Hyperg{é}om{é}trie et fonction z{ê}ta de Riemann

Details Hyperg{é}om{é}trie et fonction z{ê}ta de Riemann Hyperg{é}om{é}trie et fonction z{ê}ta de Riemann

2003-11-07

arxiv.org/abs/math/0311114v4

Mem. Amer. Math. Soc. 186, no. 875, Providence, R. I., 2007.

Mathematics

Number Theory

AmS-LaTeX, 73 pages; completely rewritten: (1) The strategy for proving the theorems for the coefficient p_0 was changed. The

Scientific paper

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.

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