The Lerch zeta function IV. Hecke operators

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 $s\in \mathbf{C}$ 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.

40 pages, preliminary version; v2 41 pages, preliminary version2, v3 42 pages, revised version for journal

References in corpus (1)

• Des equations de Dirac et de Schrodinger pour la transformation de Fourier

Cited by in corpus (1)

• Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian