An almost sharp quantitative version of the Duffin-Schaeffer conjecture - Koukoulopoulos, D; Maynard, J; Yang, D (Duke University Press, 2024)
We prove a quantitative version of the Duffin–Schaeffer conjecture with an almost sharp error term. Precisely, let ψ : N → [0, 1/2] be a function such that the series P∞ q=1 ϕ(q)ψ(q)/q diverges. In addition, given α ∈ R and Q > 1, let N(α; Q) be the number of coprime pairs (a, q) ∈ Z × N with q 6 Q and |α − a/q| < ψ(q)/q. Lastly, let Ψ(Q) = P q6Q 2ϕ(q)ψ(q)/q, which is the expected value of N(α; Q) when α is uniformly chosen from [0, 1]. We prove that N(α; Q) = Ψ(Q) + Oα,ε(Ψ(Q) 1/2+ε ) for almost all α (in the Lebesgue sense) and for every fixed ε > 0. This improves upon results of Koukoulopoulos–Maynard and of Aistleitner–Borda–Hauke.