Irrationalité de valeurs de zêta
[Irrationality of zeta values]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 910, pp. 27-62.

The values of Riemann zeta function at positive even integers are transcendental numbers, since they are rational multiples of powers of π. On the contrary, very little is known about the arithmetic nature of ζ(2k+1) for positive integers k. Apéry proved in 1978 that ζ(3) is irrational. Rivoal proved in 2000 that infinitely many ζ(2k+1) are irrational, but without being able to construct any such k2. There are several ways to see Apéry’s proof; the one using hypergeometric series yields at the same time Apéry’s and Rivoal’s theorems.

Les valeurs aux entiers pairs (strictement positifs) de la fonction ζ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de π. En revanche, on sait très peu de choses sur la nature arithmétique des ζ(2k+1), pour k1 entier. Apéry a démontré en 1978 que ζ(3) est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de ζ(2k+1) sont irrationnels, mais sans pouvoir en exhiber aucun autre que ζ(3). Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques permet d’obtenir à la fois les théorèmes d’Apéry et de Rivoal.

Classification: 11J72,  11G55,  11M06,  33C20,  41A21
Keywords: irrationality, Riemann zeta function, hypergeometric series, Padé approximation, Apéry's theorem, rational approximation, polylogarithm
@incollection{SB_2002-2003__45__27_0,
     author = {Fischler, St\'ephane},
     title = {Irrationalit\'e de valeurs de z\^eta},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:910},
     pages = {27--62},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     zbl = {1101.11024},
     mrnumber = {2111638},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__27_0/}
}
TY  - CHAP
AU  - Fischler, Stéphane
TI  - Irrationalité de valeurs de zêta
BT  - Séminaire Bourbaki : volume 2002/2003, exposés 909-923
AU  - Collectif
T3  - Astérisque
N1  - talk:910
PY  - 2004
DA  - 2004///
SP  - 27
EP  - 62
IS  - 294
PB  - Association des amis de Nicolas Bourbaki, Société mathématique de France
PP  - Paris
UR  - http://www.numdam.org/item/SB_2002-2003__45__27_0/
UR  - https://zbmath.org/?q=an%3A1101.11024
UR  - https://www.ams.org/mathscinet-getitem?mr=2111638
LA  - fr
ID  - SB_2002-2003__45__27_0
ER  - 
%0 Book Section
%A Fischler, Stéphane
%T Irrationalité de valeurs de zêta
%B Séminaire Bourbaki : volume 2002/2003, exposés 909-923
%A Collectif
%S Astérisque
%Z talk:910
%D 2004
%P 27-62
%N 294
%I Association des amis de Nicolas Bourbaki, Société mathématique de France
%C Paris
%G fr
%F SB_2002-2003__45__27_0
Fischler, Stéphane. Irrationalité de valeurs de zêta, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 910, pp. 27-62. http://www.numdam.org/item/SB_2002-2003__45__27_0/

[AO] S. Ahlgren & K. Ono - “A Gaussian hypergeometric series evaluation and Apéry number congruences”, J. reine angew. Math. 518 (2000), p. 187-212. | MR | Zbl

[AG] G. Almkvist & A. Granville - “Borwein and Bradley’s Apéry-like formulae for ζ(4n+3), Experiment. Math. 8 (1999), no. 2, p. 197-203. | MR | Zbl

[An] Y. André - G-functions and geometry, Aspects of Math., vol. E13, Vieweg, 1989. | MR | Zbl

[AnJ] R. André-Jeannin - “Irrationalité de la somme des inverses de certaines suites récurrentes”, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), p. 539-541. | MR | Zbl

[And] G. E. Andrews - “The well-poised thread : an organized chronicle of some amazing summations and their implications”, Ramanujan J. 1 (1997), no. 1, p. 7-23. | MR | Zbl

[AAR] G. E. Andrews, R. Askey & R. Roy - “Special Functions”, (G.-C. Rota, 'ed.), The Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. | MR | Zbl

[Ap1] R. Apéry - “Irrationalité de ζ(2) et ζ(3), in Journées arithmétiques (Luminy, 1978), Astérisque, vol. 61, Société Mathématique de France, 1979, p. 11-13. | Numdam | Zbl

[Ap2] -, “Interpolation de fractions continues et irrationalité de certaines constantes”, in Comité des Travaux Historiques et Scientifiques (CTHS), Bulletin de la Section des Sciences III (Mathématiques), Bibliothèque Nationale, Paris, 1981, p. 37-53. | MR | Zbl

