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 article we present several identities for the moments of $\Lambda_X'(s)$ averaged over $U(N)$, for $s \in \mathbb{C}$ as well as specialized to $|s|=1$. Additionally, we prove, for positive integer $k$, that the polynomial $\int_{U(N)} |\Lambda_X(1)|^{2k} \mathrm{dX}$ of degree $k^2$ in $N$ divides the polynomial $\int_{U(N)} |\Lambda_X'(1)|^{2k} \mathrm{dX}$ 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 non-linear Painl\'{e}ve differential equation. Finally we give some formulas in terms of the $_3F_2$ 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.