EMPIRICALLY DETERMINED APÉRY-LIKE FORMULAE FOR ζ(4n+3)

JONATHAN BORWEIN AND DAVID BRADLEY

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) that 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 nontrivial 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 formula that provides strange evaluations of a large new class of nonterminating hypergeometric series.

[Editor’s Note: The beautiful formulas in this paper are no longer conjectural.]