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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
Article

Abstract

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.

Keywords

Irrationality exponent Rational approximations Hypergeometric series 

Résumé

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 

Notes

Acknowledgments

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.

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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

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