Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2016: 0.24

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 68, Issue 3

Issues

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

Abstract

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.

References

  • [1]

    Balazard, M.: Unimodalité de la distribution du nombre de diviseurs premiers ďun entier, Ann. Inst. Fourier 40 (1990), 255–270.CrossrefGoogle Scholar

  • [2]

    Ben SaϊD, F.—Nicolas, J. L.: Sur une application de la formule de Selberg-Delange, Colloq. Math. 98 (2003), 223–247.CrossrefGoogle Scholar

  • [3]

    De Koninck, J. M.—Doyon, N.—Letendre, P.: On the proximity of additive and multiplicative functions, Funct. Approx. Comment. Math. 201 (2015), 1–18.Google Scholar

  • [4]

    De Koninck, J. M.—Luca, F.: Analytic Number Theory: Exploring the Anatomy of Integers. Grad. Stud. Math. 134, American Mathematical Society, Providence, Rhode Island, 2012.Google Scholar

  • [5]

    Hardy, G. H.—Ramanujan, S.: The normal number of prime factors of a number n, Q. J. Math. 48 (1917), 76–92.Google Scholar

  • [6]

    Spearman, B. K.—Williams, K. S.: On integers with prime factors restricted to certain congruence classes, Far East J. Math. 24 (2007), 153–161.Google Scholar

  • [7]

    Tenenbaum, G.: Introduction à la Théorie Analytique des Nombres. Collection SMF, Société Mathématique de France, 1995.Google Scholar

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.

Export Citation

© 2017 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in