Joint Moments of Higher Order Derivatives of CUE Characteristic Polynomials I: Asymptotic Formulae

We derive explicit asymptotic formulae for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the characteristic polynomials of Circular Unitary Ensemble 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.