I've been reading lots and lots of resources, texts not just blogs and wikis, about the Prime Number Theorem (PNT) and the prime counting approximations.
Some sources suggest that the PNT simply states that the number of primes thin out, but doesn't say how exactly - the asymptotic law of primes.
Some sources suggest the PNT definitely does say that $\pi(n) \sim \frac{n}{log_e(n)}$, an approximation which improves as $n \to \infty$.
But we know that there are several better approximations than $\frac{n}{log_e(n)}$, such as the logarithmic offset integral $Li(n)$.
So my question is - what does the PNT actually state? Does is simply state that the primes thin out? Or that they thin out logarithmically? Or does does it actually state the expression $\frac{n}{log_e(n)}$, which was proved in 1896, but has been improved upon since then for smaller $n$?