Multiple zeta values, Pad\'e approximation and Vasilyev's conjecture

TL;DR

This paper constructs a new Padé approximation problem related to multiple zeta values, providing a proof of Vasilyev's conjecture for odd weights and connecting multiple polylogarithms with values of the Riemann zeta function.

Contribution

It introduces a novel Padé approximation problem with a unique solution, leading to new insights into multiple zeta values and a proof of Vasilyev's conjecture for all odd weights.

Findings

Constructed a new Padé approximation problem with a unique solution.

Established a rational linear combination involving multiple zeta values and zeta at odd integers.

Provided a new proof of Vasilyev's conjecture for any odd weight.

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.