Abstract

For $x\in \mathbb{C}$, $|x|<1$, $s\in \mathbb{N}$, let ${\rm Li}_s$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha $ such that $|\alpha |<1$, the $\mathbb{Q}$-vector space spanned by $1,{\rm Li}_1,{\rm Li}_2,\dots $ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha $. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem