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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 $[0, 1]$. 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