An elemental Erdős-Kac theorem for algebraic number fields

Abstract

Fix a number field $K$. For each nonzero $\hat{I}\pm \in {\mathbb{Z}}_K$, let $\hat{I}½(\hat{I}\pm )$ denote the number of distinct, nonassociate irreducible divisors of $\hat{I}\pm$. We show that $\hat{I}½(\hat{I}\pm )$ is normally distributed with mean proportional to $(\log \log |N(\hat{I}\pm )|{)}^D$ and standard deviation proportional to $(\log \log |N(\hat{I}\pm )|{)}^{D-1/2}$. Here $D$, as well as the constants of proportionality, depend only on the class group of $K$. For example, for each fixed real $\hat{I}»$, the proportion of $\hat{I}\pm \in \mathbb{Z}[\sqrt{-5}]$ with $$\hat{I}½(\hat{I}\pm )\le \frac{1}{8}(\log \log N(\hat{I}\pm ){)}^2+\frac{\hat{I}}{»}2\sqrt{2}(\log \log N(\hat{I}\pm ){)}^{3/2}$$ is given by $\frac{1}{\sqrt{2\ddot{I}€}}{\int }_{-\infty }^{\hat{I}}»e^{-t^2/2}\mathrm{d}t$. As further evidence that "irreducibles play a game of chance", we show that the values $\hat{I}½(\hat{I}\pm )$ are equidistributed modulo $m$ for every fixed $m$.

15 pages