Integral identities and constructions of approximations to zeta-values

@article{Nesterenko2003IntegralIA,
  title={Integral identities and constructions of approximations to zeta-values},
  author={Yu. V. Nesterenko},
  journal={Journal de Theorie des Nombres de Bordeaux},
  year={2003},
  volume={15},
  pages={535-550}
}
  • Y. Nesterenko
  • Published 2003
  • Mathematics
  • Journal de Theorie des Nombres de Bordeaux
Nous presentons une construction generale de combinaisons lineaires a coefficients rationnels en les valeurs de la fonction zeta de Riemann aux entiers. Ces formes lineaires sont exprimees en termes d'integrales complexes, dites de Barnes, ce qui permet de les estimer. Nous montrons quelques identites reliant ces integrales a des integrales multiples sur le cube unite reel. 
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References

SHOWING 1-9 OF 9 REFERENCES
La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs
Resume Nous montrons que la dimension de l'espace vectoriel engendre sur les rationnels par 1 et les n premieres valeurs de la fonction zeta de Riemann aux entiers impairs croit au moins comme unExpand
Irrationality of values of the Riemann zeta function
The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We proveExpand
Generalized Hypergeometric Functions
Introduction Multiplication by Xu (Gauss contiguity) Algebraic theory Variation of Wa with g Analytic theory Deformation theory Structure of Hg Linear differential equations over a ring SingularitiesExpand
A Course of Modern Analysis
TLDR
The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis. Expand
Linear independence of vectors with polylogarithmic coordinates
  • Vestnik Moscow University Ser.1
  • 1999
A few remarks on 03B6(3)
  • Math. Notes 59
  • 1996
The irrationality of certain quantities involving 03B6(3)
  • Acta Arith. 42
  • 1983
A note on the irrationality of 03B6(2) and 03B6(3)
  • Bull. London Math. Soc. 11
  • 1979
Mathematical functions and their approximations