# zbMATH — the first resource for mathematics

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.

##### MSC:
 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
Full Text:
##### References:
 [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. Zbl1008.11028 MR1882030 · Zbl 1008.11028 [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 [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. Zbl0441.10031 MR542809 · Zbl 0441.10031 [4] T. Rivoal, Indepéndance linéaire de valeurs des polylogarithmes, J. Théor. Nombres Bordeaux 15 (2003), 551-559. Zbl1079.11038 MR2140867 · Zbl 1079.11038 [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. Zbl0904.11020 MR1695002 · Zbl 0904.11020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.