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THE DISTRIBUTION OF THE NUMBER OF SUBGROUPS OF THE MULTIPLICATIVE GROUP
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 21 December 2018, pp. 46-97
- Print publication:
- February 2020
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- Article
- Export citation
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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 
$\log I(n)$.
