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Entiers friables en progressions arithmétiques, et applications

Sary Drappeau 1 
1 Équipe de th. des nombres
IMJ - Institut de Mathématiques de Jussieu
Abstract : In this thesis, we study additive properties of integers having no large prime factors. An integer is said to be y-friable if all its prime factors are less than y. The first problem we focus on, is that of counting solutions to the equation a+b=c in y-friable integers a, b and c. The second question we address is the mean values of arithmetical functions over shifted friable numbers, of the shape n-1 where n is y-friable. The circle method reduces the first question to the problem of estimating exponential sums twisted by Dirichlet characters, over friable numbers, which is then done using classical harmonic analysis tools, and the saddle-point method. The first and second chapters study the problem respectively with and without assuming the Riemann hypothesis generalized to Dirichlet L-functions. The third and fourth chapters are dedicated to the second question, which reduces to the problem of the repartition of friable numbers in arithmetic progressions. This involves sums of Dirichlet characters over friable integers, as well as the large sieve. In the last chapter, the dispersion method is used to study the case of the average number of divisors of shifted friable integers.
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Submitted on : Thursday, January 9, 2014 - 2:42:58 PM
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  • HAL Id : tel-00926351, version 1


Sary Drappeau. Entiers friables en progressions arithmétiques, et applications. Théorie des nombres [math.NT]. Université Paris-Diderot - Paris VII, 2013. Français. ⟨NNT : ⟩. ⟨tel-00926351⟩



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