$\zeta$-functions via contour integrals and universal sum rules

TL;DR

This paper introduces a universal contour integral framework for analyzing $$-functions associated with diverse complex sequences, extending traditional spectral methods and enabling new identities and computations.

Contribution

It generalizes the contour integral approach to $$-functions for arbitrary sequences, establishing universal sum rules and connecting to Fredholm determinants.

Findings

Derived universal identities and sum rules for $$-functions.

Computed special values and residues for sequences like zeros of Airy and hypergeometric functions.

Applied rational interpolation techniques to study the Airy $$-function.

Abstract

This work develops an analytic framework for the study of the $\zeta$-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral $\zeta$-functions associated with self-adjoint differential operators, can be extended far beyond its traditional setting. In contrast to representations utilizing integrals of $\theta$-functions, our method applies to arbitrary sequences of complex numbers with minimal assumptions. This leads to a set of universal identities, including sum rules and meromorphic properties, that hold across a broad class of $\zeta$-functions. Additionally, we discuss the connection to regularized (modified) Fredholm determinants of $p$-Schatten--von Neumann class operators. We illustrate the versatility of this representation by computing special values and residues of the $\zeta$-function for a variety of sequences of complex numbers, in particular, the zeros of Airy functions, parabolic cylinder functions, and confluent hypergeometric functions. Furthermore, we employ the adaptive Antoulas--Anderson (AAA) algorithm for rational interpolation in the study of the Airy $\zeta$-function.