Zeta functions on tori using contour integration

TL;DR

This paper introduces a novel contour integration approach to zeta functions on complex tori, providing new proofs and identities, and offering insights into their functional determinants without relying on traditional formulas.

Contribution

It presents a new contour integration method for zeta functions on tori, offering alternative proofs and identities, and simplifies the understanding of functional determinants.

Findings

Agreement with Chowla-Selberg series formula

New proof of functional determinants on the torus

Derived identities involving Dedekind eta function

Abstract

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla-Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well.