The question whether the set $P\subseteq \mathbb{N}$ of perfect numbers is infinite, is famously open. I would think that everybody believes the statement below - but has it been proved?

$$\mu(P) := \lim\inf_{n\to\infty}\frac{|P\cap\{1,\ldots,n\}|}{n} = 0.$$

If yes, it would also be interesting to know if it has been established that $\mu(\log(P)) = 0$, $\mu(\log(\log(P)) = 0$ and so on, where $\log(P) = \{\lceil \log(n)\rceil: n\in P\}$.