# Linear independence of linear forms in polylogarithms

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

- Volume: 5, Issue: 1, page 1-11
- ISSN: 0391-173X

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topMarcovecchio, Raffaele. "Linear independence of linear forms in polylogarithms." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.1 (2006): 1-11. <http://eudml.org/doc/242318>.

@article{Marcovecchio2006,

abstract = {For $x\in \mathbb \{C\}$, $|x|<1$, $s\in \mathbb \{N\}$, let $\{\rm Li\}_s(x)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha $ such that $|\alpha |<1$, the $\mathbb \{Q\}(\alpha )$-vector space spanned by $1,\{\rm Li\}_1(\alpha ),\{\rm Li\}_2(\alpha ),\dots $ 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.},

author = {Marcovecchio, Raffaele},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {1},

pages = {1-11},

publisher = {Scuola Normale Superiore, Pisa},

title = {Linear independence of linear forms in polylogarithms},

url = {http://eudml.org/doc/242318},

volume = {5},

year = {2006},

}

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

PB - Scuola Normale Superiore, Pisa

VL - 5

IS - 1

SP - 1

EP - 11

AB - For $x\in \mathbb {C}$, $|x|<1$, $s\in \mathbb {N}$, let ${\rm Li}_s(x)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha $ such that $|\alpha |<1$, the $\mathbb {Q}(\alpha )$-vector space spanned by $1,{\rm Li}_1(\alpha ),{\rm Li}_2(\alpha ),\dots $ 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.

LA - eng

UR - http://eudml.org/doc/242318

ER -

## References

top- [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. Zbl1008.11028MR1882030
- [2] S. Fischler and T. Rivoal, Approximants de Padé et séries hypergéométriques équilibrées, J. Math. Pures Appl. (9) 82 (2003), 1369–1394. Zbl1064.11053MR2020926
- [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. Zbl0441.10031MR542809
- [4] T. Rivoal, Indepéndance linéaire de valeurs des polylogarithmes, J. Théor. Nombres Bordeaux 15 (2003), 551–559. Zbl1079.11038MR2140867
- [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. Zbl0904.11020MR1695002

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