A simple series representation for Apéry's constant

Abstract

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. ThusThis constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).