Erdős Problem #50
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A function \(f : \mathbb{R} \to \mathbb{R}\) is the asymptotic distribution function of the values of \(\varphi(n)/n\) if for all \(c \in [0, 1]\), the natural density of \(\{n : \varphi(n) < cn\}\) exists and equals \(f(c)\).
A function \(f : \mathbb{R} \to \mathbb{R}\) is purely singular (or singular continuous) on a set \(s \subseteq \mathbb{R}\) if it is continuous on \(s\) and its derivative within \(s\) equals zero almost everywhere on \(s\) with respect to Lebesgue measure.
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Proved, modulo Schoenberg/Mertens/Bertrand, a dense continuum-sized family of irrational points generated by an explicit recursion with infinite upper-right Dini derivative; the possibility of a finit
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