Are we only knowing prime counting function's asymptotic property but not its infinite expansion or even people could saying that there are no infinite series for the function?
If yes, what are some other function shares some similar situation?
I know beforehand that the prime number theorem only approximate the value of the function by using asymptotic notation.
The series (in $\log n$ and $\log\log n$) of $\pi(n)/n$ is known, although I believe it's a divergent expansion. It's been known since the 18th century.