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arXiv · 1203.1742

Sur les solutions friables de l'équation a+b=c

Sary Drappeau
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Abstract

Dans un récent article, Lagarias et Soundararajan étudient les solutions friables à l'équation a+b=c. Sous l'hypothèse de Riemann généralisées aux fonctions L de Dirichlet, ils obtiennent une estimation pour le nombre de solutions pondérées par un poids lisse et une minoration pour le nombre de solutions non pondérées. Le but de cet article est de présenter des arguments qui permettent d'une part de préciser les termes d'erreur et d'étendre les domaines de validité de ces estimations afin de faire le lien avec un travail de la Bretèche et Granville, d'autre part d'obtenir le comportement asymptotique exact du nombre de solutions non pondérées. In a recent paper, Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we aim to prove a more precise estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work. We also prove the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

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Sary Drappeau. 2012-03-08. Sur les solutions friables de l'équation a+b=c. https://doi.org/10.1017/s0305004112000643

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