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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 3


On the proximity of multiplicative functions to the number of distinct prime factors function

Jean-Marie De Koninck / Nicolas Doyon / François Laniel
Published Online: 2018-05-18 | DOI: https://doi.org/10.1515/ms-2017-0121


Given an additive function f and a multiplicative function g, let E(f, g;x) = #{nx: f(n) = g(n)}. We study the size of E(ω,g;x) and E(Ω,g;x), where ω(n) stands for the number of distinct prime factors of n and Ω(n) stands for the number of prime factors of n counting multiplicity. In particular, we show that E(ω,g;x) and E(Ω,g;x) are Oxloglogx for any integer valued multiplicative function g. This improves an earlier result of De Koninck, Doyon and Letendre.

MSC 2010: Primary 11N25; Secondary 11A25

Keywords: additive functions; multiplicative functions; number of distinct prime factors of an integer

The work of the first author was supported by a grant from NSERC.


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About the article

Received: 2016-05-20

Accepted: 2016-10-03

Published Online: 2018-05-18

Published in Print: 2018-06-26

Citation Information: Mathematica Slovaca, Volume 68, Issue 3, Pages 513–526, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0121.

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