An almost sharp quantitative version of the Duffin-Schaeffer conjecture

Abstract

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $\psi:\mathbb{N}\to[0,1/2]$ be a function such that the series $\sum_{q=1}^\infty \varphi(q)\psi(q)/q$ diverges. In addition, given $\alpha\in\mathbb{R}$ and $Q\geqslant1$, let $N(\alpha;Q)$ be the number of coprime pairs $(a,q)\in\mathbb{Z}\times\mathbb{N}$ with $q\leqslant Q$ and $|\alpha-a/q|<\psi(q)/q$. Lastly, let $\Psi(Q)=\sum_{q\leqslant Q}2\varphi(q)\psi(q)/q$, which is the expected value of $N(\alpha;Q)$ when $\alpha$ is uniformly chosen from $[0, 1]$. We prove that $N(\alpha;Q)=\Psi(Q)+O_{\alpha,\varepsilon}(\Psi(Q)^{1/2+\varepsilon})$ for almost all $\alpha$ (in the Lebesgue sense) and for every fixed $\varepsilon>0$. This improves upon results of Koukoulopoulos-Maynard and of Aistleitner-Borda-Hauke.