Empirically determined Apery-like formulae for zeta(4n+3)

Abstract

Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series relations of this type, at least for n<12. The formula reduces to a finite but apparently non-trivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a beautiful formula which provides strange evaluations of a large new class of non-terminating hypergeometric series. Our main results are shown to be equivalent. At the time this article was submitted for publication back in 1996, these results were only conjectures, but they have subsequently been proved as a result of work due to Gert Almkvist and Andrew Granville.