Erdős problem 50 · problems.science

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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50.lean

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Erdős [Er95] proved that the distribution function of $\varphi(n)/n$ is purely singular: it is continuous, but its derivative is zero almost everywhere.

FormalConjectures/ErdosProblems/50.leanErdos50.erdos_50_singular1 lineExact file

1∀ (f : ℝ → ℝ), Erdos50.IsDistributionOfPhiRatio f → Erdos50.IsPurelySingular f

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