The Lerch zeta function IV. Hecke operators

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 ≥ 1 } given by T m ( f ) ( a , c ) = 1 m ∑ k = 0 m - 1 f ( a + k m , m c ) acting on certain spaces of real-analytic functions, including Lerch...

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Bibliographic Details
Published in:Research in the mathematical sciences 2016-12, Vol.3 (1), p.1-39, Article 33
Main Authors: Lagarias, Jeffrey C., Li, Wen-Ching Winnie
Format: Article
Language:English
Subjects:
Analytic functions
Applications of Mathematics
Computational Mathematics and Numerical Analysis
Differential equations
Eigenvectors
Mathematical analysis
Mathematics
Mathematics and Statistics
Operators (mathematics)
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Summary: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 ≥ 1 } given by T m ( f ) ( a , c ) = 1 m ∑ k = 0 m - 1 f ( a + k m , m c ) 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.
ISSN:2197-9847
2522-0144
2197-9847
DOI:10.1186/s40687-016-0082-9