Moments of the derivative of the characteristic polynomial of unitary matrices

Let Λ_X(s) = det(I −sX^†) be the characteristic polynomial of a Haar distributed unitary matrix X. It is believed that the distribution of values of Λ_X(s) model the distribution of values of the Riemann zeta-function ζ(s). This principle motivates many avenues of study. Of particular interest is the behavior of Λ′_X(s) and the distribution of its zeros (all of which lie inside or on the unit circle). In this article we present several identities for the moments of Λ′_X(s) averaged over U(N), for s ∈ C as well as specialized to |s| = 1. Additionally, we prove, for positive integer k, that the polynomial ∫_U(N)|Λ_X(1)|^2kdX of degree k² in N divides the polynomial ∫_U(N)|Λ′X(1)|^2kdX which is of degree k² + 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 non-linear Painl´eve 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 Λ′_X(s) in that case.