Joint moments of higher order derivatives of CUE characteristic polynomials I: asymptotic formulae

TL;DR

This paper derives explicit asymptotic formulas for the joint moments of higher order derivatives of CUE characteristic polynomials, expressing them via determinants with Bessel functions and combinatorial sums, and extends these results to conjectures about the Riemann zeta-function.

Contribution

It provides the first explicit asymptotic formulas for joint moments of derivatives of CUE characteristic polynomials, linking random matrix theory with number theory conjectures.

Findings

Explicit asymptotic formulas involving determinants and Bessel functions

Representation of joint moments through combinatorial sums

Support for conjectures on Riemann zeta-function derivatives

Abstract

We derive explicit asymptotic formulae for the joint moments of the $n_1$-th and $n_2$-th derivatives of the characteristic polynomials of CUE random matrices for any non-negative integers $n_1, n_2$. These formulae are expressed in terms of determinants whose entries involve modified Bessel functions of the first kind. We also express them in terms of two types of combinatorial sums. Similar results are obtained for the analogue of Hardy's $Z$-function. We use these formulae to formulate general conjectures for the joint moments of the $n_1$-th and $n_2$-th derivatives of the Riemann zeta-function and of Hardy's $Z$-function. Our conjectures are supported by comparison with results obtained previously in the number theory literature.