The Lerch zeta function IV. Hecke operators

TL;DR

This paper explores the algebraic and analytic properties of the Lerch zeta function, introducing a family of Hecke operators acting on related function spaces, and characterizing Lerch zeta functions as eigenfunctions of these operators.

Contribution

It defines new two-variable Hecke operators for Lerch zeta functions and characterizes these functions as eigenfunctions, extending Milnor's results for the Hurwitz zeta function.

Findings

Hecke operators act on Lerch zeta functions and related spaces.

Lerch zeta functions are characterized as eigenfunctions of these operators.

A connection to a linear partial differential operator is established.

Abstract

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue 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.