2015/11/30 by Jeffrey C. Lagarias, Wen-Ching Winnie Li · 2 citations
Mathematics · #Advanced Mathematical Identities #Algebra over a field #Algebraic number #Analytic Number Theory Research #Arithmetic zeta function #Bernoulli polynomials #Differential operator #Digamma function #Mathematical functions and polynomials #Polylogarithm #Riemann zeta function #Space (punctuation) #math.NT #msc:11M35
paper · pdf · doi:10.1186/s40687-016-0082-9
published in Research in the Mathematical Sciences 3(1) (Springer Nature) · 40 pages, preliminary version; v2 41 pages, preliminary version2, v3 42 pages, revised version for journal
openalex created_date 2016/06/24 · arxiv created 2016/08/17 · openalex publication_date 2016/08/30 · arxiv updated 2017/08/07 · openalex updated_date 2026/08/06
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) = (1)/(m) ∑ k=0m-1 f((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 ∈ \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.