Number theory
Many values of the Riemann zeta function at odd integers are irrational
[Beaucoup de valeurs aux entiers impairs de la fonction zêta de Riemann sont irrationnelles]

Présenté par : Jean-Pierre Serre

Fischler, Stéphane1 ; Sprang, Johannes2 ; Zudilin, Wadim3, 4
1 Laboratoire de mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
2 Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany
3 Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, the Netherlands
4 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 707-711.

Dans cette note, on annonce le résultat suivant : au moins 2(1−ε)log⁡slog⁡log⁡s valeurs de la fonction zêta de Riemann aux entiers impairs compris entre 3 and s sont irrationnelles, où ε est un réel strictement positif et s un entier impair suffisamment grand en fonction de ε. Ceci améliore la borne 1−ε1+log⁡2log⁡s qui découle du théorème de Ball–Rivoal. On donne les idées principales de la preuve, qui est fondée sur un procédé d'élimination entre des formes linéaires en les valeurs de zêta aux entiers impairs dont les coefficients sont reliés.

In this note, we announce the following result: at least 2(1−ε)log⁡slog⁡log⁡s values of the Riemann zeta function at odd integers between 3 and s are irrational, where ε is any positive real number and s is large enough in terms of ε. This improves on the lower bound 1−ε1+log⁡2log⁡s that follows from the Ball–Rivoal theorem. We give the main ideas of the proof, which is based on an elimination process between several linear forms in odd zeta values with related coefficients.

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Accepté le :
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DOI : 10.1016/j.crma.2018.05.007
Fischler, Stéphane 1 ; Sprang, Johannes 2 ; Zudilin, Wadim 3, 4

1 Laboratoire de mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
2 Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany
3 Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, the Netherlands
4 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia 
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Fischler, Stéphane; Sprang, Johannes; Zudilin, Wadim. Many values of the Riemann zeta function at odd integers are irrational. Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 707-711. doi : 10.1016/j.crma.2018.05.007. http://archive.numdam.org/articles/10.1016/j.crma.2018.05.007/

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