Annales mathématiques du Québec

, Volume 38, Issue 1, pp 101–117 | Cite as

Two hypergeometric tales and a new irrationality measure of \(\zeta (2)\)

  • Wadim Zudilin


We prove the new upper bound \(5.095412\) for the irrationality exponent of \(\zeta (2)=\pi ^2/6\); the earlier record bound \(5.441243\) was established in 1996 by G. Rhin and C. Viola.


Irrationality exponent Rational approximations Hypergeometric series 


Nous obtenons une nouvelle borne pour l’exposant d’irrationnalité de \(\zeta (2)=\pi ^2/6\), à savoir \(5.095412\), cette dernière améliorant le record \(5.441243\) établi par G. Rhin et C. Viola.

Mathematics Subject Classification

Primary 11J82 Secondary 11Y60 33C20 33C60 



I am deeply thankful to Stéphane Fischler who has re-attracted my attention to [19] and forced me to write the details of the general construction there. This has finally grown up in a joint project with Simon Dauguet. My special thanks go to Yuri Nesterenko for many helpful comments on initial versions of the paper, and I also thank Raffaele Marcovecchio for related discussions and corrections. Finally, I acknowledge a healthy criticism of the anonymous referee that helped me to improve the presentation.


  1. 1.
    Androsenko, V.A., Salikhov, V.K.: Marcovecchio’s integral and an irrationality measure of \(\pi /\sqrt{3}\). Vestnik Bryansk State Tech. Univ. 34(4), 129–132 (2011)Google Scholar
  2. 2.
    Bailey, W.N.: Some transformations of generalized hypergeometric series, and contour-integrals of Barnes’s type. Q. J. Math. (Oxford) 3(1), 168–182 (1932)CrossRefGoogle Scholar
  3. 3.
    Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)zbMATHGoogle Scholar
  4. 4.
    Dauguet, S.; Zudilin, W.: On simultaneous diophantine approximations to \(\zeta (2)\) and \(\zeta (3)\) (2013, Preprint). arxiv:1401.5322 [math.NT]
  5. 5.
    Guillera, J.: WZ-proofs of “divergent” Ramanujan-type series. In: Kotsireas, I., Zima, E.V. (eds.) Advances in Combinatorics, Waterloo Workshop in Computer Algebra, W80 (May 26–29, 2011), pp. 187–195. Springer, Berlin (2013)Google Scholar
  6. 6.
    Hata, M.: Rational approximations to \(\pi \) and some other numbers. Acta Arith. 63(4), 335–349 (1993)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Nesterenko, Yu.V.: A few remarks on \(\zeta (3)\), Mat. Zametki 59, 865–880 (1996); English transl. Math. Notes 59, 625–636 (1996)Google Scholar
  8. 8.
    Nesterenko, Yu.V.: On the irrationality exponent of the number \(\ln 2\). Mat. Zametki 88, 549–564 (2010); English transl. Math. Notes 88, 530–543 (2010)Google Scholar
  9. 9.
    Salikhov, V.Kh.: On the irrationality measure of \(\pi \). Uspekhi Mat. Nauk 63(3), 163–164 (2008). English transl. Russ. Math. Surv. 63, 570–572 (2008)Google Scholar
  10. 10.
    Slater, L.J.: Generalized Hypergeometric Functions, 2nd edn. Cambridge University Press, Cambridge (1966)zbMATHGoogle Scholar
  11. 11.
    Stan, F.: On recurrences for Ising integrals. Adv. Appl. Math. 45, 334–345 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Pólya, G., Szegő, G.: Problems and theorems in analysis. vol. II. Grundlehren Math. Wiss., vol. 216. Springer, Berlin (1976)Google Scholar
  13. 13.
    Rhin, G., Viola, C.: On a permutation group related to \(\zeta (2)\). Acta Arith. 77(1), 23–56 (1996)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Rhin, G., Viola, C.: The group structure for \(\zeta (3)\). Acta Arith. 97(3), 269–293 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    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, 489–542 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Zudilin, W.: Well-poised hypergeometric service for diophantine problems of zeta values. J. Théorie Nombres Bordeaux 15(2), 593–626 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Zudilin, W.: Arithmetic of linear forms involving odd zeta values. J. Théorie Nombres Bordeaux 16(1), 251–291 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Zudilin, W.: Ramanujan-type formulae and irrationality measures of some multiples of \(\pi \), Mat. Sb. 196(7), 51–66 (2005). English transl. Sb. Math. 196, 983–998 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Zudilin, W.: Approximations to -, di- and tri- logarithms. J. Comput. Appl. Math. 202(2), 450–459 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Zudilin, W.: Arithmetic hypergeometric series. Uspekhi Mat. Nauk 66(2), 163–216 (2011). English transl. Russian Math. Surveys 66, 369–420 (2011)Google Scholar

Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

Personalised recommendations