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Sur deux questions de crible

Abstract : This thesis is divided into two main parts. In the first chapter, we consider the number of integers not exceeding x and admitting no divisor in an arithmetic progression a(mod q) where q is fixed. We improve here a result of Narkiewicz and Radziejewski published in 2011 by providing a different main term with a simpler expression, and we specify the term error. The main tools are the Selberg-Delange method and the Hankel contour. We also study in detail the particular case where a is a quadratic nonresidue modulo q. We also extend our result to the integers which admit no divisor in a finite set of residual classes modulo q. In the second chapter, we study the ultrafriable integers in arithmetic progressions. An integer is said to be y-ultrafriable if no prime power which divide it exceeds y. We begin with the studying of the counting function of these integers when they are coprime to q. Then we give an asymptotic formula about the number of y-ultrafriable integers which don’t exceed a number x and in an arithmetic progression a modulo q, where q is a y-friable modulus, which means that it is without a prime divisor exceeding y. Our results are valid when q, x, y are integers which verify log x « y < x, q < yc/ log log y, where c > 0 is a suitably chosen constant.
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Submitted on : Monday, January 4, 2021 - 3:49:02 PM
Last modification on : Wednesday, November 3, 2021 - 7:10:32 AM
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  • HAL Id : tel-02499408, version 1


David Feutrie. Sur deux questions de crible. Mathématiques [math]. Université de Lorraine, 2019. Français. ⟨NNT : 2019LORR0173⟩. ⟨tel-02499408⟩



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