An elemental Erdős–Kac theorem for algebraic number fields

Abstract:

Fix a number field $K$. For each nonzero $\alpha \in \mathbf {Z}_K$, 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 $\alpha \in \mathbf {Z}[\sqrt {-5}]$ 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 $m$.