[AW] R. Askey & J. A. Wilson - “A recursive relation generalizing those of Apéry”, J. Austral. Math. Soc. 36 (1984), p. 267-278. | MR | Zbl

[As] W. Van Assche - “Approximation theory and analytic number theory”, in Special Functions and Differential Equations (Madras, 1997), Allied Publishers, New Delhi, 1998, p. 336-355. | MR | Zbl

[BR] K. M. Ball & T. Rivoal - “Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs”, Invent. Math. 146 (2001), p. 193-207. | MR | Zbl

[BO] C. Batut & M. Olivier - “Sur l'accélération de la convergence de certaines fractions continues”, in Sém. de Théorie des Nombres de Bordeaux 1979-1980, 1980, exp. no 23, 25 p. | MR | Zbl

[Be1] F. Beukers - “A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11 (1979), no. 3, p. 268-272. | MR | Zbl

[Be2] -, “Padé-approximations in number theory”, in Padé approximation and its applications, (Amsterdam, 1980), Lect. Notes in Math., vol. 888, Springer, Berlin-New York, 1981, p. 90-99. | MR | Zbl

[Be3] -, “Irrationality of π 2 , periods of an elliptic curve and Γ 1 (5), in Approximations diophantiennes et nombres transcendants (Luminy, 1982) (D. Bertrand & M. Waldschmidt, éds.), Progress in Math., vol. 31, Birkhäuser, 1983, p. 47-66. | MR | Zbl

[Be4] -, “ The values of polylogarithms”, in Topics in classical number theory (Budapest, 1981), Colloq. Math. Soc. János Bolyai, vol. 34, 1984, p. 219-228. | MR | Zbl

[Be5] -, “Some congruences for the Apéry numbers”, J. Number Theory 21 (1985), p. 141-155. | MR | Zbl

[Be6] -, “Irrationality proofs using modular forms”, in Journées arithmétiques (Besançon, 1985), Astérisque, vol. 147-148, Société Mathématique de France, 1987, p. 271-283. | Numdam | MR | Zbl

[Be7] -, “Another Congruence for the Apéry Numbers”, J. Number Theory 25 (1987), p. 201-210. | MR | Zbl

[BP] F. Beukers & C. A. M. Peters - “A family of K3 surfaces and ζ(3), J. reine angew. Math. 351 (1984), p. 42-54. | MR | Zbl

[BB] J. Borwein & D. Bradley - “Empirically determined Apéry-like formulae for ζ(4n+3), Experiment. Math. 6 (1997), p. 181-194. | MR | Zbl

[BE] P. Borwein & T. Erdélyi - Polynomials and Polynomial inequalities, Graduate Texts in Math., vol. 161, Springer, 1995. | MR | Zbl

[BV] P. Bundschuh & K. Väänänen - “Arithmetical investigations of a certain infinite product”, Compositio Math. 91 (1994), p. 175-199. | Numdam | MR | Zbl

[Ca1] P. Cartier - “Démonstration automatique d'identités et fonctions hypergéométriques (d'après Zeilberger)”, in Sém. Bourbaki (1991/92), Astérisque, vol. 206, Société Mathématique de France, 1992, exp. no. 746, p. 41-91. | Numdam | MR | Zbl

[Ca2] -, “Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents”, in Sém. Bourbaki (2000/01), Astérisque, vol. 282, Société Mathématique de France, 2002, exp. no. 885, p. 137-173. | Numdam | Zbl

[CCC] S. Chowla, J. Cowles & M. Cowles - “Congruence properties of Apéry numbers”, J. Number Theory 12 (1980), p. 188-190. | MR | Zbl

[Ch] G. V. Chudnovsky - “Transcendental numbers”, in Number theory, Proc. Southern Illinois Conf. (Carbondale, 1979), Lect. Notes in Math., vol. 751, Springer, p. 45-69. | MR | Zbl

[Coh1] H. Cohen - “Démonstration de l’irrationalité de ζ(3) (d’après Apéry)”, in Sém. de Théorie des Nombres de Grenoble, octobre 1978, 9 p.

[Coh2] -, “Généralisation d'une construction de R. Apéry”, Bull. Soc. Math. France 109 (1981), p. 269-281. | Numdam | MR | Zbl

[Col] P. Colmez - “Arithmétique de la fonction zêta”, in La fonction zêta, Journées X-UPS, Éditions de l'École polytechnique, 2002.

