Linear independence of linear forms in polylogarithms

Marcovecchio, Raffaele

Marcovecchio, Raffaele

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 1-11.

Résumé

For $x \in ℂ$ , $| x | < 1$ , $s \in ℕ$ , 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 $ℚ (α)$ -vector space spanned by $1, {Li}_{1} (α), {Li}_{2} (α), \dots$ 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.

MR Zbl | 2 citations dans Numdam

Classification : 11J72, 11J17, 11J91, 33C20

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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/

Bibliographie
Cité par

[1] F. Amoroso and C. Viola, Approximation measures for logarithms of algebraic numbers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 225-249. | EuDML | Numdam | MR | Zbl

[2] S. Fischler and T. Rivoal, Approximants de Padé et séries hypergéométriques équilibrées, J. Math. Pures Appl. (9) 82 (2003), 1369-1394. | MR | Zbl

[3] E. M. Nikishin, On the irrationality of the values of the functions $F (x, s)$ , Math. Sb. (N.S.) 109 (151) (1979), 410-417 (in Russian); English translation in Math. URSS-Sb. 37 (1980), 381-388. | MR | Zbl

[4] T. Rivoal, Indepéndance linéaire de valeurs des polylogarithmes, J. Théor. Nombres Bordeaux 15 (2003), 551-559. | EuDML | Numdam | MR | Zbl

[5] C. Viola, 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