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Sommes friables d'exponentielles et applications

Published online by Cambridge University Press:  20 November 2018

Sary Drappeau*
Affiliation:
Université Paris Diderot – Paris 7, Institut de Mathé matiques de Jussieu–Paris Rive Gauche, UMR 7586, Bâtiment Chevaleret, Bureau 7C08, 75205 Paris Cedex 13 courriel: drappeau@math.jussieu.fr, sary.drappeau@gmail.com
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Abstract

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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 $y\,\ge \,\exp \left\{ c \right.\sqrt{\log \,x\,}\log \,\log \,\left. x \right\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a\,+\,b\,=\,c$ which is valid unconditionally 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

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