On the Limiting Density of a gcd Map

TL;DR

This work analyzes the gcd-map $f(a,b)=\dfrac{\gcd(a+b,ab)}{\gcd(a,b)}$, proving its integer-valuedness and surjectivity while investigating the frequency of $f(a,b)=1$. It derives a precise Euler-product for the limiting density $\rho=\prod_p\left(1-\dfrac{1}{p^2(p+1)}\right)\approx 0.88151$, obtained by conditioning on prime-adic valuations and exploiting independence across primes; the constant aligns with the quadratic class-number constant and connections to real quadratic fields and Cohen–Lenstra heuristics are discussed. For the higher-order analogue $f_r$ with $r\ge 2$, the problem collapses to the classical coprimality event, yielding the density $1/\zeta(2)=\prod_p\left(1-\dfrac{1}{p^2}\right)=\dfrac{6}{\pi^2}$. The paper also employs the Dirichlet hyperbola method to analyze related sums and situates the results within the framework of local densities and arithmetic invariants of quadratic fields.

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 $ρ=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ 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 $1/ζ(2)=6/π^2$.

Figures (2)

• Figure 1: Heat map of $f\left(a,b\right)$, where $1\le a,b\le 50$
• Figure 2: Geometric interpretation of the Dirichlet hyperbola method

Theorems & Definitions (15)

• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Theorem 2.1: limiting density
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof