# Density of perfect numbers

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 $$\mu(\log\log P)>0$$.