Asymptotic normality and greatest common divisors

TL;DR

This paper investigates the asymptotic statistical properties of greatest common divisors in large random samples, revealing normality, Fréchet distribution behavior, and extending results to r-tuples and various moments.

Contribution

It provides new insights into the distributional limits of gcd-related statistics in large samples, including normal and Fréchet approximations.

Findings

Number of coprime pairs is approximately normal.

Average gcd of pairs follows an approximate normal distribution.

Maximum gcd follows an approximate Fréchet distribution.

Abstract

We report on some statistical regularity properties of greatest common divisors: for large random samples of integers, the number of coprime pairs and the average of the gcd's of those pairs are approximately normal, while the maximum of those gcd's (appropriately normalized) follows approximately a Fr\'echet distribution approximately. We also consider $r$-tuples instead of pairs, and moments other than the average.