Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-20T02:43:04.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Hypergeometric rational approximations to ζ(4)

Part of: Diophantine approximation, transcendental number theory Computational number theory

Published online by Cambridge University Press:  03 February 2020

Raffaele Marcovecchio  and
Wadim Zudilin
Raffaele Marcovecchio
Affiliation:
Dipartimento di Ingegneria e Geologia, Università di Chieti-Pescara, Viale Pindaro, 42, 65127Pescara, Italy (raffaele.marcovecchio@unich.it)
Wadim Zudilin
Affiliation:
Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500GL Nijmegen, Netherlands (w.zudilin@math.ru.nl)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).

Keywords

irrationality measureπ4hypergeometric functionhypergeometric integralrational approximation

MSC classification

Primary: 11J82: Measures of irrationality and of transcendence
Secondary: 11Y60: Evaluation of constants 33C20: Generalized hypergeometric series, $_pF_q$ 33C60: Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 374 - 397
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Edinburgh Mathematical Society 2020

References

1.Apéry, R., Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 11–13.Google Scholar
2.Bailey, W. N., Generalized hypergeometric series, Cambridge Tracts in Mathematics, Volume 32 (Cambridge University Press, 1935).Google Scholar
3.Ball, K. and Rivoal, T., Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 146(1) (2001), 193–207.CrossRefGoogle Scholar
4.Beukers, F., A note on the irrationality of ζ(2) and ζ(3), Bull. Lond. Math. Soc. 11(3) (1979), 268–272.CrossRefGoogle Scholar
5.Cohen, H., Accélération de la convergence de certaines récurrences linéaires, Semin. Theor. Nombres Bordeaux 10 1980–1981), exp. 16, 2 pages.Google Scholar
6.Fischler, S., Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, …), Astérisque 294 (2004), 27–62.Google Scholar
7.Fischler, S., Sprang, J. and Zudilin, W., Many odd zeta values are irrational, Compos. Math. 155(5) (2019), 938–952.CrossRefGoogle Scholar
8.Gutnik, L. A., On the irrationality of certain quantities involving ζ(3), Uspekhi Mat. Nauk. [Russian Math. Surveys] 34(3) (1979), 190; Acta Arith. 42(3) (1983), 255–264.Google Scholar
9.Hata, M., Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99–125.Google Scholar
10.Krattenthaler, C. and Rivoal, T., Hypergéométrie et fonction zêta de Riemann, Memoirs of the American Mathematical Society, Volume 185 (American Mathematical Society, 2007).CrossRefGoogle Scholar
11.Krattenthaler, C. and Zudilin, W., Hypergeometry inspired by irrationality questions, Kyushu J. Math. 73(1) (2019), 189–203.CrossRefGoogle Scholar
12.Marcovecchio, R., The Rhin–Viola method for log 2, Acta Arith. 139(2) (2009), 147–184.CrossRefGoogle Scholar
13.Nesterenko, Yu. V., A few remarks on ζ(3), Mat. Zametki [Math. Notes] 59(6) (1996), 865–880.Google Scholar
14.Rhin, G. and Viola, C., On a permutation group related to ζ(2), Acta Arith. 77(3) (1996), 23–56.CrossRefGoogle Scholar
15.Rhin, G. and Viola, C., The group structure for ζ(3), Acta Arith. 97(3) (2001), 269–293.CrossRefGoogle Scholar
16.Rivoal, T., La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Math. Acad. Sci. Paris Ser. I 331(4) (2000), 267–270.CrossRefGoogle Scholar
17.Sorokin, V. N., On algorithm for fast calculation of π4, preprint (Russian Academy of Sciences, M.V. Keldysh Institute for Applied Mathematics, Moscow 2002), 46 pages, available at http://mi.mathnet.ru/eng/ipmp1002.Google Scholar
18.van der Poorten, A., A proof that Euler missed…Apéry's proof of the irrationality of ζ(3) (an informal report), Math. Intelligencer 1(4) (1978/79), 195–203.CrossRefGoogle Scholar
19.Zudilin, W., Well-poised hypergeometric service for diophantine problems of zeta values, J. Théor. Nombres Bordeaux 15(2) (2003), 593–626.CrossRefGoogle Scholar
20.Zudilin, W., Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux 16(1) (2004), 251–291.CrossRefGoogle Scholar
21.Zudilin, W., Well-poised hypergeometric transformations of Euler-type multiple integrals, J. Lond. Math. Soc. (2) 70(1) (2004), 215–230.CrossRefGoogle Scholar
22.Zudilin, W., Two hypergeometric tales and a new irrationality measure of ζ(2), Ann. Math. Qué. 38(1) (2014), 101–117.CrossRefGoogle Scholar