高斯域上一种加权形式的Erdős-Kac定理

A Weighted Form of the Erdős-Kac Theorem over Gaussian Fields

摘要: Erdős-Kac定理是数论中的一个经典结果，它描述了在自然数范围内，整数的不同素因子个数的分布渐进服从正态分布。本文主要目的是将Erdős-Kac定理在高斯域中进行推广，令$K$ 是高斯域，$O_K$ 是其整数环。设$a\in O_K$ ，$\omega \left(a\right)$ 表示其不同的素因子个数，${\tau }_k\left(a\right)$ 是高斯域上$k$ 重除数函数。我们用围道积分法，推导出$\omega \left(a\right)$ 的加权均值和$m$ 阶中心矩，并由此推导出高斯域上权重为${\tau }_k\left(a\right)$ 的Erdős-Kac定理。这一结果不仅丰富了数论中的分布理论，也为进一步研究高斯域中的数论问题提供了新的工具和方法。

Abstract: The Erdős-Kac theorem is a classical result in number theory, which describes that the distribution of the number of distinct prime factors of integers asymptotically follows a normal distribution. The primary aim of this paper is to extend the Erdős-Kac theorem to Gaussian fields. Let $K$ be a Gaussian field and $O_K$ be its ring of integers. Let $a\in O_K$ , and $\omega \left(a\right)$ denote the number of distinct prime factors of $a$ . Let ${\tau }_k\left(a\right)$ be the -fold divisor function on the Gaussian field. Using the method of contour integration, we derive the weighted mean and the $k$ -th central moment of$\omega \left(a\right)$ , and from these, we deduce a weighted form of the Erdős-Kac theorem on Gaussian fields with weight ${\tau }_k\left(a\right)$ . This result not only enriches the distribution theory in number theory but also provides new tools and methods for further research on number-theoretical problems in Gaussian fields.