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. We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively. We prove a two-part theorem on local ideal zeta functions of Lie rings of upper-triangular matrices. First, we prove that these local zeta functions display a strong uniformity. Secondly, we prove that these zeta functions satisfy a local functional equation. Some explicit examples of these zeta functions are also presented. Finally, we consider certain quotients of these Lie rings, showing that the strong uniformity continues to hold, and that under certain circumstances the functional equation does too.