Sur les facteurs premiers milieux d'un entier
|Advisor:||De Koninck, Jean-Marie; Doyon, Nicolas|
|Abstract:||The aim of this thesis is the study of some sums of the prime factors that are between the smallest and the largest ones, called the middle prime factors. In particular, this is an extended version of published and prepublished articles on this subject. The first chapter develops all the preliminary notions necessary for the good understanding of this document. In particular, arithmetic and asymptotic notations are established. Moreover, some classical analytic number theory results, such as the Abel summation formula, Mertens' formula and Dirichlet series, and an introduction to the theory of uniform distribution mod 1 are mentioned. The second chapter is about some problems on the asymptotic behavior of sums and series of integers that have a given number of prime factors, with or without multiplicity, and that have other properties concerning their smallest and biggest prime factors. In the case of smooth numbers, one of the result was obtained by Erdõs and Tenenbaum by the use of the saddle-point method. For the integers without small prime factors, the results were obtained by Alladi by the use of the Selberg-Delange method. The third chapter exposes the first main result of this thesis, namely the study of the asymptotic behavior of the sum of the reciprocals of the β-positioned prime factors of the integers n ≤ x. The proofs improve and generalize previous work of De Koninck and Luca about the middle prime factor. This was possible by the use of Alladi's and Erdõs and Tenenbaum's results which are given in the second chapter. This third chapter ends with the study of the distribution of the β-positioned prime factor. The fourth chapter presents the second main result, which is about the study of the asymptotic behavior of the sum of the reciprocals of the β-positioned prime factors with multiplicity of the integers n ≤ x. The methods used are different from those in the third chapter and allow for much more precise estimates. Moreover, this chapter ends by showing that the proof can be improved in the case of the middle prime factor with multiplicity.|
|Document Type:||Thèse de doctorat|
|Open Access Date:||21 December 2018|
|Collection:||Thèses et mémoires|
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