Multidimensional extension of the generalized Chowla-Selberg formula

TL;DR

This paper extends the generalized Chowla-Selberg formula to multidimensional inhomogeneous Epstein zeta functions, providing an exponentially convergent expression valid across the entire complex plane, with applications in physics.

Contribution

It introduces a multidimensional extension of the Chowla-Selberg formula for Epstein zeta functions, including explicit residue calculations and variations.

Findings

Derived an exponentially convergent formula valid on the whole complex plane.

Reduced the formula to the classical case of p=2, b=0, q=0.

Discussed physical applications of the extended formula.

Abstract

After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form \zeta_{A,\vec{b},q} (s) = \sum_{\vec{n}\in Z^p (\vec{n}^T A \vec{n} +\vec{b}^T \vec{n}+q)^{-s}, with $A$ the $p\times p$ matrix of a quadratic form, $\vec{b}$ a $p$ vector and $q$ a constant, is obtained. It is valid on the whole complex $s$-plane, is exponentially convergent and provides the residua at the poles explicitly. It reduces to the famous formula of Chowla and Selberg in the particular case $p=2$, $\vec{b}= \vec{0}$, $q=0$. Some variations of the formula and physical applications are considered.