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.) (English. Abridged French version) Zbl 1398.11109

Fischler, Stéphane; Sprang, Johannes; Zudilin, Wadim

C. R., Math., Acad. Sci. Paris 356, No. 7, 707-711 (2018).

Authors’ abstract: In this note, we announce the following result: at least \(2^{(1 - \varepsilon) \frac{\log s}{\log \log s}}\) values of the Riemann zeta function at odd integers between 3 and \(s\) are irrational, where \(\varepsilon\) is any positive real number and \(s\) is large enough in terms of \(\varepsilon\). This improves on the lower bound \(\frac{1 - \varepsilon}{1 + \log 2} \log s\) 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.

Reviewer: Giovanni Coppola (Avellino)

MSC:

11M06

\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

Riemann zeta function; Ball-Rivoal theorem; irrational values

References:

[1]	Apéry, R., Irrationalité de \(\zeta(2)\) et \(\zeta(3)\), (Journées Arithmétiques, Luminy, 1978, Astérisque, vol. 61, (1979)), 11-13 · Zbl 0401.10049
[2]	Ball, K.; Rivoal, T., Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., 146, 1, 193-207, (2001) · Zbl 1058.11051
[3]	Fischler, S., Shidlovsky’s multiplicity estimate and irrationality of zeta values, J. Aust. Math. Soc., (2018), in press
[4]	Fischler, S.; Zudilin, W., A refinement of Nesterenko’s linear independence criterion with applications to zeta values, Math. Ann., 347, 739-763, (2010) · Zbl 1206.11088
[5]	Fischler, S.; Sprang, J.; Zudilin, W., Many odd zeta values are irrational, (2018), preprint · Zbl 1398.11109
[6]	Krattenthaler, C.; Zudilin, W., Hypergeometry inspired by irrationality questions, (2018), preprint
[7]	Nesterenko, Y., On the linear independence of numbers, Vestn. Mosk. Univ., Ser. Filos., 40, 1, 46-49, (1985), (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, 331, 4, 267-270, (2000) · Zbl 0973.11072
[9]	Rivoal, T.; Zudilin, W., A note on odd zeta values, (2018), preprint
[10]	Sprang, J., Infinitely many odd zeta values are irrational. by elementary means, (2018), preprint
[11]	Zudilin, W., Irrationality of values of the Riemann zeta function, Izv. Akad. Nauk SSSR, Ser. Mat., 66, 3, 49-102, (2002), (489-542) · Zbl 1114.11305
[12]	Zudilin, W., One of the odd zeta values from \(\zeta(5)\) to \(\zeta(25)\) is irrational. by elementary means, SIGMA, 14, 028, (2018), (8 pages) · Zbl 1445.11063