RAMANUJAN’S FORMULA FOR ζ(2n+ 1)

not only provides an elegant formula for evaluating ζ(2n), but it also tells us of the arithmetical nature of ζ(2n). In contrast, we know very little about ζ(2n + 1). One of the major achievements in number theory in the past half-century is R. Apéry’s proof that ζ(3) is irrational [2], but for n ≥ 2, the arithmetical nature of ζ(2n+ 1) remains open. Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for ζ(2n + 1). To be sure, Ramanujan’s formula does not possess the elegance of (1.1), nor does it provide any arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. Theorem 1.1 (Ramanujan’s formula for ζ(2n+1)). Let Br, r ≥ 0, denote the r-th Bernoulli number. As usual, set σk(n) = ∑ d|n d . If α and β are positive numbers such that αβ = π, and if n is a positive integer, then