Riemann, Hurwitz and Hurwitz-Lerch Zeta Functions and Associated Series and Integrals

Abstract

The main object of this article is to present a survey-cum-expository account of some recent developments involving the Riemann Zeta function \(\zeta (s)\), the Hurwitz (or generalized) Zeta function \(\zeta (s,a)\), and the Hurwitz-Lerch Zeta function \(\Phi (z,s,a)\) as well as its various interesting extensions and generalizations. We first investigate the problems associated with the evaluations and representations of \(\zeta \left (s\right )\) when \(s \in \mathbb{N} \setminus \left \{1\right \}\), \(\mathbb{N}\) being the set of natural numbers, emphasizing upon several interesting classes of rapidly convergent series representations for \(\zeta \left (2n + 1\right )\) \(\left (n \in \mathbb{N}\right )\) which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that \(\zeta \left (3\right )\) can be represented by means of series which converge much more rapidly than that in Euler’s celebrated formula as well as the series which was used more recently by Roger Apery (1916–1994) in his proof of the irrationality of \(\zeta \left (3\right )\). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places. We also consider a variety of series and integrals associated with the Hurwitz-Lerch Zeta function \(\Phi (z,s,a)\) as well as its various interesting extensions and generalizations.