Hypergeometric series and the Riemann zeta function

Hypergeometric Functions that Generate Series Acceleration Formulae for Values of the Riemann Zeta Function

was first obtained by A. Markov in 1890, but became more widely known after its appearance in connection with Apéry’s proof of the irrationality of ζ(3). We outline here the role that hypergeometric functions play in generating more general series acceleration formulae for values of the Riemann zeta function at the positive integers. At odd positive integers, we have the formulae of Koecher, an...

Moments of the Riemann Zeta Function and Eisenstein Series I

Hypergeometric Series Associated with the Hurwitz-lerch Zeta Function

The present work is a sequel to the papers [3] and [4], and it aims at introducing and investigating a new generalized double zeta function involving the Riemann, Hurwitz, Hurwitz-Lerch and Barnes double zeta functions as particular cases. We study its properties, integral representations, differential relations, series expansion and discuss the link with known results.

Notes Relating to Newton Series for the Riemann Zeta Function

This paper consists of the extended working notes and observations made during the development of a joint paper[?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably ...

q-Riemann zeta function

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...