## Experimental Mathematics

- Experiment. Math.
- Volume 6, Issue 3 (1997), 181-194.

### Empirically determined Apéry-like formulae for {$\zeta(4n+3)$}

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 \zet$(4n+3)$ that generalizes Apéry's series for \zet$(3)$, and appears to give the best possible series relations of this type, at least for n{\mathversion{normal}$\,<\,$}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. See note on page 194.]

#### Article information

**Source**

Experiment. Math., Volume 6, Issue 3 (1997), 181-194.

**Dates**

First available in Project Euclid: 17 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1047920419

**Mathematical Reviews number (MathSciNet)**

MR1481588

**Zentralblatt MATH identifier**

0887.11037

**Subjects**

Primary: 11Y60: Evaluation of constants

Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

#### Citation

Borwein, Jonathan; Bradley, David. Empirically determined Apéry-like formulae for {$\zeta(4n+3)$}. Experiment. Math. 6 (1997), no. 3, 181--194. https://projecteuclid.org/euclid.em/1047920419