On the Distribution of the Greatest Common Divisor of Gaussian Integers

TL;DR

This paper investigates the asymptotic behavior of the average norm of the greatest common divisor of pairs of random Gaussian integers, providing explicit error terms and supporting computational data for higher moments.

Contribution

It extends the understanding of gcd distribution from integers to Gaussian integers, offering new asymptotic formulas and moment analyses with computational validation.

Findings

Asymptotic formulas for average gcd norm in Gaussian integers

Explicit error terms in the asymptotic analysis

Computational data supporting higher moment results

Abstract

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erd\"os.