Hyperg{\'e}om{\'e}trie et fonction z{\^e}ta de Riemann

Abstract

We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over $\mathbb Q$ spanned by $1,\zeta(m),\zeta(m+2),...,\zeta(m+2h)$, where $m$ and $h$ are integers such that $m\ge2$ and $h\ge0$. In particular, we immediately get the following results as corollaries: at least one of the eight numbers $\zeta(5),\zeta(7),...,\zeta(19)$ is irrational, and there exists an odd integer $j$ between 5 and 165 such that 1, $\zeta(3)$ and $\zeta(j)$ are linearly independent over $\mathbb{Q}$. This strengthens some recent results in [41] and [8], respectively. We also prove a related conjecture, due to Vasilyev [49], and as well a conjecture, due to Zudilin [55], on certain rational approximations of $\zeta(4)$. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews [3]. We hope that it will be possible to apply our construction to the more general linear forms constructed by Zudilin [56], with the ultimate goal of strengthening his result that one of the numbers $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational.