Corpus ID: 118935111

Linear independence of linear forms in polylogarithms

@article{Marcovecchio2006LinearIO,
  title={Linear independence of linear forms in polylogarithms},
  author={Raffaele Marcovecchio},
  journal={Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze},
  year={2006},
  volume={5},
  pages={1-11}
}
  • Raffaele Marcovecchio
  • Published 2006
  • Mathematics
  • Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze
For x ∈ C, |x | < 1, s ∈ N, let Lis(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, Li1(α), Li2(α), . . . 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 Pade approximation problem. Mathematics… Expand
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References

SHOWING 1-6 OF 6 REFERENCES
Approximation measures for logarithms of algebraic numbers
Given a number field K and a number ) lll we say that A > 0 is a K-irrationality measure of 03BE if, for any e > 0, > - ( 1 + e) p h(o) for all p E K with sufficiently large Weil logarithmic heightExpand
Indépendance linéaire des valeurs des polylogarithmes
Nous montrons que pour tout rationnel a de [-1,1], l'ensemble des valeurs des polylogarithmes {Li,(a),s E N, s > 1} contient une infinite de nombres Q-lineairement independants.
Approximants de Padé et séries hypergéométriques équilibrées
Resume Dans cet article, nous enoncons et resolvons des problemes d'approximation de Pade nouveaux et tres generaux dont les solutions s'expriment a l'aide de series hypergeometriques : parExpand
Indepéndance linéaire de valeurs des polylogarithmes , J . Théor
  • 2003
Analytic Number Theory: Hypergeometric Functions and Irrationality Measures
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