On the Probability of Relative Primality in the Gaussian Integers

TL;DR

This paper explores the probability that two Gaussian integers are coprime, linking number theory, probability, and geometry, and provides a lattice-based proof for this probability involving Dedekind zeta functions.

Contribution

It introduces a lattice-theoretic approach to determine the probability of coprimality in Gaussian integers and generalizes the result to principal ideal domains in number fields.

Findings

Probability that two Gaussian integers are coprime equals 1 over the Dedekind zeta function at 2.

Provides a lattice-theoretic proof technique for number field generalizations.

Establishes a connection between probability, geometry, and algebraic number theory.

Abstract

This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{\zeta_{\Q(i)}(2)}$$ where $z_{1},z_{2} \in \mathbb{Z}[i]$ are randomly chosen and $\zeta_{\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theorem to arbitrary number fields that are principal ideal domains.