Some properties and results involving the zeta and associated functions

In this research-cum-expository article, we aim at presenting a systematic account of some recent developments involving the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta function ζ(s, a), and the Hurwitz-Lerch Zeta function Φ(z, s, a) as well as its various interesting extensions and generalizations. In particular, we begin by looking into the problems associated with the evaluations and representations of ζ (s) when s ∈ N \ {1}, N being the set of natural numbers, emphasizing upon various potentially useful and computationally friendly classes of rapidly convergent series representations for ζ (2n+ 1) (n ∈ N) which have been developed in recent years. We then turn toward some other investigations involving certain general classes of Goldbach-Euler type sums. Finally, we present a systematic investigation of various properties and results involving several families of generating functions and their partial sums which are associated with the aforementioned general classes of the extended Hurwitz-Lerch Zeta functions. References to some of latest developments in the theory and applications of several families of the extended Hurwitz-Lerch zeta functions are also provided for the interested researchers on these and other related topics in Analytic Number Theory, Geometric Function Theory of Complex Analysis, and so on.