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
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- [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 , 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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