The Ramanujan Journal

, Volume 27, Issue 3, pp 387–408

On Plouffe’s Ramanujan identities

Article

DOI: 10.1007/s11139-011-9335-9

Cite this article as:
Vepštas, L. Ramanujan J (2012) 27: 387. doi:10.1007/s11139-011-9335-9

Abstract

Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry’s constant given by Ramanujan:
$$\zeta(3)=\frac{7\pi^{3}}{180}-2\sum_{n=1}^{\infty}\frac {1}{n^{3}(e^{2\pi n}-1)}.$$
Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe’s identities as corollaries. The resemblance of the general relation to the structure of theta functions and modular forms is briefly sketched.

Keywords

Apéry’s constant Theta function Modular form 

Mathematics Subject Classification (2000)

11M06 

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.OpenCog ProjectAustinUSA