Arithmetic of linear forms involving odd zeta values
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 251-291.

A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of ζ(2) and ζ(3), as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ(5), ζ(7), ζ(9), and ζ(11) is irrational.

Une construction hypergéométrique générale de formes linéaires de valeurs de la fonction zéta aux entiers impairs est présentée. Cette construction permet de retrouver les records de Rhin et Violla pour les mesures d’irrationnalité de ζ(2) et ζ(3), ainsi que d’expliquer les résultats récents de Rivoal sur l’infinité des valeurs irrationnelles de la fonction zéta aux entiers impairs et de prouver qu’au moins un des quatre nombres ζ(5), ζ(7), ζ(9) et ζ(11) est irrationnel.

DOI: 10.5802/jtnb.447
Wadim Zudilin 1

1 Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, Russia
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Wadim Zudilin. Arithmetic of linear forms involving odd zeta values. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 251-291. doi : 10.5802/jtnb.447. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.447/

[Ap] R. Apéry, Irrationalité de ζ(2) et ζ(3). Astérisque 61 (1979), 11–13. | Zbl

[Ba1] W. N. Bailey, Some transformations of generalized hypergeometric series, and contour integrals of Barnes’s type. Quart. J. Math. Oxford 3:11 (1932), 168–182. | Zbl

[Ba2] W. N. Bailey, Transformations of well-poised hypergeometric series. Proc. London Math. Soc. II Ser. 36:4 (1934), 235–240. | Zbl

[Ba3] W. N. Bailey, Generalized hypergeometric series. Cambridge Math. Tracts 32 (Cambridge University Press, Cambridge, 1935); 2nd reprinted edition (Stechert-Hafner, New York, NY, 1964). | MR | Zbl

[BR] K. Ball, T. Rivoal, Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146:1 (2001), 193–207. | MR | Zbl

[Be] F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11:3 (1979), 268–272. | MR | Zbl

[Ch] G. V. Chudnovsky, On the method of Thue–Siegel. Ann. of Math. II Ser. 117:2 (1983), 325–382. | MR | Zbl

[DV] R. Dvornicich, C. Viola, Some remarks on Beukers’ integrals. Colloq. Math. Soc. János Bolyai 51 (North-Holland, Amsterdam, 1987), 637–657. | MR | Zbl

[FN] N. I. Fel’dman, Yu. V. Nesterenko, Transcendental numbers. (Number theory IV), Encyclopaedia Math. Sci. 44 (Springer-Verlag, Berlin, 1998). | MR | Zbl

[Gu] L. A. Gutnik, 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. | MR | Zbl

[Ha1] M. Hata, Legendre type polynomials and irrationality measures. J. Reine Angew. Math. 407:1 (1990), 99–125. | EuDML | MR | Zbl

[Ha2] M. Hata, Irrationality measures of the values of hypergeometric functions. Acta Arith. 60:4 (1992), 335–347. | EuDML | MR | Zbl

[Ha3] M. Hata, Rational approximations to the dilogarithm. Trans. Amer. Math. Soc. 336:1 (1993), 363–387. | MR | Zbl

[Ha4] M. Hata, A note on Beukers’ integral. J. Austral. Math. Soc. Ser. A 58:2 (1995), 143–153. | MR | Zbl

[Ha5] M. Hata, A new irrationality measure for ζ(3). Acta Arith. 92:1 (2000), 47–57. | EuDML | MR | Zbl

[HMV] A. Heimonen, T. Matala-Aho, K. Väänänen, On irrationality measures of the values of Gauss hypergeometric function. Manuscripta Math. 81:1/2 (1993), 183–202. | EuDML | MR | Zbl

[He] T. G. Hessami Pilerhood, Arithmetic properties of values of hypergeometric functions. Ph. D. thesis (Moscow University, Moscow, 1999); Linear independence of vectors with polylogarithmic coordinates. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 6 (1999), 54–56. | Zbl

[Lu] Yu. L. Luke, Mathematical functions and their approximations. (Academic Press, New York, NY, 1975). | MR | Zbl

[Ne1] Yu. V. Nesterenko, A few remarks on ζ(3). Mat. Zametki [Math. Notes] 59:6 (1996), 865–880. | MR | Zbl

[Ne2] Yu. V. Nesterenko, Integral identities and constructions of approximations to zeta values. Actes des 12èmes rencontres arithmétiques de Caen (June 29–30, 2001), J. Théorie Nombres Bordeaux 15:2 (2003), 535–550. | EuDML | Numdam | MR | Zbl

[Ne3] Yu. V. Nesterenko, Arithmetic properties of values of the Riemann zeta function and generalized hypergeometric functions. Manuscript (2001).

