Sommes friables d'exponentielles et applications

Drappeau, Sary

Canadian journal of mathematics, Tome 67 (2015) no. 3, pp. 597-638

Résumé

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.

Détail

DOI : 10.4153/CJM-2014-036-5

Mots-clés : 12N25, 11L07, théorie analytique des nombres, entiers friables, méthode du col