The Lerch zeta function IV. Hecke operators

Jeffrey C. Lagarias, Wen Ching Winnie Li

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators {Tm:m≥1} given by Tm(f)(a,c)=1m∑k=0m-1f(a+km,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∈ 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.

Original languageEnglish (US)
Article number33
JournalResearch in Mathematical Sciences
Volume3
Issue number1
DOIs
StatePublished - Dec 1 2016

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Mathematics (miscellaneous)
  • Computational Mathematics
  • Applied Mathematics

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