Asymptotic inverses of asymptotic functions

The prime number theorem states that two functions are asymptotic. Their inverses (as functions of an integral variable) are also asymptotic. In general, under what conditions are the inverses of asymptotic functions themselves asymptotic?

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One of the functions in the prime number theorem is $\pi(x)$, the number of primes less than or equal to $x$. That function certainly is not invertible: every $x\in[3,5)$ satisfies $\pi(x)=2$, for example. So I can't figure out what question you're trying to ask. – Greg Martin Mar 11 '13 at 19:06
@Greg: formally it is not invertible, but it has semi-inversed, which are all equivalent. So, probably the two statements are "$\pi(x) \sim x/ \log(x)$" and "$p_n \sim n \log n$". – Fedor Petrov Mar 11 '13 at 21:49
I don't know what "semi-inversed" means, nor what "equivalent" means in this context. – Gerry Myerson Mar 11 '13 at 23:08
@Fedor: Thank you. Yes, those are the two statements. Cf. Hardy & Wright, Section 1.8. – user32024 Mar 12 '13 at 15:54