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 $\log(P) = \{\lceil \log(n)\rceil: n\in P\}$.
Yes - Wirsing has shown that the number of odd perfect numbers in $[1,n]$ is at most $n^{O(1/\log\log n)}$. By Euler's correspondence between even perfect numbers and Mersenne primes it is certainly true that the number of even perfect numbers in $[1,n]$ is $\ll \log n$.
Therefore $\lvert P\cap [1,n]\rvert \leq n^{O(1/\log\log n)}$ (and in particular $P$ has zero density).
Wirsing's bound is the best known for odd perfect numbers, and hence it is unknown whether e.g. $\mu(\log P)=0$.
Furthermore, since conjecturally there are $\gg \log\log n$ many Mersenne primes in $[1,n]$, it should in fact be true that $\mu(\log\log P)>0$.