[Di] J. Dieudonné - Calcul infinitésimal, Collection Méthodes, Hermann, 1968. | Zbl

[Dw1] B. Dwork - “On Apéry's differential operator”, in Groupe d'étude d'analyse ultramétrique, 1979-1981, exp. 25, 6 p. | Numdam | MR | Zbl

[Dw2] -, “Arithmetic theory of differential equations”, in Symposia Math. (INDAM, Rome, 1979), vol. 24, Academic Press, 1981, p. 225-243. | MR

[Dw3] B. Dwork, G. Gerotto & F. J. Sullivan - An introduction to G-functions, Annals of Math. Studies, vol. 133, Princeton Univ. Press, 1994. | MR | Zbl

[FN] N. I. Fel'Dman & Yu. V. Nesterenko - “Transcendental numbers”, in Number theory, IV (A.N. Parshin & I.R. Shafarevich, éds.), Encyclopaedia of Mathematical Sciences, vol. 44, Springer, Berlin, 1998. | MR | Zbl

[Fi1] S. Fischler - “Formes linéaires en polyzêtas et intégrales multiples”, C. R. Acad. Sci. Paris Sér. I Math. 335 (2002), p. 1-4. | MR | Zbl

[Fi2] -, “Groupes de Rhin-Viola et intégrales multiples”, J. Théor. Nombres Bordeaux 15 (2003), no. 2, p. 479-534. | Numdam | MR | Zbl

[FR] S. Fischler & T. Rivoal - “Approximants de Padé et séries hypergéométriques équilibrées”, J. Math. Pures Appl. 82 (2003), p. 1369-1394. | MR | Zbl

[Gel] A. O. Gel'Fond - Calcul des différences finies, Dunod, Paris, 1963. | MR | Zbl

[Ges] I. Gessel - “Some congruences for Apéry numbers”, J. Number Theory 14 (1982), p. 362-368. | MR | Zbl

[Gu1] L. A. Gutnik - “The irrationality of certain quantities involving ζ(3), Uspekhi Mat. Nauk 34 (1979), no. 3, 190 [207]. | MR | Zbl

[Gu2] -, “On the irrationality of some quantities containing ζ(3), Acta Arith. 42 (1983), no. 3, p. 255-264, en russe ; traduction en anglais dans Amer. Math. Soc. Transl., 140 (1988), p. 45-55. | MR | Zbl

[Hab] L. Habsieger - “Introduction to diophantine approximation”, notes de cours.

[HW] G. H. Hardy & E. M. Wright - An introduction to the theory of numbers, 3e 'ed., Oxford Univ. Press, 1954. | JFM | MR | Zbl

[Hat1] M. Hata - “On the linear independence of the values of polylogarithmic functions”, J. Math. Pures Appl. 69 (1990), no. 2, p. 133-173. | MR | Zbl

[Hat2] -, “Rational approximations to the dilogarithm”, Trans. Amer. Math. Soc. 336 (1993), no. 1, p. 363-387. | MR | Zbl

[Hat3] -, “A new irrationality measure for ζ(3), Acta Arith. 92 (2000), no. 1, p. 47-57. | MR | Zbl

[Haz] M. Hazewinkel - Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, 1978. | MR | Zbl

[He] T. G. Hessami Pilehrood - “Linear independence of vectors with polylogarithmic coordinates”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 54 (1999), no. 6, p. 54-56, Moscow Univ. Math. Bull., p. 40-42. | MR | Zbl

[Hu1] M. Huttner - “Équations différentielles fuchsiennes. Approximations du dilogarithme, de ζ(2) et de ζ(3), Pub. IRMA Lille 43 (1997).

[Hu2] -, “Constructible sets of linear differential equations and effective rational approximations of G-functions”, Pub. IRMA Lille 59 (2002). | Zbl

[Inc] E. L. Ince - Ordinary differential equations, Dover Publ., 1926. | JFM | MR | Zbl

[Ing] A. E. Ingham - The distribution of prime numbers, Cambridge Univ. Press, 1932. | JFM | MR | Zbl

[Is] T. Ishikawa - “On Beukers' conjecture”, Kobe J. Math. 6 (1989), p. 49-52. | MR | Zbl

[Ko] M. Koecher - “Letter”, Math. Intelligencer 2 (1980), p. 62-64. | MR | Zbl

[Kr] C. Krattenthaler - Communication personnelle du 28 octobre 2002.

[La] S. Lang - Algebra, 3e 'ed., Addison-Wesley, 1993. | MR | Zbl

[Le] D. Leshchiner - “Some new identities for ζ(k), J. Number Theory 13 (1981), p. 355-362. | MR | Zbl

[Li] J. Liouville - “Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques”, J. Math. Pures Appl. 16 (1851), p. 133-142.

