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SMOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN ARITHMETIC PROGRESSIONS BEYOND THE $x^{1/2}$-BARRIER

Part of: Multiplicative number theory

Published online by Cambridge University Press:  29 November 2017

Sary Drappeau ,
Andrew Granville  and
Xuancheng Shao
Sary Drappeau
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France email sary-aurelien.drappeau@univ-amu.fr
Andrew Granville
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K. email andrew@dms.umontreal.ca
Xuancheng Shao
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, U.K. email xuancheng.shao@maths.ox.ac.uk
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Abstract

We show that smooth-supported multiplicative functions $f$ are well distributed in arithmetic progressions $a_{1}a_{2}^{-1}\;(\text{mod}~q)$ on average over moduli $q\leqslant x^{3/5-\unicode[STIX]{x1D700}}$ with $(q,a_{1}a_{2})=1$.

MSC classification

Primary: 11N56: Rate of growth of arithmetic functions
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 895 - 918
Copyright
Copyright © University College London 2017 

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References

de la Bretèche, R. and Tenenbaum, G., Propriétés statistiques des entiers friables. Ramanujan J. 9 2005, 13–202.CrossRefGoogle Scholar
Drappeau, S., Théorèmes de type Fouvry–Iwaniec pour les entiers friables. Compos. Math. 151 2015, 828–862.CrossRefGoogle Scholar
Fouvry, É. and Tenenbaum, G., Entiers sans grand facteur premier en progressions arithmetiques. Proc. Lond. Math. Soc. (3) 63 1991, 449–494.Google Scholar
Fouvry, É. and Tenenbaum, G., Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques. Proc. Lond. Math. Soc. (3) 72 1996, 481–514.Google Scholar
Granville, A., Harper, A. and Soundararajan, K., A new proof of Halász’s Theorem, and some consequences. Preprint.Google Scholar
Granville, A. and Shao, X., Bombieri–Vinogradov for multiplicative functions, and beyond the $x^{1/2}$ -barrier. Preprint.Google Scholar
Harper, A., Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers. Preprint.Google Scholar
Hildebrand, A., Integers free of large prime divisors in short intervals. Quart. J. Math. Oxford 36 1985, 57–69.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society Colloquium Publications 53 ), American Mathematical Society (Providence, RI, 2004).Google Scholar
Roth, K. F., On the large sieves of Linnik and Rényi. Mathematika 12 1965, 1–9.Google Scholar