Distribution of irrational zeta values

Stéphane Fischler (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Linear independence; irrationality; Riemann zeta function; series of hypergeometric type; saddle point method.; saddle point method

Domaines: Théorie des nombres [math.NT]

Stéphane Fischler. Distribution of irrational zeta values. Bulletin de la société mathématique de France, 2017, 145 (3), pp.381-409. ⟨hal-00870284⟩