Abstract
For x ∈ C, |x | < 1, s ∈ N, let Lis(x) be the s-th polylogarithm of x . We prove that for any non-zero algebraic number α such that |α| < 1, the Q(α)-vector space spanned by 1, Li1(α), Li2(α), . . . has infinite dimension. This result extends a previous one by Rivoal for rational α. 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 Pade approximation problem. Mathematics Subject Classification (2000): 11J72 (primary); 11J17, 11J91, 33C20 (secondary).
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