Multiple zeta values, Padé approximation and Vasilyev's conjecture

• Stéphane Fischler ^[1] ; Tanguy Rivoal ^[2]
  1. [1] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  2. [2] Institut Fourier, Université Grenoble 1, France
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 15, Nº Extra 1 (Special volume), 2016, págs. 1-24
• Idioma: inglés
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• Resumen

  • Sorokin gave in 1996 a new proof that ⇡ is transcendental. It is based on a simultaneous Padé approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of ⇡. In this paper we construct a Padé approximation problem of the same flavour, and prove that it has a unique solution up to proportionality. At the point 1, this provides a rational linear combination of 1 and multiple zeta values in an extended sense that turn out to be values of the Riemann zeta function at odd integers. As an application, we obtain a new proof of Vasilyev’s conjecture for any odd weight, which concerns the explicit evaluation of certain hypergeometric multiple integrals, first proved by Zudilin in 2003.