M ay 2 00 5 Empirically Determined Apéry - Like Formulae for ζ ( 4 n + 3 )

Abstract. Some rapidly convergent formulae for special values of the Riemann Zeta function are given. We obtain a generating function formula for ζ(4n+3) which generalizes Apéry’s series for ζ(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. It should be emphasized that our main results are shown equivalent but are still only conjectures.