An elemental Erd\H{o}s-Kac theorem for algebraic number fields

Abstract

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