The Group Structure for Ζ(3)

1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. .. and µ(ζ(3)) < 13.41782. .. Several improvements on such irrationality measures were subsequently given, and we refer to the introductions of the papers [3] and [4] for an account of these results. As usual, we denote here by µ(α) the least irrationality measure of an irrational number α, i.e., the least exponent λ such that for any ε > 0 there exists a constant q 0 = q 0 (ε) > 0 for which α −