On Plouffe's Ramanujan Identities

Details On Plouffe's Ramanujan Identities On Plouffe's Ramanujan Identities

2006-09-28

arxiv.org/abs/math/0609775v4

Mathematics

Number Theory

19 pages, 3 figures; v4: minor corrections; modified intro; revised concluding statements

Scientific paper

10.1007/s11139-011-9335-9

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 Apery'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.

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