Moments of the derivative of the characteristic polynomial of unitary matrices
Emilia Alvarez, Brian Conrey, Michael O. Rubinstein, Nina C. Snaith
TL;DR
The paper studies the moments of the derivative $Λ'_X(s)$ of the characteristic polynomial $Λ_X(s)$ of Haar-distributed unitary matrices. It derives exact finite-$N$ determinant formulas for $\int_{U(N)}|Λ'_X(x)|^{2k}dX$ for general $x$, including simplifications on the unit circle $|x|=1$, and reveals a factorization of the $2k$-th moment into the $2k$-th moment of $|Λ_X(1)|$ times a polynomial $f(N,k)$ of degree $2k$. It further establishes a congruence modulo primes of the form $4k-1$ and connects the moment structure to a nonlinear Painlevé differential equation, offering efficient computation via a differential equation. The authors provide explicit finite-$N$ expressions for $N=2$ in terms of $_3F_2$ hypergeometric functions and study the radial distribution of zeros of $Λ'_X(s)$ using Jensen’s formula, highlighting deep connections between random matrix theory and special functions. These results enrich the random matrix model for zeta-function value distributions and offer concrete tools for exploring zeros of derivatives of characteristic polynomials.
Abstract
Let $Λ_X(s)=\det(I-sX^{\dagger})$ be the characteristic polynomial of a Haar distributed unitary matrix $X$. It is believed that the distribution of values of $Λ_X(s)$ 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 $s \in \mathbb{C}$ as well as specialized to $|s|=1$. Additionally, we prove, for positive integer $k$, that the polynomial $\int_{U(N)} |Λ_X(1)|^{2k} \mathrm{dX}$ of degree $k^2$ in $N$ divides the polynomial $\int_{U(N)} |Λ_X'(1)|^{2k} \mathrm{dX}$ 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 Painléve 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 $Λ_X'(s)$ in that case.