Multiple zeta values, Pad\'e approximation and Vasilyev's conjecture
Abstract
Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Pad\'e 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 Pad\'e 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.
Cite
@article{arxiv.1309.2534,
title = {Multiple zeta values, Pad\'e approximation and Vasilyev's conjecture},
author = {Stephane Fischler and Tanguy Rivoal},
journal= {arXiv preprint arXiv:1309.2534},
year = {2013}
}