Mazur’s refined question on the order of S_{α,β,γ}(X) when α+β+γ>1

Determine whether S_{α,β,γ}(X), the number of coprime triples (a, b, c) with a + b = c, a, b, c ∈ [1, X], and radical constraints rad(a) ≤ a^α, rad(b) ≤ b^β, rad(c) ≤ c^γ for fixed α, β, γ ∈ (0,1], has order X^{α+β+γ−1} whenever α + β + γ > 1.

Background

Mazur introduced a refined counting function S_{α,β,γ}(X) measuring triples with controlled radicals and asked about its precise asymptotic order when the sum of exponents exceeds 1. The paper reviews this question and notes partial progress by Kane, particularly when α+β+γ≥2, but the general case remains unresolved in certain ranges.