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Entiers friables dans des progressions arithmétiques de grand module

Part of: Multiplicative number theory Sequences and sets

Published online by Cambridge University Press:  20 March 2019

R. DE LA BRETÈCHE  and
D. FIORILLI
R. DE LA BRETÈCHE
Affiliation:
Institut de Mathématiques de Jussieu–Paris Rive Gauche, Université Paris Diderot, Sorbonne Paris Cité, UMR 7586, Case Postale 7012, F-75251 Paris CEDEX 13, France. e-mails: regis.delabreteche@imj-prg.fr, daniel.fiorilli@math.u-psud.fr
D. FIORILLI
Affiliation:
Institut de Mathématiques de Jussieu–Paris Rive Gauche, Université Paris Diderot, Sorbonne Paris Cité, UMR 7586, Case Postale 7012, F-75251 Paris CEDEX 13, France. e-mails: regis.delabreteche@imj-prg.fr, daniel.fiorilli@math.u-psud.fr
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Résumé

We study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ y ⩽ x and on average over q ⩽ x/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11B25: Arithmetic progressions 11N37: Asymptotic results on arithmetic functions 11N69: Distribution of integers in special residue classes

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 75 - 102
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Bourgain, J.. Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc. à paraitre. 30 (2017), no. 1, 205–224.CrossRefGoogle Scholar
de la Bretèche, R. et Tenenbaum, G.. Propriétés statistiques des entiers friables. Ramanujan Journal 9 (2005), n° 1-2, 139–202.CrossRefGoogle Scholar
Drappeau, S.. Théorème de Fouvry–Iwaniec pour les entiers friables. Compositio Math. 151 (2015), pp. 828–862.CrossRefGoogle Scholar
Fiorilli, D.. Residue classes containing an unexpected number of primes. Duke Math. J. 161 (2012), no. 15, 2923–2943.CrossRefGoogle Scholar
Fiorilli, D.. The influence of the first term of an arithmetic progression. Proc. London Math. Soc. (3) 106 (2013), no. 4, 819–858.CrossRefGoogle Scholar
Fouvry, É. et Tenenbaum, G.. Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques. Proc. London Math. Soc. (3) 72 (1996), no. 3, p. 481–514.CrossRefGoogle Scholar
Harper, A. J.. On a paper of K. Soundararajan on smooth numbers in arithmetic progressions. J. Number Theory 132 (2012), no. 1, 182–199.CrossRefGoogle Scholar
Harper, A. J.. Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers, pré-publication (2012).Google Scholar
Hildebrand, A.. Integers free of large prime factors and the Riemann hypothesis. Mathematika 31 (1984), no. 2, (1985), 258–271.CrossRefGoogle Scholar
Hildebrand, A.. Integers free of large prime divisors in short intervals. Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 57–69.CrossRefGoogle Scholar
Hildebrand, A.. On the number of positive integers ⩽ x and free of prime factors > y. J. Number Theory 22 (1986), 289–307.CrossRefGoogle Scholar
Hildebrand, A. et Tenenbaum, G.. On integers free of large prime factors. Trans. Amer. Math. Soc. 296 (1986), 265–290.CrossRefGoogle Scholar
Lachand, A. et Tenenbaum, G.. Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques. Quart. J. Math. (Oxford), 66 (2015), 245–250.CrossRefGoogle Scholar
Saias, E.. Sur le nombre des entiers sans grand facteur premier. J. Number Theory 32 (1989), no. 1, 78–99.CrossRefGoogle Scholar
Soundararajan, K.. The distribution of smooth numbers in arithmetic progressions, in: Anatomy of Integers, in: CRM Proc. Lect. Notes, vol. 46, Amer. Math. Soc. (Providence, RI, 2008), pp. 115–128.CrossRefGoogle Scholar
Tenenbaum, G.. Introduction à la théorie analytique et probabiliste des nombres, troisième édition (coll. Échelles, Belin, 2008), 592 pp.Google Scholar
Wolke, D.. Über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen. I. Math. Ann. 202 (1973), p. 1–25.CrossRefGoogle Scholar