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Linear forms in zeta values arising from certain Sorokin-type integrals

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This paper deals with certain multiple integrals which can be represented as linear forms of zeta values with rational coefficients.

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References

  1. R. Apéry, “Irrationalité de ζ(2) et ζ(3),” Astérisque, 61, 11–13 (1979).

    MATH  Google Scholar 

  2. K. Ball and T. Rivoal, “Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs,” Invent. Math., 146, No. 1, 193–207 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  3. F. Beukers, “A note on the irrationality of ζ(2) and ζ(3),” Bull. London Math. Soc., 11, 268–272 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  4. F. Beukers, “Padé-approximations in number theory,” in: M. G. de Bruin and H. van Rossum, eds., Padé Approximation and Its Applications: Amsterdam 1980. Proc. Conference Held in Amsterdam, The Netherlands, October 29–31, 1980, Lect. Notes Math., Vol. 888, Springer, Berlin (1981), pp. 90–99. Padé approximation and its applications, Amsterdam 1980 (Amsterdam, 1980), pp. 90–99, Lecture Notes in Math. 888, 1981.

  5. J. Cresson, S. Fischler, and T. Rivoal, “Phénomènes de symétrie dans des formes linéaires en polyzêtas,” J. Reine Angew. Math., 617, 109–151 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Cresson, S. Fischler, and T. Rivoal, “Séries hypergéométriques multiples et polyzêtas,” Bull. Soc. Math. France, 136, No. 1, 97–145 (2008).

    MATH  MathSciNet  Google Scholar 

  7. J. Cresson, S. Fischler, and T. Rivoal, http://www.math.u-psud.fr/~fischler/algo.html.

  8. S. Fischler, “Groupes de Rhin–Viola et intégrales multiples,” J. Théor. Nombres Bordeaux, 15, No. 2, 479–534 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Fischler, “Multiple series connected to Hoffman’s conjecture on multiple zeta values,” J. Algebra, 320, 1682–1703 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  10. L. A. Gutnik, “Irrationality of some quantities that contain ζ(3),” Acta Arith., 42, No. 3, 255–264 (1983).

    MATH  MathSciNet  Google Scholar 

  11. C. Krattenthaler and T. Rivoal, “An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series,” Ramanujan J., 13, No. 1-3, 203–219 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  12. C. Krattenthaler and T. Rivoal, Hypergéomé et fonction zêta de Riemann, Mem. Amer. Math. Soc., Vol. 186, Amer. Math. Soc., 2007.

  13. Yu. V. Nesterenko, “Some remarks on ζ(3),” Math. Notes, 59, No. 5-6, 625–636 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  14. Yu. V. Nesterenko, “Integral identities and constructions of approximations to zeta-values,” J. Théor. Nombres Bordeaux, 15, No. 2, 535–550 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Rhin and C. Viola, “On a permutation group related to ζ(2),” Acta Arith., 77, 23–56 (1996).

    MATH  MathSciNet  Google Scholar 

  16. G. Rhin and C. Viola, “The group structure for ζ(3),” Acta Arith., 97, No. 3, 269–293 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Rhin and C. Viola, “Multiple integrals and linear forms in zeta-values,” Funct. Approx., 37, 429–444 (2007).

    MATH  MathSciNet  Google Scholar 

  18. T. Rivoal, “La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs,” C. R. Acad. Sci. Paris, Sér. I. Math., 331, No. 4, 267–270 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  19. T. Rivoal, “Irrationalité d’au moins un des neuf nombres ζ(5), ζ(7),…, ζ(21),” Acta Arith., 103, No. 2, 157–167 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Rivoal, “Valeurs aux entiers de la fonction zêta de Riemann,” Quadrature, 49 (2003), http://www-fourier.ujf-grenoble.fr/~rivoal/articles/quaddefi.pdf.

  21. V. N. Sorokin, “Apéry’s theorem,” Moscow Univ. Math. Bull., No. 3, 48–52 (1998).

    MathSciNet  Google Scholar 

  22. D. V. Vasilyev, “Approximations of zero by linear forms in values of the Riemann zeta-function,” Dokl. Nats. Akad. Nauk Belarusi, 45, No. 5, 36–40 (2001).

    MathSciNet  Google Scholar 

  23. S. Zlobin, “Integrals presented as linear forms in generalized polylogarithms,” Mat. Zametki, 71, No. 5, 782–787 (2002).

    MathSciNet  Google Scholar 

  24. S. Zlobin, “On some integral identities,” Russ. Math. Surv., 57, No. 3, 617–618 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  25. W. Zudilin, “Well-poised hypergeometric service for diophantine problems of zeta values,” J. Théor. Nombres Bordeaux, 15, No. 2, 593–626 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  26. W. Zudilin, “Arithmetic of linear forms involving odd zeta values,” J. Théor. Nombres Bordeaux, 16, 251–291 (2004).

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Tanguy Rivoal.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 5, pp. 161–172, 2010.

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Rivoal, T. Linear forms in zeta values arising from certain Sorokin-type integrals. J Math Sci 180, 641–649 (2012). https://doi.org/10.1007/s10958-012-0662-1

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