Complete determination of the singularity structure of zeta functions

TL;DR

This paper fully characterizes the singularity structure of Epstein-type zeta functions using the Wodzicki residue, revealing all pole residues and advancing the understanding of their analytic continuation in physical applications.

Contribution

It provides a complete determination of the singularities of these zeta functions via the generalized Wodzicki residue, extending previous partial results.

Findings

Residues of all poles are obtained using the Wodzicki residue.

The regular part of the analytic continuation is a convergent or asymptotic series.

The methods are applied to physically relevant Epstein-type zeta functions.

Abstract

Series of extended Epstein type provide examples of non-trivial zeta functions with important physical applications. The regular part of their analytic continuation is seen to be a convergent or an asymptotic series. Their singularity structure is completely determined in terms of the Wodzicki residue in its generalized form, which is proven to yield the residua of all the poles of the zeta function, and not just that of the rightmost pole (obtainable from the Dixmier trace). The calculation is a very down-to-earth application of these powerful functional analytical methods in physics.