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

Published online by Cambridge University Press:  16 January 2013

SARY DRAPPEAU
SARY DRAPPEAU*
Affiliation:
Université Denis Diderot - Paris VII, Institut de Mathématiques de Jussieu (UMR 7586) Bâtiment Chevaleret, Bureau 7C08, 75205 Paris Cedex 13, France. e-mail: drappeau@math.jussieu.fr
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 439 - 463
Copyright
Copyright © Cambridge Philosophical Society 2013

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References

REFERENCES

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