Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-29T19:50:55.158Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]de la Bretèche, R. and Tenenbaum, G.Propriétés statistiques des entiers friables. The Ramanujan J. 9, 1 (2005), 139–202.CrossRefGoogle Scholar
[2]de la Bretèche, R. and Granville, A. Densité des friables. Bull. Soc. Math. France, à paraître.Google Scholar
[3]Hildebrand, A. and Tenenbaum, G.On integers free of large prime factors. Trans. Amer. Math. Soc. 296, 1 (1986), 265–290.CrossRefGoogle Scholar
[4]Lagarias, J. C. and Soundararajan, K.Smooth solutions to the abc equation: the xyz Conjecture. J. Théor. Nombres Bordeaux 23, 1 (2009), 209–234.CrossRefGoogle Scholar
[5]Lagarias, J. C. and Soundararajan, K.Counting smooth solutions to the equation A+B=C. Proc. London Math. Soc. 104, 4 (2012), 770–798.CrossRefGoogle Scholar
[6]Oesterlé, J.Nouvelles aproches du “théoreme” de Fermat, Séminaire Bourbaki no. 694 (1987-88). Astérisque 161–162 (1988), 165–186.Google Scholar