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THE DISTRIBUTION OF THE NUMBER OF SUBGROUPS OF THE MULTIPLICATIVE GROUP

Part of: Multiplicative number theory

Published online by Cambridge University Press:  21 December 2018

GREG MARTIN  and
LEE TROUPE
GREG MARTIN
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2 email gerg@math.ubc.ca
LEE TROUPE*
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, C526 University Hall, 4401 University Drive West, Lethbridge, AB, Canada T1K 3M4 email lee.troupe@uleth.ca
*
✉lee.troupe@uleth.ca
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Abstract

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Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$, and let $G(n)$ denote the number of subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erdős–Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that $\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of $\log G(n)$ and $\log I(n)$.

Keywords

additive functionsdistribution of valuesmaximal ordermethod of momentscounting subgroups

MSC classification

Primary: 11N60: Distribution functions associated with additive and positive multiplicative functions 11N45: Asymptotic results on counting functions for algebraic and topological structures
Secondary: 11N37: Asymptotic results on arithmetic functions

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 1 , February 2020 , pp. 46 - 97
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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