Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-09-06T00:35:25.133Z Has data issue: false hasContentIssue false
  • English
  • Français

Une nouvelle approche dans la théorie des entiers friables

Part of: Multiplicative number theory

Published online by Cambridge University Press:  20 February 2017

Régis de la Bretèche  and
Gérald Tenenbaum
Régis de la Bretèche
Affiliation:
Université Paris Diderot – Paris 7, Sorbonne Paris Cité, UMR 7586, Institut de Mathématiques de Jussieu-PRG, Case 7012, F-75013 Paris, France email regis.delabreteche@imj-prg.fr
Gérald Tenenbaum
Affiliation:
Institut Élie Cartan, Université de Lorraine, BP 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France email gerald.tenenbaum@univ-lorraine.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a new approach starting with a residue computation, we sharpen some of the known estimates for the counting function of friable integers. The improved accuracy turns out to be crucial for various applications, some of which concern fundamental questions in probabilistic number theory.

Grâce à une nouvelle approche, dont le point de départ est un calcul de résidu, nous précisons certaines des estimations connues pour la fonction de comptage des entiers friables. Le gain se révèle crucial pour diverses applications, dont certaines concernent des questions fondamentales de la théorie probabiliste des nombres.

Keywords

Entiers friablesméthode du colinégalité de Montgomery–Wirsingprincipe de transfertrégion sans zérosommes d’exponentielles sur les nombres premiers

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints 11N37: Asymptotic results on arithmetic functions

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 3 , March 2017 , pp. 453 - 473
Copyright
© The Authors 2017 

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.)

Article purchase

Temporarily unavailable

References

de la Bretèche, R. and Tenenbaum, G., Propriétés statistiques des entiers friables , Ramanujan J. 9 (2005), 139–202.CrossRefGoogle Scholar
de la Bretèche, R. and Tenenbaum, G., Entiers friables : inégalité de Turán–Kubilius et applications , Invent. Math. 159 (2005), 531–588.Google Scholar
de la Bretèche, R. and Tenenbaum, G., On the friable Turán–Kubilius inequality , in Analytic and probabilistic methods in number theory, eds Manstavičius, E. et al. (TEV, Vilnius, 2012), 259–265.Google Scholar
de la Bretèche, R. and Tenenbaum, G., Sur l’inégalité de Turán–Kubilius friable , J. Lond. Math. Soc. (2) 93 (2016), 175–193.Google Scholar
de Bruijn, N. G., On the number of positive integers ⩽x and free of prime factors > y , Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50–60; Indag. Math. 13 (1951), 50–60.CrossRefGoogle Scholar
de Bruijn, N. G., On the number of positive integers ⩽x and free of prime factors > y, II , Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 239–247.CrossRefGoogle Scholar
Ennola, V., On numbers with small prime divisors , Ann. Acad. Sci. Fenn. Ser. A I 440 (1969), 3–16.Google Scholar
Granville, A., Smooth numbers: computational number theory and beyond , in Algorithmic number theory: lattices, number fields, curves and cryptography, Mathematical Sciences Research Institute Publications, vol. 44 (Cambridge University Press, Cambridge, 2008).Google Scholar
Hanrot, G., Martin, B. and Tenenbaum, G., Constantes de Turán–Kubilius friables : étude numérique , Exp. Math. 19 (2010), 345–361.Google Scholar
Hildebrand, A., Integers free of large prime factors and the Riemann hypothesis , Mathematika 31 (1984), 258–271.Google Scholar
Hildebrand, A., On the number of positive integers ⩽x and free of prime factors > y , J. Number Theory 22 (1986), 289–307.Google Scholar
Hildebrand, A. and Tenenbaum, G., On integers free of large prime factors , Trans. Amer. Math. Soc. 296 (1986), 265–290.Google Scholar
Hildebrand, A. and Tenenbaum, G., Integers without large prime factors , J. Théor. Nombres Bordeaux 5 (1993), 411–484.CrossRefGoogle Scholar
De Koninck, J.-M., Granville, A. and Luca, F. (eds), Anatomy of integers , inPapers from the CRM Workshop held at the Université de Montréal, Montréal, QC, 13–17 March 2006, CRM Proceedings and Lecture Notes, vol. 46 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Martin, B. and Tenenbaum, G., Sur l’inégalité de Turán–Kubilius friable , J. reine angew. Math. 647 (2010), 175–234.Google Scholar
Montgomery, H. L., Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, RI, 1994).Google Scholar
Pomerance, C., The role of smooth numbers in number-theoretic algorithms , in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994) (Birkhäuser, Basel, 1995), 411–422.Google Scholar
Robert, O. and Tenenbaum, G., Sur la répartition du noyau d’un entier , Indag. Math. 24 (2013), 802–914.Google Scholar
Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres, quatrième édn, Collection Échelles (Belin, Paris, 2015). Engl. transl. Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, vol. 163 (American Mathematical Society, Providence, RI, 2015).Google Scholar