The Lerch Zeta Function II. Analytic Continuation

TL;DR

This paper extends the Lerch zeta function to a multivariable complex domain, analyzing its analytic continuation, monodromy, and multivalued properties on a complex manifold.

Contribution

It provides the analytic continuation of the Lerch zeta function as a multivalued function on C^3 and characterizes its monodromy on the maximal abelian cover.

Findings

Lerch zeta function is well-defined as a multivalued function on C^3 minus certain hyperplanes.

It becomes single-valued on the maximal abelian cover of the manifold.

Monodromy functions are explicitly computed and their properties analyzed.

Abstract

This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. In this paper we analytically continue it as a function of three complex variables. We that it is well defined as a multivalued function on the manifold M equal to C^3 with the hyperplanes corresponding to integer values of the two variables a and c removed. We show that it becomes single valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of their properties.