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Number of prime factors with a given multiplicity

Part of: Elementary number theory

Published online by Cambridge University Press:  03 May 2021

Ertan Elma  and
Yu-Ru Liu
Ertan Elma*
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada yrliu@uwaterloo.ca
Yu-Ru Liu
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada yrliu@uwaterloo.ca
*
e-mail: eelma@uwaterloo.ca
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Abstract

Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$. Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.

Keywords

Prime divisorsnormal ordernormal distribution

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11A41: Primes

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 253 - 269
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Copyright
© Canadian Mathematical Society 2021

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