Nesterenko's linear independence criterion for vectors

Stéphane Fischler (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, 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 $\frac{2 \log a}{1+\log 2}(1+o(1))$ odd integers $i\in\{3,5,\ldots,a\}$ are such that either $\zeta(i)$ or $\zeta(i+2)$ is irrational, where $a$ is an odd integer, $a\to\infty$. This refines upon Ball-Rivoal's theorem, namely $\zeta(i) \not\in\Q$ for at least $\frac{ \log a}{1+\log 2}(1+o(1))$ such $i$.

Keywords: linear independence; Riemann zeta function; irrationality; measure of simultaneous approximation.; measure of simultaneous approximation

Domains: Number Theory [math.NT]

Stéphane Fischler. Nesterenko's linear independence criterion for vectors. 2012. ⟨hal-00665913v1⟩