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

Abstract

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)}) $.

Keywords

• selberg-delange method
• the distribution
• gaussian ring
• beta distribution