Nesterenko's linear independence criterion for vectors

摘要

In this paper we deduce a lower bound for the rank of a family of $p$ vectorsin $\R^k$ (considered as a vector space over the rationals) from the existenceof a sequence of linear forms on $\R^p$, with integer coefficients, which aresmall at $k$ points. This is a generalization to vectors of Nesterenko's linearindependence criterion (which corresponds to $k=1$), used by Ball-Rivoal toprove that infinitely many values of Riemann zeta function at odd integers areirrational. The proof is based on geometry of numbers, namely Minkowski'stheorem on convex bodies.