An elemental Erdős-Kac theorem for algebraic number fields

Abstract

Fix a number field $K$. For each nonzero $α\in \mathbb{Z}_K$, let $ν(α)$ denote the number of distinct, nonassociate irreducible divisors of $α$. We show that $ν(α)$ is normally distributed with mean proportional to $(\log\log |N(α)|)^{D}$ and standard deviation proportional to $(\log\log{|N(α)|})^{D-1/2}$. Here $D$, as well as the constants of proportionality, depend only on the class group of $K$. For example, for each fixed real $λ$, the proportion of $α\in \mathbb{Z}[\sqrt{-5}]$ with $$ ν(α) \le \frac{1}{8}(\log\log{N(α)})^2 + \fracλ{2\sqrt{2}} (\log\log{N(α)})^{3/2} $$ is given by $\frac{1}{\sqrt{2π}} \int_{-\infty}^λ e^{-t^2/2}\, \mathrm{d}t$. As further evidence that "irreducibles play a game of chance", we show that the values $ν(α)$ are equidistributed modulo $m$ for every fixed $m$.

15 pages