Multiple zeta values, Pade approximation and Vasilyev's conjecture

作者：Fischler Stephane^*; Rivoal Tanguy^*

来源：Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2016, 15: 1-24.

Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Pade 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 Pade 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.

• 出版日期2016