An Extension of the Dirichlet Density for Sets of Gaussian Integers

TL;DR

This paper introduces a new extension of the Dirichlet density specifically for sets of Gaussian integers, exploring its properties and potential applications in higher-dimensional number theory.

Contribution

The paper proposes a novel extension of the Dirichlet density to Gaussian integers, expanding the concept to higher dimensions and analyzing its fundamental properties.

Findings

The extended Dirichlet density is well-defined for Gaussian integers.

The properties of the extended density are consistent with classical density measures.

Potential applications in number theory and complex analysis are discussed.

Abstract

Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and investigate some of its properties.