On the Limiting Density of a gcd Map

TL;DR

This paper investigates the density of pairs (a,b) for which a specific gcd-based function equals one, revealing a surprising connection to quadratic class number constants.

Contribution

It derives the limiting density of the gcd map function and explores its higher-order analogue, linking number theory concepts.

Findings

The limiting density is approximately 0.88151, matching a quadratic class number constant.

The higher-order analogue's density simplifies to 1/ζ(2)=6/π².

The density is expressed as an Euler product over primes.

Abstract

The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $\rho=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/\zeta(2)=6/\pi^2$.