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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 [email protected]
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 [email protected]
Xuancheng Shao
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, U.K. email [email protected]
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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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