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Irrationality of some p-adic L-values

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Abstract

We give a proof of the irrationality of p-adic zeta-values ζ p (k) for p = 2,3 and k = 2, 3. Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.

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References

  1. Rivoal, T., Ball, R.: Irrationalité de d’une infinité de valeurs de la function zêta aux entiers impairs. Inv. Math., 146, 193–207 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zudilin, W.: Arithmetic of linear forms involving odd zeta values. J. Théorie des Nombres Bordeaux, 16, 251–291 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Rivoal, T., Zudilin, W.: Diophantine properties of numbers related to catalan’s constant. Math. Ann., 326, 705–721 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beukers, F.: Irrationality proofs using modular forms, Journées Arithmétiques de Besanco, volume 147–148 of Astérisque, pages 271–284, 1987

  5. Calegari, F.: Irrationality of certain p-adic periods for small p. Int. Math. Research Notices, 20, 1235–1249 (2005)

    Article  MathSciNet  Google Scholar 

  6. Prévost, M.: A new proof of the irrationality of ζ(2) and ζ(3) using padé approximants. J. Comp. Appl. Math., 67, 219–235 (1996)

    Article  MATH  Google Scholar 

  7. Rivoal, T.: Nombres d’euler, approximants de padet constant de catalan. Ramanujan Math. J., 11, 199–214, (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Washington, L.: Introduction to cyclotomic Fields, Springer-Verlag, New York, 1997

    MATH  Google Scholar 

  9. Diamond, J.: On the values of p-adic l-functions at positive integers. Acta Arith., 35, 223–237 (1979)

    MathSciNet  MATH  Google Scholar 

  10. Stieltjes, T. J.: Sur quelques intégrales définies et leur devéloppement en fractions continues. Quarterly J. Math. London, 24, 370–382 (1890)

    Google Scholar 

  11. Touchard, J.: Nombres exponentiels et nombres de bernoulli. Canad. J. Math., 8, 305–320 (1956)

    MathSciNet  MATH  Google Scholar 

  12. Carlitz, L.: Some polynomials of touchard connected with the bernoulli numbers. Canad. J. Math., 9, 188–190 (1957)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Universitry of Utrecht, P. O. Box 80.010, 3508 TA, Utrecht, Netherlands

    Frits Beukers

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  1. Frits Beukers
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Correspondence to Frits Beukers.

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Beukers, F. Irrationality of some p-adic L-values. Acta. Math. Sin.-English Ser. 24, 663–686 (2008). https://doi.org/10.1007/s10114-007-1029-2

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  • DOI: https://doi.org/10.1007/s10114-007-1029-2

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