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# Sommes friables d'exponentielles et applications

Published:2015-02-17
Printed: Jun 2015
• Sary Drappeau,
Université Paris Diderot - Paris 7, Institut de Mathématiques de Jussieu--Paris Rive Gauche, UMR 7586, Bâtiment Chevaleret, Bureau 7C08, 75205 Paris Cedex 13
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## Abstract

An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a+b=c$ which is valid unconditionnally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.
 Keywords: théorie analytique des nombres, entiers friables, méthode du col
 MSC Classifications: 12N25 - unknown classification 12N2511L07 - Estimates on exponential sums

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