Multiple zeta values, Padé approximation and Vasilyev's conjecture

Stephane Fischler; Tanguy Rivoal

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arXiv · 1309.2534

Multiple zeta values, Padé approximation and Vasilyev's conjecture

Stephane Fischler·Tanguy Rivoal

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Abstract

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.

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Stephane Fischler, Tanguy Rivoal. 2013-09-10. Multiple zeta values, Padé approximation and Vasilyev's conjecture. https://arxiv.org/abs/1309.2534

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