Well-poised hypergeometric service for diophantine problems of zeta values

Wadim Zudilin

Wadim Zudilin

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 2, pp. 593-626.

Résumé
Abstract

On montre comment les concepts classiques de séries et intégrales hypergéométriques bien équilibrées devient crucial dans l’étude des propriétés arithmétiques des valeurs de la fonction zêta de Riemann. Par ces arguments, on obtient (1) un groupe de permutations pour les formes linéaires en $1$ et $ζ (4) = π^{4} / 90$ donnant une majoration conditionnelle de la mesure d’irrationalité de $ζ (4)$ ; (2) une récurrence d’ordre deux pour $ζ (4)$ semblable à celles introduites par Apéry pour $ζ (2)$ et $ζ (3)$ , ainsi que des récurrences d’ordre réduit pour les formes linéaires en des valeurs de la fonction zêta aux entiers impairs ; (3) un gros groupe de permutations pour une famille d’intégrales multiples généralisant les intégrales dites de Beukers pour $ζ (2)$ et $ζ (3)$ .

It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in $1$ and $ζ (4) = π^{4} / 90$ yielding a conditional upper bound for the irrationality measure of $ζ (4)$ ; (2) a second-order Apéry-like recursion for $ζ (4)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for $ζ (2)$ and $ζ (3)$ .

MR Zbl

@article{JTNB_2003__15_2_593_0,
     author = {Wadim Zudilin},
     title = {Well-poised hypergeometric service for diophantine problems of zeta values},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {593--626},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {2},
     year = {2003},
     zbl = {02184613},
     mrnumber = {2140869},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2003__15_2_593_0/}
}

TY  - JOUR
AU  - Wadim Zudilin
TI  - Well-poised hypergeometric service for diophantine problems of zeta values
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2003
SP  - 593
EP  - 626
VL  - 15
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2003__15_2_593_0/
LA  - en
ID  - JTNB_2003__15_2_593_0
ER  -

%0 Journal Article
%A Wadim Zudilin
%T Well-poised hypergeometric service for diophantine problems of zeta values
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 593-626
%V 15
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2003__15_2_593_0/
%G en
%F JTNB_2003__15_2_593_0

Wadim Zudilin. Well-poised hypergeometric service for diophantine problems of zeta values. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 2, pp. 593-626. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_2_593_0/

Bibliographie
Cité par

[A1] Yu M. Aleksentsev, On the measure of approximation for the number π by algebraic numbers. Mat. Zametki [Math. Notes] 66:4 (1999), 483-493.

[An] G.E. Andrews, The well-poised thread: An organized chronicle of some amazing summations and their implications. The Ramanujan J. 1:1 (1997), 7-23. | MR | Zbl

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

[Ba] W.N. Bailey, Generalized hypergeometric series. Cambridge Math. Tracts 32 (Cambridge Univ. Press, Cambridge, 1935); 2nd reprinted edition Stechert-Hafner, New York-London, 1964. | JFM | 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

[Be1] F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11:3 (1979), 268-272. | Zbl

[Be2] F. Beukers, Padé approximations in number theory. Lecture Notes in Math. 888, Springer-Verlag, Berlin, 1981, 90-99. | MR | Zbl

[Be3] F. Beukers, Irrationality proofs using modular forms. Astérisque 147-148 (1987), 271-283. | MR | Zbl

[Be4] F. Beukers, On Dwork's accessory parameter problem. Math. Z. 241:2 (2002), 425-444. | MR | Zbl

[Br] N.G. De Bruijn, Asymptotic methods in analysis. North-Holland Publ., Amsterdam, 1958. | Zbl

[Co] H. Cohen, Accélération de la convergence de certaines récurrences linéaires, Séminaire de Théorie des nombres de Bordeaux (Année 1980-81), exposé 16, 2 pages. | 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. | Zbl

[Han] J. Hancl, A simple proof of the irrationality of π4. Amer. Math. Monthly 93 (1986), 374-375. | Zbl

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

[JT] W.B. Jones, W.J. Thron, Continued fractions. Analytic theory and applications. Encyclopaedia Math. Appl. Section: Analysis 11, Addison-Wesley, London, 1980. | MR | Zbl

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

[Ne2] Yu V. Nesterenko, Integral identities and constructions of approximations to zeta val-. ues. J. Théor. Nombres Bordeaux 15 (2003), ?-?. | Numdam | MR | Zbl

[Ne3] Yu V. Nesterenko, Arithmetic properties of values of the Riemann zeta function and generalized hypergeometric functions in preparation (2002).

[PWZ] M. Petkovšek, H.S. Wilf, D. Zeilberger, A = B. A. K. Peters, Ltd., Wellesley, MA, 1997. | MR

[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. | Zbl

[RV1] G. Rhin, C. Viola, On a permutation group related to ζ(2). Acta Arith. 77:1 (1996), 23-56. | Zbl

[RV2] G. Rhin, C. Viola, The group structure for ζ(3). Acta Arith. 97:3 (2001), 269-293. | 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, Propriétés diophantiennes des valeurs de la fonction zêta de Riemann aux entiers impairs. Thèse de Doctorat, Univ. de Caen, 2001.

[Ri3] T. Rivoal, Séries hypergéométriques et irrationalité des valeurs de la fonction zêta. J. Théor. Nombres Bordeaux 15 (2003), 351-365. | Numdam | MR | Zbl

[So1] V.N. Sorokin, Hermite-Padé approximations for Nikishin's systems and irrationality of ζ(3). Uspekhi Mat. Nauk [Russian Math. Surveys] 49:2 (1994), 167-168. | Zbl

[So2] V.N. Sorokin, A transcendence measure of π2. Mat. Sb. [Russian Acad. Sci. Sb. Math.] 187:12 (1996), 87-120. | Zbl

[So3] V.N. Sorokin, Apéry's theorem. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 53:3 (1998), 48-52. | MR | Zbl

[So4] V.N. Sorokin, One algorithm for fast calculation of π4. Preprint (Russian Academy of Sciences, M. V. Keldysh Institute for Applied Mathematics, Moscow, 2002), 59 pages; http://www.wis.kuleuven.ac.be/applied/intas/Art5.pdf.

[VaO] O.N. Vasilenko, Certain formulae for values of the Riemann zeta-function at integral points. Number theory and its applications, Proceedings of the science-theoretic conference (Tashkent, September 26-28, 1990), 27 (Russian).

[VaD] 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, Birational transformations and values of the Riemann zeta-function. J. Théor. Nombres Bordeaux 15 (2003), ?-? | Numdam | MR | Zbl

[WZ] H.S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities. Invent. Math. 108:3 (1992), 575-633. | MR | Zbl

[Zl1] S.A. Zlobin, Integrals expressible as linear forms in generalized polylogarithms. Mat. Zametki [Math. Notes] 71:5 (2002), 782-787. | MR | Zbl

[Zl2] S.A. Zlobin, On some integral identities. Uspekhi Mat. Nauk [Russian Math. Surveys] 57:3 (2002), 153-154. | MR | Zbl

[Zu1] W. Zudilin, Difference equations and the irrationality measure of numbers. Collection of papers: Analytic number theory and applications, Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.] 218 (1997), 165-178. | MR | Zbl

[Zu2] 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

[Zu3] W.V. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. Uspekhi Mat. Nauk [Russian Math. Surveys] 56:4 (2001), 149-150. | Zbl

[Zu4] W. Zudilin, Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux, to appear. | Numdam | MR | Zbl

[Zu5] W. Zudilin, An elementary proof of Apérys theorem. E-print math.NT/0202159 (February 2002).