TY - JOUR

T1 - The Lerch zeta function IV. Hecke operators

AU - Lagarias, Jeffrey C.

AU - Li, Wen Ching Winnie

N1 - Funding Information:
The authors thank Paul Federbush for helpful remarks regarding Fourier expansions in Theorem 8. The authors thank the two reviewers for many helpful comments and corrections. In particular, we thank one of them for the observation (41) which strengthened Theorem 2.1 (3). This project was initiated at AT&T Labs-Research when the first author worked there and the second author consulted there; they thank AT&T for support. The first author received support from the Mathematics Research Center at Stanford University in 2009–2010. The second author received support from the National Center for Theoretical Sciences and National Tsing Hua University in Taiwan in 2009–2014. To these institutions the authors express their gratitude. The research of the first author was supported by NSF Grants DMS-0801029, DMS-1101373 and DMS-1401224, that of the second author by NSF Grant DMS-1101368 and Simons Foundation Grant #355798.
Publisher Copyright:
© 2016, The Author(s).

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

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U2 - 10.1186/s40687-016-0082-9

DO - 10.1186/s40687-016-0082-9

M3 - Article

AN - SCOPUS:85043504602

VL - 3

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

SN - 2522-0144

IS - 1

M1 - 33

ER -