic polynomial of unitary matrices

Let Lambda X(s) = det(I - sX dagger) be the characteristic polynomial of a Haar distributed unitary matrix X. It is believed that the distribution of values of Lambda(X )(s) model the distribution of values of the Riemann zeta-function zeta(s). This principle motivates many avenues of study. Of particular interest is the behavior of Lambda(X)' (s) 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 Lambda(X)' (s) averaged over U(N), for s is an element of & Copf; as well as specialized to |s| = 1. Additionally, we prove, for positive integer k, that the polynomial integral(U (N))|Lambda(X) (1)|(2k )dX of degree k(2) in N divides the polynomial integral(U (N) )|Lambda(X)'(1)|(2k )dX which is of degree k(2 )+ 2k in N and that the ratio, f(N, k), of these moments factors into linear factors modulo 4k - 1 if 4k - 1 is prime. We also discuss the relationship of these moments to a solution of a second-order nonlinear Painl & eacute;ve differential equation. Finally we give some formulas in terms of the 3F2 hypergeometric series for the moments in the simplest case when N = 2, and also study the radial distribution of the zeros of Lambda(X)'(s) in that case.