What upper bounds exist for the frequency of perfect numbers. I.e. what functions $f(x)$ are there where we can say that as x tends to infinity, there exists so value n such that if $x>n$, the number of perfect numbers less than x is always less then $f(x)$?
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1Assuming that there is no odd perfect number , $\pi(log_4n)$ is an upper bound but still far too large based on the expression $2^{n-1}\cdot (2^n-1)$ which is larger than $4^{n-1}$ and $n$ must be prime, therefore the $\pi$-function. We know $51$ perfect numbers , the largest having roughly $50$ million digits – Peter Jul 03 '23 at 13:30