An elemental Erdős–Kac theorem for algebraic number fields

Abstract

Fix a number field K K . For each nonzero α ∈ Z K \alpha \in \mathbf {Z}_K , let ν ( α ) \nu (\alpha ) denote the number of distinct, nonassociate irreducible divisors of α \alpha . We show that ν ( α ) \nu (\alpha ) is normally distributed with mean proportional to ( log ⁡ log ⁡ | N ( α ) | ) D (\log \log |N(\alpha )|)^{D} and standard deviation proportional to ( log ⁡ log ⁡ | N ( α ) | ) D − 1 / 2 (\log \log {|N(\alpha )|})^{D-1/2} . Here D D , as well as the constants of proportionality, depend only on the class group of K K . For example, for each fixed real λ \lambda , the proportion of α ∈ Z [ − 5 ] \alpha \in \mathbf {Z}[\sqrt {-5}] with \[ ν ( α ) ≤ 1 8 ( log ⁡ log ⁡ N ( α ) ) 2 + λ 2 2 ( log ⁡ log ⁡ N ( α ) ) 3 / 2 \nu (\alpha ) \le \frac {1}{8}(\log \log {N(\alpha )})^2 + \frac {\lambda }{2\sqrt {2}} (\log \log {N(\alpha )})^{3/2} \] is given by 1 2 π ∫ − ∞ λ e − t 2 / 2 d t \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 ν ( α ) \nu (\alpha ) are equidistributed modulo m m for every fixed m m .