[Lu] Y. L. Luke - The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, vol. 53, Academic Press, 1969. | MR | Zbl

[Me] M. Mendès-France - “Roger Apéry et l'irrationnel”, La Recherche 97 (1979), p. 170-172. | Zbl

[Ne1] Yu. V. Nesterenko - “On the linear independence of numbers”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 40 (1985), no. 1, p. 46-49, Moscow Univ. Math. Bull., p. 69-74. | MR | Zbl

[Ne2] -, “A few remarks on ζ(3), Mat. Zametki 59 (1996), no. 6, p. 865-880, Math. Notes, p. 625-636. | MR | Zbl

[Ne3] -, “Integral identities and constructions of approximations to zeta-values”, J. Théor. Nombres Bordeaux 15 (2003), no. 2, p. 535-550. | Numdam | MR | Zbl

[Ni] E. M. Nikishin - “On the irrationality of the values of the functions F(x,s)”, Mat. Sbornik 109 (1979), no. 3, p. 410-417, Math. USSR-Sb. 37, p. 381-388. | MR | Zbl

[NS] E. M. Nikishin & V. N. Sorokin - Rational approximations and orthogonality, Translations of Math. Monographs, vol. 92, American Mathematical Society, 1991. | MR | Zbl

[NZM] I. Niven, H. S. Zuckerman & H. L. Montgomery - An introduction to the theory of numbers, 5e 'ed., J. Wiley, 2000. | MR | Zbl

[Oe] J. Oesterlé - “Polylogarithmes”, in Sém. Bourbaki (1992/93), Astérisque, vol. 216, Société Mathématique de France, 1993, exp. no 762, p. 49-67. | Numdam | MR | Zbl

[PS] C. Peters & J. Stienstra - “A pencil of K3-surfaces related to Apéry’s recurrence for ζ(3) and Fermi surfaces for potential zero”, in Arithmetics of complex manifolds (Erlangen, 1988) (W.P. Barth & H. Lange, éds.), Lect. Notes in Math., vol. 1399, Springer, 1989, p. 110-127. | MR | Zbl

[PWZ] M. Petkovšek, H. S. Wilf & D. Zeilberger - A=B, A.K. Peters, 1996. | MR

[Po1] A. Van Der Poorten - “A proof that Euler missed... Apéry’s proof of the irrationality of ζ(3), Math. Intelligencer 1 (1978-79), no. 4, p. 195-203. | MR | Zbl

[Po2] -, “Some wonderful formulae... footnotes to Apéry’s proof of the irrationality of ζ(3), in Sém. Delange-Pisot-Poitou, 20e année, 1978-79, exp. 29, 7 p. | Numdam | Zbl

[Po3] -, “Some wonderful formulas... an introduction to polylogarithms”, in Proceedings of the Queen's Number Theory Conference (Kingston, 1979), Queen's Papers in Pure and Applied Mathematics, vol. 54, 1980, p. 269-286. | MR | Zbl

[Pr1] M. Prévost - “A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants”, J. Comp. Appl. Math. 67 (1996), p. 219-235. | MR | Zbl

[Pr2] -, “On the irrationality of t n Aα n +Bβ n , J. Number Theory 73 (1998), p. 139-161. | MR

[Re1] E. Reyssat - “Irrationalité de ζ(3) selon Apéry”, in Sém. Delange-Pisot-Poitou, 20e année, 1978-79, exp. 6, 6 p. | Numdam | Zbl

[Re2] -, “Mesures de transcendance pour les logarithmes de nombres rationnels”, in Approximations diophantiennes et nombres transcendants (Luminy, 1982) (D. Bertrand & M. Waldschmidt, éds.), Progress in Math., vol. 31, Birkhäuser, 1983, p. 235-245. | MR | Zbl

[RV] G. Rhin & C. Viola - “The group structure for ζ(3), Acta Arith. 97 (2001), no. 3, p. 269-293. | 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 (2000), no. 4, p. 267-270. | MR | Zbl

[Ri2] -, “Propriétés diophantiennes des valeurs de la fonction zêta de Riemann aux entiers impairs”, Thèse, Univ. de Caen, 2001, disponible sur http://theses-EN-ligne.in2p3.fr.

