TITLE: Probability that a positive integer is the euler phi function of another positive integer
QUESTION [6 upvotes]: Define $f(n) =  |\{m : m\le n,  \exists k \text{ s.t. }\phi(k) = m\}|$.
Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. 
Is $\limsup_{n\rightarrow\infty} \frac{f(n)}{n} > 0$. It is less than $\frac{1}{2}$ by my previous statement, but I don't know how to proceed.

REPLY [14 votes]: See Erick Wong's response here. In particular, Kevin Ford proved (in more precise form) that
$$ f(n) = \frac{n}{\log n} \exp\left(O(\log \log \log n)^2\right),$$
whence $f(n)/n$ tends to zero. The same consequence also follows from an earlier result of Pillai (1929), available online here.