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

For x, |x|<1, s, let Li s (x) be the s-th polylogarithm of x. We prove that for any non-zero algebraic number α such that |α|<1, the (α)-vector space spanned by 1, Li 1 (α), Li 2 (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 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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     pages = {1-11},
     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, Serie 5, Volume 5 (2006) no. 1, pp. 1-11. http://www.numdam.org/item/ASNSP_2006_5_5_1_1_0/

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