×

zbMATH — the first resource for mathematics

A note on odd zeta values. (English) Zbl 1470.11203
Summary: Using a new construction of rational linear forms in odd zeta values [the second author, SIGMA, Symmetry Integrability Geom. Methods Appl. 14, Paper 028, 8 p. (2018; Zbl 1445.11063)] and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values \(\zeta(5), \zeta (7), \ldots, \zeta (69)\).

MSC:
11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
Citations:
Zbl 1445.11063
PDF BibTeX XML Cite
Full Text: Link
References:
[1] R. Ap´ery,Irrationalit´e deζ(2)etζ(3), Ast´erisque61(1979), 11-13.
[2] K. M. Ball, T. Rivoal,Irrationalit´e d’une infinit´e de valeurs de la fonction zˆeta aux entiers impairs, Invent. Math.146.1 (2001), 193-207.
[3] S. Fischler, W. Zudilin,A refinement of Nesterenko’s linear independence criterion with applications to zeta values, Math. Ann.347(2010), 739-763. · Zbl 1206.11088
[4] S. Fischler, T. Rivoal,Linear independence of values ofG-functions, preprint 2017, 46 pages, to appear in J. Europ. Math. Soc., doi: 10.4171/JEMS/950.
[5] C. Krattenthaler, T. Rivoal,Hyperg´eom´etrie et fonction zˆeta de Riemann, Mem. Amer. Math. Soc. 186(2007), 93 pages.
[6] C. Krattenthaler, W. Zudilin,Hypergeometry inspired by irrationality questions, Kyushu J. Math.73 (2019), 189-203. · Zbl 1450.11072
[7] T. Rivoal,La fonction zˆeta de Riemann prend une infinit´e de valeurs irrationnelles aux entiers impairs, Comptes Rendus Acad. Sci. Paris S´er. I Math.331.4 (2000), 267-270. · Zbl 0973.11072
[8] T. Rivoal,Irrationalit´e d’au moins un des neuf nombresζ(5), ζ(7), . . . , ζ(21), Acta Arith.103.2 (2002), 157-167. · Zbl 1015.11033
[9] J. Sprang,Infinitely many odd zeta values are irrational. By elementary means, preprint 2018, 11 pages, arXiv: 1802.09410 [math.NT]. · Zbl 1430.11097
[10] W. Zudilin,One of the numbersζ(5), ζ(7), ζ(9), ζ(11)is irrational, Russian Math. Surveys56.4 (2001), 774-776. · Zbl 1047.11072
[11] W. Zudilin,Irrationality of values of the Riemann zeta function, Izv. Math.66.3 (2002), 489-542. · Zbl 1114.11305
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.