A note on linear independence of polylogarithms over the rationals

Abstract

In this article, we give a new lower bound for the dimension of the linear space over the rationals spanned by 1 and values of polylogarithmic functions at a non-zero rational number. Our proof uses Padé approximation following the argument of T. Rivoal, however we adapt a new linear independence criterion due to S. Fischler and W. Zudilin. We also present an example of the linear space of dimension $\geqslant 3$ over $\mathbf{Q}$, which is generated by 1 and polylogarithms.