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Linear independence of linear forms in polylogarithms. (English) Zbl 1114.11063
For a complex number \(x\) with \(|x|<1\) and a positive integer \(s\), the \(s\)-th polylogarithms of \(x\) are \({\text Li}_s(x)=\sum_{k=1}^\infty x^k/k^s\). The purpose of this paper is to prove that for any non-zero algebraic number \(\alpha\) with \(|\alpha|<1\) in the sequence of \(1\), \(\text{Li}_1(\alpha), \text{Li}_2(\alpha), \dots\) infinitely many terms are linearly independent over \(\mathbb Q(\alpha)\). This result extends a previous one for rational \(\alpha\) by T. Rivoal [J. Théor. Nombres Bordx. 15, No. 2, 551–559 (2003; Zbl 1079.11038)]. The main tool is a method introduced by S. Fischler and T. Rivoal [J. Math. Pures Appl. (9) 82, No. 10, 1369–1394 (2003; Zbl 1064.11053)], which shows that the vector of the coefficients of the polylogarithms in the relevant series is the unique non-zero solution (up to a multiplicative constant) of a suitable Padé approximation problem.

11J72 Irrationality; linear independence over a field
11J17 Approximation by numbers from a fixed field
11J91 Transcendence theory of other special functions
11G55 Polylogarithms and relations with \(K\)-theory
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[2] S. Fischler and T. Rivoal, Approximants de Padé et séries hypergéométriques équilibrées, J. Math. Pures Appl. (9) 82 (2003), 1369-1394. Zbl1064.11053 MR2020926 · Zbl 1064.11053
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