staple
With cedram.org

Search the site

Table of contents for this issue | Previous article
Wadim Zudilin
Arithmetic of linear forms involving odd zeta values
Journal de théorie des nombres de Bordeaux, 16 no. 1 (2004), p. 251-291, doi: 10.5802/jtnb.447
Article PDF | Reviews MR 2145585 | Zbl 02184645 | 2 citations in Cedram

Résumé - Abstract

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 $\zeta (2)$ and $\zeta (3)$, 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 $\zeta (5)$, $\zeta (7)$, $\zeta (9)$, and $\zeta (11)$ is irrational.

Bibliography

[Ap] R. Apéry, Irrationalité de $\zeta (2)$ et $\zeta (3)$. Astérisque 61 (1979), 11–13.  Zbl 0401.10049
[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 0005.40001 |  JFM 58.0385.03
[Ba2] W. N. Bailey, Transformations of well-poised hypergeometric series. Proc. London Math. Soc. II Ser. 36:4 (1934), 235–240.  Zbl 0008.07101 |  JFM 59.0376.01
[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 185155 |  Zbl 0011.02303 |  JFM 61.0406.01
[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 1859021 |  Zbl 1058.11051
[Be] F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$. Bull. London Math. Soc. 11:3 (1979), 268–272.  MR 554391 |  Zbl 0421.10023
[Ch] G. V. Chudnovsky, On the method of Thue–Siegel. Ann. of Math. II Ser. 117:2 (1983), 325–382.  MR 690849 |  Zbl 0518.10038
[DV] R. Dvornicich, C. Viola, Some remarks on Beukers’ integrals. Colloq. Math. Soc. János Bolyai 51 (North-Holland, Amsterdam, 1987), 637–657.  MR 1058238 |  Zbl 0755.11019
[FN] N. I. Fel’dman, Yu. V. Nesterenko, Transcendental numbers. (Number theory IV), Encyclopaedia Math. Sci. 44 (Springer-Verlag, Berlin, 1998).  MR 1603608 |  Zbl 0885.11004
[Gu] L. A. Gutnik, On the irrationality of certain quantities involving $\zeta (3)$. Uspekhi Mat. Nauk [Russian Math. Surveys] 34:3 (1979), 190; Acta Arith. 42:3 (1983), 255–264.  MR 729735 |  Zbl 0437.10015
[Ha1] M. Hata, Legendre type polynomials and irrationality measures. J. Reine Angew. Math. 407:1 (1990), 99–125.  MR 1048530 |  Zbl 0692.10034
[Ha2] M. Hata, Irrationality measures of the values of hypergeometric functions. Acta Arith. 60:4 (1992), 335–347. Article |  MR 1159350 |  Zbl 0760.11020
[Ha3] M. Hata, Rational approximations to the dilogarithm. Trans. Amer. Math. Soc. 336:1 (1993), 363–387.  MR 1147401 |  Zbl 0768.11022
[Ha4] M. Hata, A note on Beukers’ integral. J. Austral. Math. Soc. Ser. A 58:2 (1995), 143–153.  MR 1323987 |  Zbl 0830.11026
[Ha5] M. Hata, A new irrationality measure for $\zeta (3)$. Acta Arith. 92:1 (2000), 47–57. Article |  MR 1739738 |  Zbl 0955.11023
[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. Article |  MR 1247597 |  Zbl 0801.11032
[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 0983.11044
[Lu] Yu. L. Luke, Mathematical functions and their approximations. (Academic Press, New York, NY, 1975).  MR 501762 |  Zbl 0318.33001
[Ne1] Yu. V. Nesterenko, A few remarks on $\zeta (3)$. Mat. Zametki [Math. Notes] 59:6 (1996), 865–880.  MR 1445472 |  Zbl 0888.11028
[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. Cedram |  MR 2140866 |  Zbl 02184610
[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 542809 |  Zbl 0414.10032
[Po] A. van der Poorten, A proof that Euler missed... Apéry’s proof of the irrationality of $\zeta (3)$. An informal report, Math. Intelligencer 1:4 (1978/79), 195–203.  MR 547748 |  Zbl 0409.10028
[RV1] G. Rhin, C. Viola, On the irrationality measure of $\zeta (2)$. Ann. Inst. Fourier (Grenoble) 43:1 (1993), 85–109. Cedram |  MR 1209696 |  Zbl 0776.11036
[RV2] G. Rhin, C. Viola, On a permutation group related to $\zeta (2)$. Acta Arith. 77:1 (1996), 23–56. Article |  MR 1404975 |  Zbl 0864.11037
[RV3] G. Rhin, C. Viola, The group structure for $\zeta (3)$. Acta Arith. 97:3 (2001), 269–293.  MR 1826005 |  Zbl 1004.11042
[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 1787183 |  Zbl 0973.11072
[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). Article
[Ri4] T. Rivoal, Irrationalité d’au moins un des neuf nombres $\zeta (5),\zeta (7),\dots , \zeta (21)$. Acta Arith. 103 (2001), 157–167.  MR 1904870 |  Zbl 1015.11033
[Ru] E. A. Rukhadze, A lower bound for the approximation of $\ln 2$ by rational numbers. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 6 (1987), 25–29.  MR 922879 |  Zbl 0635.10025
[Sl] L. J. Slater, Generalized hypergeometric functions. 2nd edition (Cambridge University Press, Cambridge, 1966).  MR 201688 |  Zbl 0135.28101
[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 1695002 |  Zbl 0904.11020
[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 1859714 |  Zbl 1037.11048
[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. arXiv
[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 1921809 |  Zbl 01930695
[Zu4] W. V. Zudilin, One of the eight numbers $\zeta (5),\zeta (7),\dots ,\zeta (17),\zeta (19)$ is irrational. Mat. Zametki [Math. Notes] 70:3 (2001), 472–476.  MR 1882257 |  Zbl 1022.11035
[Zu5] W. V. Zudilin, Cancellation of factorials. Mat. Sb. [Russian Acad. Sci. Sb. Math.] 192:8 (2001), 95–122.  MR 1862246 |  Zbl 1030.11032
[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. Cedram |  MR 2140869 |  Zbl 02184613