Distribution of irrational zeta values

Abstract

In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer $a$ sufficiently large in terms of $\epsilon>0$, there exist $[ \frac{(1-\epsilon)\log a}{1+\log 2}]$ odd integers $s$ between 3 and $a$, with distance at least $a^{\epsilon}$ from one another, at which Riemann zeta function takes $\Q$-linearly independent values. As a consequence, if there are very few integers $s$ such that $\zeta(s)$ is irrational, then they are rather evenly distributed. The proof involves series of hypergeometric type estimated by the saddle point method, and the generalization to vectors of Nesterenko's linear independence criterion.