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A formula of S. Ramanujan. (English) Zbl 0606.10032
While editing the notebooks of Ramanujan, Berndt and Joshi discovered an incorrect formula. The present author, in an attempt to obtain a correct version, was led to consider 12 classes of infinite series of the form \(\sum^{\infty}_{r=1}\sum^{r}_{k=1}\) or \(\sum^{\infty}_{r=1}\sum^{2r-1}_{k=1}\) where typical summands are \(r^{-s} k^{-1},\quad r^{-s} (k+r)^{-1},\) and \(r^{-s} (-1)^{k- 1} k^{-1}.\) The author provides a very useful review of the literature on these and similar series and gives elementary proofs of a variety of new and old results. He evaluates several of the series explicitly in terms of \(\zeta\) (2), \(\zeta\) (3) and Catalan’s constant and derives a number of expressions for Ramanujan’s series.
Reviewer: W.E.Briggs

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M99 Zeta and \(L\)-functions: analytic theory
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