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

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∈ O K ， ω( a ) 表示其不同的素因子个数， τ k ( a ) 是高斯域上 k 重除数函数。我们用围道积分法，推导出 ω( a ) 的加权均值和 m 阶中心矩，并由此推导出高斯域上权重为 τ k ( a ) 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。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∈ O K , and ω( a ) denote the number of distinct prime factors of a . Let τ k ( a ) 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 ω( a ) , and from these, we deduce a weighted form of the Erdős-Kac theorem on Gaussian fields with weight τ k ( a ) . 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.