The Lerch zeta function II. Analytic continuation

Jeffrey C. Lagarias; Wen-Ching Winnie Li

doi:10.1515/form.2011.048

Article

The Lerch zeta function II. Analytic continuation

Jeffrey C. Lagarias and Wen-Ching Winnie Li

Published/Copyright: January 19, 2012

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Forum Mathematicum Volume 24 Issue 1

Abstract.

This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. The Lerch zeta function (s,a,c):=n=0e2ina(n+c)s was introduced by Lipschitz in 1857, and is named after Lerch, who showed in 1887 that it satisfied a functional equation. Here we analytically continue (s,a,c) as a function of three complex variables. We show that it is well-defined as a multivalued function on the manifold :=(s,a,c)()(), and that this analytic continuation becomes single-valued on the maximal abelian cover of . We compute the monodromy functions describing the multivalued nature of this function on , and determine various of its properties.

Keywords.: Hurwitz zeta function; Lerch zeta function; polylogarithm

Received: 2008-10-05

Revised: 2010-02-07

Published Online: 2012-01-19

Published in Print: 2012-January

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/form.2011.048

Keywords for this article

Hurwitz zeta function; Lerch zeta function; polylogarithm