Arithmetic of linear forms involving odd zeta values

Wadim Zudilin

doi:10.5802/jtnb.447

Wadim Zudilin ¹

¹ Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, Russia

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

Résumé
Abstract

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 $ζ (2)$ et $ζ (3)$ , 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 $ζ (5)$ , $ζ (7)$ , $ζ (9)$ et $ζ (11)$ 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 $ζ (2)$ and $ζ (3)$ , 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 $ζ (5)$ , $ζ (7)$ , $ζ (9)$ , and $ζ (11)$ is irrational.

MR Zbl

DOI : 10.5802/jtnb.447

Affiliations des auteurs :

Wadim Zudilin ¹

¹ Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, Russia

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Wadim Zudilin. Arithmetic of linear forms involving odd zeta values. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 251-291. doi : 10.5802/jtnb.447. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.447/

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