Integral identities and constructions of approximations to zeta-values

Yuri V. Nesterenko

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 2, page 535-550
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Nesterenko, Yuri V.. "Integral identities and constructions of approximations to zeta-values." Journal de théorie des nombres de Bordeaux 15.2 (2003): 535-550. <http://eudml.org/doc/249071>.

@article{Nesterenko2003,
abstract = {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.},
author = {Nesterenko, Yuri V.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {zeta function; linear forms in polylogarithms; hypergoemetric integrals},
language = {eng},
number = {2},
pages = {535-550},
publisher = {Université Bordeaux I},
title = {Integral identities and constructions of approximations to zeta-values},
url = {http://eudml.org/doc/249071},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Nesterenko, Yuri V.
TI - Integral identities and constructions of approximations to zeta-values
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 2
SP - 535
EP - 550
AB - 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.
LA - eng
KW - zeta function; linear forms in polylogarithms; hypergoemetric integrals
UR - http://eudml.org/doc/249071
ER -

References

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  1. [1] F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc.11 (1979), 268-272. Zbl0421.10023
  2. [2] L.A. Gutnik, The irrationality of certain quantities involving ζ(3). Acta Arith.42 (1983), 255-264. Zbl0474.10026
  3. [3] Yu L. Luke, Mathematical functions and their approximations. Academic Press, New York, 1975. Zbl0318.33001MR501762
  4. [4] YU. Nesterenko, A few remarks on ζ(3). Math. Notes59 (1996), 625-636. Zbl0888.11028
  5. [5] T. Hessami- Pilerhood, Linear independence of vectors with polylogarithmic coordinates. Vestnik Moscow University Ser.1 (1999), no6, 54-56. Zbl0983.11044MR1735148
  6. [6] T. Rivoal, La fonction Zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris331 (2000), 267-270. Zbl0973.11072MR1787183
  7. [7] L.J. Slater, Generalized hypergeometric functions. Cambridge Univ. Press, 1966. Zbl0135.28101MR201688
  8. [8] E.T. Whittaker, G.N. Watson, A course of modern analysis. CambdidgeUniversity Press, 1927. Zbl45.0433.02MR1424469JFM53.0180.04
  9. [9] V.V. Zudilin, On irrationality of values of Riemann zeta function. Izvestia of Russian Acad. Sci.66, 2002, 1-55. Zbl1114.11305MR1921809

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