Mathematische Annalen

, Volume 347, Issue 4, pp 739–763 | Cite as

A refinement of Nesterenko’s linear independence criterion with applications to zeta values

  • Stéphane Fischler
  • Wadim Zudilin
Open Access


We refine (and give a new proof of) Nesterenko’s famous linear independence criterion from 1985, by making use of the fact that some coefficients of linear forms may have large common divisors. This is a typical situation appearing in the context of hypergeometric constructions of \({\mathbb{Q}}\)-linear forms involving zeta values or their q-analogs. We apply our criterion to sharpen previously known results in this direction.


Linear Form Rational Approximation Acta Arith Positive Divisor Viola Method 
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Open Access

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  1. 1.
    Apéry R.: Irrationalité de ζ(2) et ζ(3). Astérisque 61, 11–13 (1979)zbMATHGoogle Scholar
  2. 2.
    Ball K., Rivoal T.: Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146(1), 193–207 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bedulev, E.V.: On the linear independence of numbers over number fields. Mat. Zametki 64(4), 506–517 (1998); English transl., Math. Notes 64(3–4), 440–449 (1998)Google Scholar
  4. 4.
    Colmez, P.: Arithmétique de la fonction zêta. La fonction zêta, pp. 37–164 (Journées X-UPS 2002)Google Scholar
  5. 5.
    Davenport H., Schmidt W.M.: Approximation to real numbers by quadratic irrationals. Acta Arith. 13, 169–176 (1967)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Davenport H., Schmidt W.M.: Approximation to real numbers by algebraic integers. Acta Arith. 15, 393–416 (1969)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Fischler, S.: Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, . . .). Séminaire Bourbaki 2002–2003, exp. no. 910. Astérisque 294, 27–62 (2004)Google Scholar
  8. 8.
    Fischler, S.: Restricted rational approximation and Apéry-type constructions. Indag. Math. (2010, in press)Google Scholar
  9. 9.
    Fischler, S., Rivoal, T.: Irrationality exponent and rational approximations with prescribed growth. Proc. Am. Math. Soc. (2010, in press)Google Scholar
  10. 10.
    Jouhet, F., Mosaki, E.: Irrationalité aux entiers impairs positifs d’un q-analogue de la fonction zêta de Riemann. Int. J. Number Theory (2007, in press). arXiv:0712.1762 [math.CO]Google Scholar
  11. 11.
    Krattenthaler, C., Rivoal, T.: Hypergéométrie et fonction zêta de Riemann. Mem. Am. Math. Soc. 186(875) (2007)Google Scholar
  12. 12.
    Krattenthaler C., Rivoal T., Zudilin W.: Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zêta et séries d’Eisenstein. J. Inst. Math. Jussieu 5(1), 53–79 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Marcovecchio R.: The Rhin–Viola method for log 2. Acta Arith. 139(2), 147–184 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nesterenko, Yu.V.: On the linear independence of numbers. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1, 46–49 (1985); English transl., Moscow Univ. Math. Bull. 40(1), 69–74 (1985)Google Scholar
  15. 15.
    Rhin G., Viola C.: The group structure for ζ(3). Acta Arith. 97(3), 269–293 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rivoal T.: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris Sér. I Math. 331(4), 267–270 (2000)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Slater L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)zbMATHGoogle Scholar
  18. 18.
    Töpfer T.: Über lineare Unabhängigkeit in algebraischen Zahlkörpern. Results Math. 25(1–2), 139–152 (1994)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Viola, C.: Hypergeometric functions and irrationality measures. Analytic number theory (Kyoto, 1996). In: London Mathematical Society Lecture Note Series, vol. 247, pp. 353–360. Cambridge University Press, Cambridge (1997)Google Scholar
  20. 20.
    Zudilin, W.: On the irrationality of the values of the Riemann zeta function. Izv. Ross. Akad. Nauk Ser. Mat. 66(3), 49–102 (2002); English transl., Izv. Math. 66(3), 489–542 (2002)Google Scholar
  21. 21.
    Zudilin, W.: On the irrationality measure for a q-analogue of ζ(2). Mat. Sb. 193(8), 49–70 (2002); English transl., Sb. Math. 193(7–8), 1151–1172 (2002)Google Scholar
  22. 22.
    Zudilin W.: Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux 16(1), 251–291 (2004)zbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’OrsayUniversité Paris-SudOrsay CedexFrance
  2. 2.CNRSOrsay CedexFrance
  3. 3.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia

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