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Répartition d'entiers avec contraintes sur les diviseurs

Abstract : This thesis deals with the distribution of three sets of integers characterized by some properties on their divisors. In the first part we use the one-dimensional saddle point method to study the density of square free integers whose k-th prime factor is fixed. The second part is concerned with the estimate of the number [pi subscript k] (x) of integers lower than x which have exactly k prime factors, obtained by using the saddle point method in dimension 2.We get a uniform estirnate for [pi subscript k](X) in a domain of parameters (x, k) near to the optimum, from which we deduce new informations on the local behavior of k ---> [pi subscrip tk](X). This provides a new proof of the Erdôs conjecture about the unimodality of k ---> [pi subscript k](X), proved in 1990 by Balazard, and even an asymptotic estimate of the quotient [pi subscript k+1] (X)/[pi](x).At last,we study the normal divisorial distribution of vd modulo 1, where v is a real number and d exhausts the divisors of any normal integer: we prove in this setting that the real numbers vd have a regular distribution modulo1, for a large class of irrational numbers v.
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Sébastien Kerner. Répartition d'entiers avec contraintes sur les diviseurs. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2002. Français. ⟨NNT : 2002NAN10239⟩. ⟨tel-01746530⟩



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