zbMATH — the first resource for mathematics

Many odd zeta values are irrational. (English) Zbl 1430.11097
The main result of this paper is that if \(\varepsilon >0\) and \(s\geq 3\) is an odd number sufficiently large with respect to \(\varepsilon\), then among the numbers \(\zeta(3), \zeta(5), \zeta(7),\dots ,\zeta(s)\) at least \[ 2^{\frac{(1-\varepsilon)(\log s)}{\log \log s}} \] are irrational. The proof is based on the suitable estimations of specific linear forms.

11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
Full Text: DOI arXiv
[1] Apéry, R., Irrationalité de 𝜁(2) et 𝜁(3), Journées Arithmétiques (Luminy, 1978), 11-13, (1979), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 0401.10049
[2] Ball, K.; Rivoal, T., Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., 146, 193-207, (2001) · Zbl 1058.11051
[3] De Bruijn, N. G., Asymptotic methods in analysis, (1981), Dover Publications: Dover Publications, New York · Zbl 0556.41021
[4] Colmez, P., Arithmétique de la fonction zêta, Journées mathématiques X-UPS 2002, 37-164, (2003), éditions de l’école Polytechnique: éditions de l’école Polytechnique, Palaiseau
[5] Fekete, M.; Pólya, G., Über ein Problem von Laguerre, Rend. Circ. Mat. Palermo, 34, 89-120, (1912) · JFM 43.0145.02
[6] Fischler, S., Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, …), Sém. Bourbaki 2002/03, (2004), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 1101.11024
[7] Fischler, S., Shidlovsky’s multiplicity estimate and irrationality of zeta values, J. Aust. Math. Soc., 105, 145-172, (2018) · Zbl 1432.11084
[8] Fischler, S.; Sprang, J.; Zudilin, W., Many values of the Riemann zeta function at odd integers are irrational, C. R. Math. Acad. Sci. Paris, 356, 707-711, (2018) · Zbl 1398.11109
[9] 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
[10] Fulton, W.; Harris, J., Representation theory: a first course, (1991), Springer: Springer, New York · Zbl 0744.22001
[11] Gantmacher, F.; Krein, M., Oscillation matrices and kernels and small vibrations of mechanical systems, (2002), AMS Chelsea Publishing: AMS Chelsea Publishing, Providence, RI · Zbl 1002.74002
[12] Hardy, G.; Wright, E., An introduction to the theory of numbers, (1979), Oxford University Press: Oxford University Press, Oxford · Zbl 0423.10001
[13] Krattenthaler, C., Advanced determinant calculus, Sém. Lotharingien Combin., 42, (1999)
[14] Krattenthaler, C.; Rivoal, T., Hypergéométrie et fonction zêta de Riemann, Mem. Amer. Math. Soc., 186, (2007) · Zbl 1113.11039
[15] Krattenthaler, C.; Zudilin, W., Hypergeometry inspired by irrationality questions, Kyushu J. Math., 73, 189-203, (2019)
[16] Laurent, M.; Mignotte, M.; Nesterenko, Y., Formes linéaires en deux logarithmes et déterminants d’interpolation, J. Number Theory, 55, 285-321, (1995) · Zbl 0843.11036
[17] Macdonald, G., Symmetric functions and Hall polynomials, (1979), Clarendon Press: Clarendon Press, Oxford · Zbl 0487.20007
[18] Nash, M. H.
[19] Nesterenko, Y., On the linear independence of numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], 40, 46-49, (1985)
[20] Nishimoto, M.
[21] Proctor, R., Equivalence of the combinatorial and the classical definitions of Schur functions, J. Combin. Theory, Ser. A, 51, 135-137, (1989) · Zbl 0691.20012
[22] Rivoal, T., La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris, Ser. I, 331, 267-270, (2000) · Zbl 0973.11072
[23] Rivoal, T., Irrationalité d’au moins un des neuf nombres 𝜁(5), 𝜁(7), …, 𝜁(21), Acta Arith., 103, 157-167, (2002) · Zbl 1015.11033
[24] Rivoal, T.; Zudilin, W., Diophantine properties of numbers related to Catalan’s constant, Math. Ann., 326, 705-721, (2003) · Zbl 1028.11046
[25] Rivoal, T.; Zudilin, W.
[26] Sprang, J.
[27] Zudilin, W., One of the numbers 𝜁(5), 𝜁(7), 𝜁(9), 𝜁(11) is irrational, Uspekhi Mat. Nauk [Russian Math. Surveys], 56, 149-150, (2001) · Zbl 1047.11072
[28] Zudilin, W., Irrationality of values of the Riemann zeta function, Izvestiya Ross. Akad. Nauk Ser. Mat. [Izv. Math.], 66, 49-102, (2002)
[29] Zudilin, W., One of the odd zeta values from 𝜁(5) to 𝜁(25) is irrational. By elementary means, SIGMA Symmetry Integrability Geom. Methods Appl., 14, (2018) · Zbl 1445.11063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.