DISTRIBUTION OF IRRATIONAL ZETA VALUES

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