This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch @math -functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group @math , called here the sub-Jacobi group, which is a semi-direct product of @math with the Heisenberg group @math . The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold @math decomposes as @math , where each space @math consists of @math copies of an irreducible representation of @math with central character @math . The paper shows that show one can further decompose @math into irreducible @math -modules @math indexed by Dirichlet characters @math for @math , each of which carries an irreducible @math -action. On each @math there is an action of certain two-variable Hecke operators @math ; these Hecke operators have a natural global definition on all of @math , including the space of one-dimensional representations @math . For @math with @math suitable Lerch @math -functions on the critical line @math form a complete family of generalized eigenfunctions (pure continuous spectrum) for a certain linear partial differential operator @math . These Lerch @math -functions are also simultaneous eigenfunctions for all two-variable Hecke operators @math and their adjoints @math , provided @math . Lerch @math -functions are characterized by this Hecke eigenfunction property.
This work grew out of a joint project with Winnie Li @cite_18 - @cite_37 on the Lerch zeta function. I think her for helpful discussions. I thank E. M. Kiral for useful references. Initial work on this project was done at AT & T Labs-Research. The author's work related to this project has been supported by several NSF grants, currently on grant DMS-1401224.
@cite_18 @cite_37
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This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1 2. If λ = 1 2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given. 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 , 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.