# The Lerch zeta function IV. Hecke operators

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Research in the Mathematical Sciences

, 3:33

First Online: 12 December 2016Received: 16 December 2015Accepted: 17 August 2016

Abstract

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators \\{ \mathrm{T} m: \, m \ge 1\}\ given by \\mathrm{T} mfa, c = \frac{1}{m} \sum {k=0}^{m-1} f\frac{a+k}{m}, mc\ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. The actions of various related operators on these function spaces are determined. It is shown that, for each \s \in {\mathbb C}\, there is a two-dimensional vector space spanned by linear combinations of Lerch zeta functions characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This is an analog of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the a, c-variables having the Lerch zeta function as an eigenfunction.

KeywordsFunctional equation Hurwitz zeta function Lerch zeta function Mathematics Subject ClassificationPrimary 11M35  Download fulltext PDF

Autor: Jeffrey C. Lagarias - Wen-Ching Winnie Li