Hankel determinants of Dirichlet series

TL;DR

This paper derives a general formula for Hankel determinants of Dirichlet series, analyzes their asymptotics for the Riemann zeta function, and explores their connections to integrals and measures in number theory.

Contribution

It introduces a general expression for Hankel determinants of Dirichlet series and examines their asymptotic behavior specifically for the Riemann zeta function.

Findings

Hankel determinants relate to discrete analogues of Selberg integrals.

Asymptotic behavior characterized for the Riemann zeta function case.

Connections made to matrix integrals and Plancherel measures.

Abstract

We derive a general expression for the Hankel determinants of a Dirichlet series F(s) and derive the asymptotic behavior for the special case that F(s) is the Riemann zeta function. In this case the Hankel determinant is a discrete analogue of the Selberg integral and can be viewed as a matrix integral with discrete measure. We briefly comment on its relation to Plancherel measures.