Généralisations du critère d’indépendance linéaire de Nesterenko

Abstract : This Ph.D. thesis lies in the path opened by Apéry who proved the irrationality of ζ(3) andalready followed by Ball-Rivoal who proved that there are infinitely many odd integers at which Riemann zeta function takes irrational values. A fundamental tool in the proof of Ball-Rivoal is Nesterenko’s linear independence criterion. This criterion has been generalized by Fischler and Zudilin to use common divisors of the coefficients of linear forms, under some restrictive assumptions. Then Fischler gave another generalization for simultaneous approximations (instead of small Z-linear combinations).In this Ph.D. thesis, we improve this last result by greatly weakening the assumption on thedivisors. We prove also an analogous linear independence criterion in the spirit of Siegel. Inanother part joint with Zudilin, we construct simultaneous linear approximations to ζ(2) and ζ(3) using hypergeometric identitites. These linear approximations allow one to prove at thesame time the irrationality of ζ(2) and that of ζ(3). Then, using a criterion from the previouspart, we deduce a lower bound on Z-linear combinations of 1, ζ(2) and ζ(3), under somestrong divisibility hypotheses on the coefficients (so that the Q-linear independence of thesethree numbers still remains an open problem).
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Simon Dauguet. Généralisations du critère d’indépendance linéaire de Nesterenko. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2014. Français. ⟨NNT : 2014PA112085⟩. ⟨tel-01084492⟩



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