Une étude asymptotique probabiliste des coefficients d'une série entière

Bernard Candelpergher 1 Michel Miniconi 1
1 JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : Following the ideas of Rosenbloom [7] and Hayman [5], Luis B ́aez-Duarte gives in [1] a probabi- listic proof of Hardy-Ramanujan's asymptotic formula for the partitions of an integer. The main principle of the method relies on the convergence in law of a family of random variables to a gaussian variable. In our work we prove a theorem of the Liapounov type (Chung [2]) that justifies this convergence. To obtain simple asymptotic formulæ a condition of the so-called strong Gaussian type defined by Luis B ́aez-Duarte is required; we demonstrate this in a situation that make it possible to obtain a classical asymptotic formula for the partitions of an integer with distinct parts (Erd ̈os-Lehner [4], Ingham [6]).
Keywords : number theory probability theory
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Bernard Candelpergher, Michel Miniconi. Une étude asymptotique probabiliste des coefficients d'une série entière. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 2014, 26 (1), p. 45-67. ⟨10.5802/jtnb.858⟩. ⟨hal-00720010v2⟩

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