Multiple zeta values, Padé approximation and Vasilyev's conjecture

Stephane Fischler (1) , Tanguy Rivoal (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IF - Institut Fourier

Sorokin gave in 1996 a new proof that pi 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 pi. 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, concerning the explicit evaluation of certain hypergeometric multiple integrals; it was first proved by Zudilin in 2003.

Mots clés: {Multiple zeta value; Padé approximation; Hypergeometric integrals; Polylogarithm

Domaines: Théorie des nombres [math.NT]

Stephane Fischler, Tanguy Rivoal. Multiple zeta values, Padé approximation and Vasilyev's conjecture. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2016, 15, pp.1-24. ⟨hal-00860302⟩