Integral identities and constructions of approximations to zeta-values
Journal de Théorie des Nombres de Bordeaux, Volume 15 (2003) no. 2, pp. 535-550.

Some general construction of linear forms with rational coefficients in values of Riemann zeta-function at integer points is presented. These linear forms are expressed in terms of complex integrals of Barnes type that allows to estimate them. Some identity connecting these integrals and multiple integrals on the real unit cube is proved.

Nous présentons une construction générale de combinaisons linéaires à coefficients rationnels en les valeurs de la fonction zêta de Riemann aux entiers. Ces formes linéaires sont exprimées en termes d'intégrales complexes, dites de Barnes, ce qui permet de les estimer. Nous montrons quelques identités reliant ces intégrales à des intégrales multiples sur le cube unité réel.

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     title = {Integral identities and constructions of approximations to zeta-values},
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Yuri V. Nesterenko. Integral identities and constructions of approximations to zeta-values. Journal de Théorie des Nombres de Bordeaux, Volume 15 (2003) no. 2, pp. 535-550. doi : 10.5802/jtnb.412. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.412/

[1] F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11 (1979), 268-272. | Zbl: 0421.10023

[2] L.A. Gutnik, The irrationality of certain quantities involving ζ(3). Acta Arith. 42 (1983), 255-264. | Zbl: 0474.10026

[3] Yu L. Luke, Mathematical functions and their approximations. Academic Press, New York, 1975. | MR: 501762 | Zbl: 0318.33001

[4] Yu. Nesterenko, A few remarks on ζ(3). Math. Notes 59 (1996), 625-636. | Zbl: 0888.11028

[5] T. Hessami- Pilerhood, Linear independence of vectors with polylogarithmic coordinates. Vestnik Moscow University Ser.1 (1999), no6, 54-56. | MR: 1735148 | Zbl: 0983.11044

[6] T. Rivoal, La fonction Zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris 331 (2000), 267-270. | MR: 1787183 | Zbl: 0973.11072

[7] L.J. Slater, Generalized hypergeometric functions. Cambridge Univ. Press, 1966. | MR: 201688 | Zbl: 0135.28101

[8] E.T. Whittaker, G.N. Watson, A course of modern analysis. Cambdidge University Press, 1927. | JFM: 53.0180.04 | MR: 1424469

[9] V.V. Zudilin, On irrationality of values of Riemann zeta function. Izvestia of Russian Acad. Sci. 66, 2002, 1-55. | MR: 1921809

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