Schoenberg [Sch38] proved that the asymptotic distribution function of $\varphi(n)/n$ exists. That is, for any $c \in [0, 1]$, the proportion of integers $n \le N$ satisfying $\varphi(n)/n < c$ approaches a limit as $N \to \infty$. This limit function is the cumulative distribution function of the values of $\varphi(n)/n$.

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