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

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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.

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.

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Notes

  1. This formula appears as equation (26) below.

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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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  1. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW , 2308, Australia

    Wadim Zudilin

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  1. Wadim Zudilin
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Correspondence to Wadim Zudilin.

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The author is supported by the Australian Research Council.

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Zudilin, W. Two hypergeometric tales and a new irrationality measure of \(\zeta (2)\) . Ann. Math. Québec 38, 101–117 (2014). https://doi.org/10.1007/s40316-014-0016-0

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