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Christian Krattenthaler and Tanguy Rivoal

Hypergéométrie et fonction zêta de Riemann

(73 pages)

English 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 Q spanned by 1,\zeta(m),\zeta(m+2),...,\zeta(m+2h), where m and h are integers such that m>=2 and h>=0. 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 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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