Sommes friables d'exponentielles et applications

TL;DR

This paper develops estimates for exponential sums over y-friable numbers and applies these to count solutions to the equation a+b=c, advancing understanding of friable numbers in additive problems.

Contribution

It provides new non-trivial estimates for exponential sums over y-friable numbers and derives an asymptotic formula for solutions to a+b=c in this set, unconditionally under certain conditions.

Findings

Established estimates for exponential sums over y-friable numbers.

Derived an asymptotic count for y-friable solutions to a+b=c.

Applied saddle point method to analyze friable numbers in additive equations.

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 & Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.