[Ni] E. M. Nikishin, On irrationality of values of functions F(x,s). Mat. Sb. [Russian Acad. Sci. Sb. Math.] 109:3 (1979), 410–417. | MR | Zbl

[Po] A. van der Poorten, 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. | MR | Zbl

[RV1] G. Rhin, C. Viola, On the irrationality measure of ζ(2). Ann. Inst. Fourier (Grenoble) 43:1 (1993), 85–109. | EuDML | Numdam | MR | Zbl

[RV2] G. Rhin, C. Viola, On a permutation group related to ζ(2). Acta Arith. 77:1 (1996), 23–56. | EuDML | MR | Zbl

[RV3] G. Rhin, C. Viola, The group structure for ζ(3). Acta Arith. 97:3 (2001), 269–293. | EuDML | MR | Zbl

[Ri1] T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris Sér. I Math. 331:4 (2000), 267–270. | MR | Zbl

[Ri2] T. Rivoal, Irrationnalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Rapport de recherche SDAD no. 2000-9 (Université de Caen, Caen, 2000).

[Ri3] T. Rivoal, Propriétés diophantiennes des valeurs de la fonction zêta de Riemann aux entiers impairs. Thèse de doctorat (Université de Caen, Caen, 2001).

[Ri4] T. Rivoal, Irrationalité d’au moins un des neuf nombres ζ(5),ζ(7),,ζ(21). Acta Arith. 103 (2001), 157–167. | EuDML | MR | Zbl

[Ru] E. A. Rukhadze, A lower bound for the approximation of ln2 by rational numbers. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 6 (1987), 25–29. | MR | Zbl

[Sl] L. J. Slater, Generalized hypergeometric functions. 2nd edition (Cambridge University Press, Cambridge, 1966). | MR | Zbl

[Va] D. V. Vasilyev, On small linear forms for the values of the Riemann zeta-function at odd points. Preprint no. 1 (558) (Nat. Acad. Sci. Belarus, Institute Math., Minsk, 2001).

[Vi] C. Viola, Hypergeometric functions and irrationality measures. Analytic Number Theory (ed. Y. Motohashi), London Math. Soc. Lecture Note Ser. 247 (Cambridge University Press, Cambridge, 1997), 353–360. | MR | Zbl

[Zu1] W. V. Zudilin, Irrationality of values of zeta function at odd integers. Uspekhi Mat. Nauk [Russian Math. Surveys] 56:2 (2001), 215–216. | MR | Zbl

[Zu2] W. Zudilin, Irrationality of values of zeta-function. Contemporary research in mathematics and mechanics, Proceedings of the 23rd Conference of Young Scientists of the Department of Mechanics and Mathematics (Moscow State University, April 9–14, 2001), part 2 (Publ. Dept. Mech. Math. MSU, Moscow, 2001), 127–135; E-print math.NT/0104249.

[Zu3] W. Zudilin, Irrationality of values of Riemann’s zeta function. Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.] 66:3 (2002), 49–102. | MR | Zbl

[Zu4] W. V. Zudilin, One of the eight numbers ζ(5),ζ(7),,ζ(17),ζ(19) is irrational. Mat. Zametki [Math. Notes] 70:3 (2001), 472–476. | MR | Zbl

[Zu5] W. V. Zudilin, Cancellation of factorials. Mat. Sb. [Russian Acad. Sci. Sb. Math.] 192:8 (2001), 95–122. | MR | Zbl

[Zu6] W. Zudilin, Well-poised hypergeometric service for diophantine problems of zeta values. Actes des 12èmes rencontres arithmétiques de Caen (June 29–30, 2001), J. Théorie Nombres Bordeaux 15:2 (2003), 593–626. | EuDML | Numdam | MR | Zbl

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