Problem #417: Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq...

Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq x\}\]and\[V(x)=\#\{\phi(m) \leq x : 1\leq m\}.\]Does $\lim V(x)/V'(x)$ exist? Is it $>1$?

SOLVED

Problem Statement

Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq x\}\]and\[V(x)=\#\{\phi(m) \leq x : 1\leq m\}.\]Does $\lim V(x)/V'(x)$ exist? Is it $>1$?

Categories: Number Theory

It is trivial that $V'(x) \leq V(x)$. In [Er98] Erdős suggests the limit may be infinite. See also [416].

Source: erdosproblems.com/417 | Last verified: January 15, 2026