Linear independence of linear forms in polylogarithms
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 1-11.

For $x\in ℂ$, $|x|<1$, $s\in ℕ$, let ${\mathrm{Li}}_{s}\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $|\alpha |<1$, the $ℚ\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_{1}\left(\alpha \right),{\mathrm{Li}}_{2}\left(\alpha \right),\cdots$ 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.

Classification : 11J72,  11J17,  11J91,  33C20
@article{ASNSP_2006_5_5_1_1_0,
author = {Marcovecchio, Raffaele},
title = {Linear independence of linear forms in polylogarithms},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {1--11},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 5},
number = {1},
year = {2006},
zbl = {1114.11063},
mrnumber = {2240162},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2006_5_5_1_1_0/}
}
Marcovecchio, Raffaele. Linear independence of linear forms in polylogarithms. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 1-11. http://www.numdam.org/item/ASNSP_2006_5_5_1_1_0/

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