Mazur’s question on the exact order of N_λ(X) for λ > 1

Determine whether N_λ(X), the number of coprime integer triples (a, b, c) with a + b = c and rad(abc) < c^λ in the box [1, X]^3, has exact order X^{λ−1} for every fixed λ > 1 as X tends to infinity.

Background

In connection with counting abc-type triples above the critical exponent 1, Mazur posed a question about the precise asymptotic growth of N_λ(X). The authors’ main theorem applies slightly beyond λ = 1 and they discuss this question, noting existing partial results and bounds but not a complete resolution.