For , , , let be the -th polylogarithm of . We prove that for any non-zero algebraic number such that , the -vector space spanned by has infinite dimension. This result extends a previous one by Rivoal for rational . 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.
@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}, mrnumber = {2240162}, zbl = {1114.11063}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2006_5_5_1_1_0/} }
TY - JOUR AU - Marcovecchio, Raffaele TI - Linear independence of linear forms in polylogarithms JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 SP - 1 EP - 11 VL - 5 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2006_5_5_1_1_0/ LA - en ID - ASNSP_2006_5_5_1_1_0 ER -
%0 Journal Article %A Marcovecchio, Raffaele %T Linear independence of linear forms in polylogarithms %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 1-11 %V 5 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2006_5_5_1_1_0/ %G en %F 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://archive.numdam.org/item/ASNSP_2006_5_5_1_1_0/
[1] Approximation measures for logarithms of algebraic numbers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 225-249. | EuDML | Numdam | MR | Zbl
and ,[2] Approximants de Padé et séries hypergéométriques équilibrées, J. Math. Pures Appl. (9) 82 (2003), 1369-1394. | MR | Zbl
and ,[3] On the irrationality of the values of the functions , Math. Sb. (N.S.) 109 (151) (1979), 410-417 (in Russian); English translation in Math. URSS-Sb. 37 (1980), 381-388. | MR | Zbl
,[4] Indepéndance linéaire de valeurs des polylogarithmes, J. Théor. Nombres Bordeaux 15 (2003), 551-559. | EuDML | Numdam | MR | Zbl
,[5] Hypergeometric functions and irrationality measures, In: “Analytic Number Theory", Y. Motohashi (ed.), London Math. Soc. Lecture Note Series 247, Cambridge Univ. Press, 1997, 353-360. | MR | Zbl
,