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arXiv · 1310.1685

Distribution of irrational zeta values

Stéphane Fischler
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Abstract

In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer $a$ sufficiently large in terms of $ε>0$, there exist $[ \frac{(1-ε)\log a}{1+\log 2}]$ odd integers $s$ between 3 and $a$, with distance at least $a^ε$ 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 $ζ(s)$ 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.

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Stéphane Fischler. 2013-10-07. Distribution of irrational zeta values. https://arxiv.org/abs/1310.1685

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