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

2023/07/04 by Jonathan P. Keating, Fei Wei, Keating, Jonathan P. +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.01625

openalex publication_date 2023/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive explicit asymptotic formulae for the joint moments of the n₁-th and n₂-th derivatives of the characteristic polynomials of CUE random matrices for any non-negative integers n₁, n₂. 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 n₁-th and n₂-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.

Cited by

• Higher order linear differential equations for unitary matrix integrals: applications and generalisations
• Joint Moments of Characteristic Polynomials from the Orthogonal and Unitary Symplectic Groups

Related

• Joint moments of higher order derivatives of CUE characteristic polynomials II: Structures, recursive relations, and applications
• Moments of the derivative of the Riemann zeta-function and of characteristic polynomials
• On moments of the derivative of CUE characteristic polynomials and the Riemann zeta function
• Joint Moments of Characteristic Polynomials from the Orthogonal and Unitary Symplectic Groups
• On the joint moments of the characteristic polynomials of random unitary matrices