no. 7
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.

Reçu le :
Accepté le :
Publié le :
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://www.numdam.org/articles/10.1016/j.crma.2018.05.007/

[1] Apéry, R. Irrationalité de ζ(2) et ζ(3), Luminy, 1978 (Astérisque), Volume vol. 61 (1979), pp. 11-13

[2] Ball, K.; Rivoal, T. Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., Volume 146 (2001) no. 1, pp. 193-207

[3] Fischler, S. Shidlovsky's multiplicity estimate and irrationality of zeta values, J. Aust. Math. Soc. (2018) (preprint in press) | arXiv

[4] Fischler, S.; Zudilin, W. A refinement of Nesterenko's linear independence criterion with applications to zeta values, Math. Ann., Volume 347 (2010), pp. 739-763

[5] Fischler, S.; Sprang, J.; Zudilin, W. Many odd zeta values are irrational, 2018 (preprint) | arXiv

[6] Krattenthaler, C.; Zudilin, W. Hypergeometry inspired by irrationality questions, 2018 (preprint) | arXiv

[7] Nesterenko, Y. On the linear independence of numbers, Vestn. Mosk. Univ., Ser. Filos. (Mosc. Univ. Math. Bull.), Volume 40 (1985) no. 1, pp. 46-49 (69–74)

[8] Rivoal, T. La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000) no. 4, pp. 267-270

[9] Rivoal, T.; Zudilin, W. A note on odd zeta values, 2018 (preprint) | arXiv

[10] Sprang, J. Infinitely many odd zeta values are irrational. By elementary means, 2018 (preprint) | arXiv

[11] Zudilin, W. Irrationality of values of the Riemann zeta function, Izv. Akad. Nauk SSSR, Ser. Mat. (Izv. Math.), Volume 66 (2002) no. 3, pp. 49-102 (489–542)

[12] Zudilin, W. One of the odd zeta values from ζ(5) to ζ(25) is irrational. By elementary means, SIGMA, Volume 14 (2018) no. 028 (8 pages)

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