Arithmetic of linear forms involving odd zeta values

Zudilin, Wadim

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 251-291.

Une construction hypergéométrique générale de formes linéaires de valeurs de la fonction zéta aux entiers impairs est présentée. Cette construction permet de retrouver les records de Rhin et Violla pour les mesures d’irrationnalité de $\zeta \left(2\right)$ et $\zeta \left(3\right)$, ainsi que d’expliquer les résultats récents de Rivoal sur l’infinité des valeurs irrationnelles de la fonction zéta aux entiers impairs et de prouver qu’au moins un des quatre nombres $\zeta \left(5\right)$, $\zeta \left(7\right)$, $\zeta \left(9\right)$ et $\zeta \left(11\right)$ est irrationnel.

A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $\zeta \left(2\right)$ and $\zeta \left(3\right)$, as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers $\zeta \left(5\right)$, $\zeta \left(7\right)$, $\zeta \left(9\right)$, and $\zeta \left(11\right)$ is irrational.

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