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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
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

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