The distribution of ideals whose norm divides $n$ in the Gaussian ring

Let $O_K=\mathbb{Z}[i]$. For each positive integer $n$, denote ${\xi }_K(n)$ as the number of integral ideals whose norm divides $n$ in $O_K$. In this paper, we studied the distribution of ideals whose norm divides $n$ in $O_K$ by using the Selberg-Delange method. This is a natural variant of a result studied by Deshouillers, Dress, and Tenenbaum (often called the DDT Theorem), and we found that the distribution function was subject to beta distribution with density $\sqrt{3}/(2\pi \sqrt[3]{u^2(1-u)})$.