CONTRIBUTIONS TO THE THEORY OF THE HURWITZ-LERCH ZETA-FUNCTION

Author:

Masahiro HASHIMOTO

AMS Classification:

11R29, 33B15, 11R11

Keywords:

Dirichlet L-function, integral representation, Hurwitz-Lerch transcendent, periodic summation formulas

Abstract:

In this paper, we shall prove useful analogues of known results on the Hurwitz zeta-function as presented by [14]. Naturally there can be two natural analogues, one with an additive character (the exponential function) and the other with a multiplicative character (Dirichlet character) both to the fixed modulus tex:q. It turns out that in the case of the perturbed Dirichlet tex:L-function, there appear the generalized Bernoulli polynomials introduced by Leopoldt and the results obtained are in conformity with what has been known, while the remaining case of the Lerch zeta-function, what appears is a generalization of the cotangent function in the context of [12] and presents new aspects comparable to those of [15]. The main results are of the same form as before and manifest the principle that an essential part already describes the whole, i.e., from the integral representation for the partial sum, we may immediately deduce that of the whole zeta-function and its approximation by the partial sum, together with the Ewald expansion which in turn implies the functional equation. The main tool is an analogue of the Euler-Maclaurin summation formula for periodic sequences similar to that developed by Berndt [4] and Berndt and Schoenfeld [5], but here we use a unified form due to Rane ([23], [24]), in simplified and handier form, whose proof is to appear in the author thesis.

Download paper:

hashimoto-08.pdf

Vol. 3 (11), 2008