Generalizations of the Lerch zeta function.

Abstract

The Lerch zeta function is a three-variable generalization of the Riemann zeta function and the Hurwitz zeta function. In this thesis, we study generalizations and analogues of the Lerch zeta function. Our approaches proceed along three directions: local, global and classical.

In the local study, we construct a new family of local zeta integrals, called local Lerch-Tate zeta integrals. These local zeta integrals have analytic continuation and functional equation which resembles the functional quation of the Lerch zeta function. Our local zeta integrals generalize Tate's local zeta integrals.

In the global investigation, we introduce a family of global zeta integrals over global number field and a family of global zeta integrals over global function field; these families are called global Lerch-Tate zeta integrals. These global zeta integrals converge absolutely on a right half-plane; they have meromorphic continuation and functional equation. Our global zeta integrals generalize Tate's global zeta integrals. In the special case when the number field is the field of rational numbers, specializations of these global zeta integrals give the symmetrized Lerch zeta functions.

In the classical approach, we study a generalized Lerch zeta function with four variables. We compute its Fourier expansion and establish its analytic continuation.