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

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

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

Source Link
Thomas Bloom
  • 7k
  • 1
  • 39
  • 59

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