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Sur les solutions friables de l'équation a+b=c
- Source :
- Mathematical Proceedings, Mathematical Proceedings, Cambridge University Press (CUP), 2013, 154 (3), pp.439-463. ⟨10.1017/S0305004112000643⟩, Mathematical Proceedings of the Cambridge Philosophical Society, Mathematical Proceedings of the Cambridge Philosophical Society, 2013, 154 (3), pp.439-463. ⟨10.1017/S0305004112000643⟩
- Publication Year :
- 2013
- Publisher :
- HAL CCSD, 2013.
-
Abstract
- Dans un r\'ecent article, Lagarias et Soundararajan \'etudient les solutions friables \`a l'\'equation a+b=c. Sous l'hypoth\`ese de Riemann g\'en\'eralis\'ees aux fonctions L de Dirichlet, ils obtiennent une estimation pour le nombre de solutions pond\'er\'ees par un poids lisse et une minoration pour le nombre de solutions non pond\'er\'ees. Le but de cet article est de pr\'esenter des arguments qui permettent d'une part de pr\'eciser les termes d'erreur et d'\'etendre les domaines de validit\'e de ces estimations afin de faire le lien avec un travail de la Bret\`eche et Granville, d'autre part d'obtenir le comportement asymptotique exact du nombre de solutions non pond\'er\'ees. 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 Bret\`eche 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.<br />Comment: 20 pages
- Subjects :
- Mathematics - Number Theory
General Mathematics
010102 general mathematics
0102 computer and information sciences
01 natural sciences
Upper and lower bounds
Dirichlet distribution
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Entier friable
Combinatorics
Riemann hypothesis
symbols.namesake
010201 computation theory & mathematics
somme d'exponentielle
Bounded function
symbols
11N25, 11M06
0101 mathematics
méthode du cercle
Mathematics
Subjects
Details
- Language :
- French
- ISSN :
- 03050041 and 14698064
- Database :
- OpenAIRE
- Journal :
- Mathematical Proceedings, Mathematical Proceedings, Cambridge University Press (CUP), 2013, 154 (3), pp.439-463. ⟨10.1017/S0305004112000643⟩, Mathematical Proceedings of the Cambridge Philosophical Society, Mathematical Proceedings of the Cambridge Philosophical Society, 2013, 154 (3), pp.439-463. ⟨10.1017/S0305004112000643⟩
- Accession number :
- edsair.doi.dedup.....1136ba133e138dcb40e1ae3956f74857