The universal profile of the invariant factors of $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$

Abstract

The structure of the multiplicative group $M_n=(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ encodes a great deal of arithmetic information about the integer $n$ (examples include $\varphi (n)$, the Carmichael function $\lambda (n)$, and the number $\omega (n)$ of distinct prime factors of $n$). We examine the invariant factor structure of $M_n$ for typical integers $n$, that is, the decomposition $M_n\cong \mathbb{Z}/d_1\mathbb{Z}\times \mathbb{Z}/d_2\mathbb{Z}\times \cdots \times \mathbb{Z}/d_k\mathbb{Z}$ where $d_1\mid d_2\mid \cdots \mid d_k$. We show that almost all integers have asymptotically the same invariant factors for all but the largest factors; for example, asymptotically $1/2$ of the invariant factors equal $\mathbb{Z}/2\mathbb{Z}$, asymptotically $1/4$ of them equal $\mathbb{Z}/12\mathbb{Z}$, asymptotically $1/12$ of them equal $\mathbb{Z}/120\mathbb{Z}$, and so on. Furthermore, for positive integers $k$, we establish a theorem of Erdős-Kac type for the number of invariant factors of $M_n$ that equal $\mathbb{Z}/k\mathbb{Z}$, except that the distribution is not a normal distribution but rather a skew-normal or related distribution.

69 pages