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

Published online by Cambridge University Press:  16 January 2013

Université Denis Diderot - Paris VII, Institut de Mathématiques de Jussieu (UMR 7586) Bâtiment Chevaleret, Bureau 7C08, 75205 Paris Cedex 13, France. e-mail:


In a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally 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 prove a more precise conditional 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 [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

Research Article
Copyright © Cambridge Philosophical Society 2013

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