The distribution of the number of subgroups of the multiplicative group

TL;DR

This paper proves that the logarithms of the number of subgroups and isomorphism classes of subgroups of the multiplicative group modulo n follow Erdős-Kac laws, revealing their probabilistic distribution and magnitude.

Contribution

It establishes Erdős-Kac laws for both nd G(n), and analyzes their maximal orders, providing new insights into subgroup distribution in multiplicative groups.

Findings

nd nd G(n) satisfy ErdF6s-Kac laws with normal distributions.

nd G(n) have explicitly determined orders of magnitude.

nd G(n) exhibit probabilistic behavior similar to additive functions, despite nd G(n) not being additive.

Abstract

Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subgroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erd\"os-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 $\log I(n)$.