Nesterenko’s linear independence criterion for vectors

In this paper we deduce a lower bound for the rank of a family of p vectors in Rk (considered as a vector space over the rationals) from the existence of a sequence of linear forms on Rp, with integer coefficients, which are small at k points. This is a generalization to vectors of Nesterenko’s linear independence criterion (which corresponds to k = 1). It enables one to make use of some known constructions of linear forms small at several points, related to Padé approximation. As an application, we prove that at least 2 log a 1+log 2(1+ o(1)) odd integers i ∈ {3, 5, . . . , a} are such that either ζ(i) or ζ(i + 2) is irrational, where a is an odd integer, a → ∞. This refines upon Ball-Rivoal’s theorem, namely ζ(i) 6∈ Q for at least log a 1+log 2(1 + o(1)) such i.