On the Limiting Density of a gcd Map

2025/12/27 by Thang Pang Ern, Ern, Thang Pang, Malcolm Tan Jun Xi +3
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2512.22494

openalex publication_date 2025/12/27 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

Abstract

The function f(a,b)=(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=∏_p(1-(1)/(p²(p+1)))≈ 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/π².

Related

• On the Asymptotic Density of a GCD-based Map
• On the Density of naturals n coprime to \lfloor P(n) \rfloor for certain Classes of Polynomials
• On GCD(Φ_N(aⁿ),Φ_N(bⁿ))
• Quadratic Euler-Kronecker constants in positive characteristic
• On the Asymptotic Density of k-tuples of Positive Integers Satisfying Arbitrary GCD Conditions