[Ri3] -, “Irrationalité d’au moins un des neuf nombres ζ(5),ζ(7),...,ζ(21), Acta Arith. 103 (2002), no. 2, p. 157-167. | MR | Zbl

[Ri4] -, “Séries hypergéométriques et irrationalité des valeurs de la fonction zêta de Riemann”, J. Théor. Nombres Bordeaux 15 (2003), no. 1, p. 351-365. | Numdam | MR | Zbl

[RZ] T. Rivoal & W. Zudilin - “Diophantine properties of numbers related to Catalan's constant”, Math. Annalen 326 (2003), p. 705-721. | MR | Zbl

[Se] J.-P. Serre - Cours d'arithmétique, Presses Universitaires de France, 1970. | MR | Zbl

[Sl] I. J. Slater - Generalized hypergeometric functions, Cambridge Univ. Press, 1966. | MR | Zbl

[So1] V. N. Sorokin - “Hermite-Padé approximations for Nikishin systems and the irrationality of ζ(3), Uspekhi Mat. Nauk 49 (1994), no. 2, p. 167-168, Russian Math. Surveys, p. 176-177. | MR | Zbl

[So2] -, “A transcendence measure for π 2 , Mat. Sbornik 187 (1996), no. 12, p. 87-120, Sb. Math., p. 1819-1852. | Zbl

[So3] -, “Apéry's theorem”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 53 (1998), no. 3, p. 48-53, Moscow Univ. Math. Bull., p. 48-52. | MR

[SB] J. Stienstra & F. Beukers - “On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces”, Math. Ann. 271 (1985), p. 269-304. | MR | Zbl

[Su] B. Sury - “On a conjecture of Chowla et al.”, J. Number Theory 72 (1998), p. 137-139. | MR | Zbl

[V] O. N. Vasilenko - “Certain formulae for values of the Riemann zeta function at integral points”, in Number theory and its applications, Proceedings of the science-theoretical conference (Tashkent), 1990, en russe, p. 27.

[Va1] D. V. Vasilyev - “Some formulas for Riemann zeta-function at integer points”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 51 (1996), no. 1, p. 81-84, Moscow Univ. Math. Bull., p. 41-43. | MR | Zbl

[Va2] -, “On small linear forms for the values of the Riemann zeta-function at odd integers”, Doklady NAN Belarusi (Reports of the Belarus National Academy of Sciences) 45 (2001), no. 5, p. 36-40, en russe. | MR

[Wa] M. Waldschmidt - “Valeurs zêta multiples : une introduction”, J. Théor. Nombres Bordeaux 12 (2000), p. 581-595. | Numdam | MR | Zbl

[We] A. Weil - “Remarks on Hecke's lemma and its use”, in Œuvres scientifiques - Collected Papers III, Springer, 1979, p. 405-412. | MR | Zbl

[WW] E. T. Whittaker & G. N. Watson - A course of modern analysis, 4e 'ed., Cambridge Univ. Press, 1927. | JFM | MR

[Za1] D. Zagier - “Introduction to modular forms”, in From number theory to physics (Les Houches, 1989) (M. Waldschmidt, P. Moussa, J.M. Luck & C. Itzykson, éds.), Springer, 1992, p. 238-291. | MR | Zbl

[Za2] -, “Cours au Collège de France”, mai 2001.

[Ze1] D. Zeilberger - “Closed form (pun intended !)”, in A tribute to Emil Grosswald : Number theory and related analysis (M. Knopp & M. Sheingorn, éds.), Comtemporary Math., vol. 143, American Mathematical Society, 1993, p. 579-607. | MR | Zbl

[Ze2] -, “Computerized deconstruction”, Adv. Applied Math. 31 (2003), p. 532-543. | MR

[Zl] S. A. Zlobin - “Integrals expressible as linear forms in generalized polylogarithms”, Mat. Zametki 71 (2002), no. 5, p. 782-787, Math. Notes, p. 711-716. | MR | Zbl

[Zu1] W. Zudilin - “One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational”, Uspekhi Mat. Nauk 56 (2001), no. 4, p. 149-150, Russian Math. Surveys, p. 774-776. | MR | Zbl

[Zu2] -, “Irrationality of values of the Riemann zeta function”, Izvestiya RAN Ser. Mat. 66 (2002), no. 3, p. 49-102, Izv. Math., p. 489-542. | MR | Zbl

[Zu3] -, “Well-poised hypergeometric service for diophantine problems of zeta values”, J. Théor. Nombres Bordeaux 15 (2003), no. 2, p. 593-626. | Numdam | MR

[Zu4] -, “Arithmetic of linear forms involving odd zeta values”, preprint, math.NT/0206176, 2002.

[Zu5] -, “An elementary proof of Apéry's theorem”, preprint, math.NT/0202159, 2002.