Assuming the first Hardy--Littlewood conjecture for twin primes, what is an asymptotic formula for the $n$th twin prime $t_n$ (which, let us say, enumerates the first of the each twin prime pair, so that the sequence $t_n$ is $3,5,11,17,29,\ldots$)? I imagine it will take the form $$t_n \sim C n \,(\log n)^2 \ (n \to \infty)$$ for some constant $C$ expressible in terms of the twin prime constant. Also, is the Hardy--Littlewood conjecture for twin primes equivalent to such an asymptotic expression for the $n$th twin prime $t_n$? If so, how would you deduce the Hardy--Littlewood conjecture for twin primes from such an asymptotic relation for $t_n$?
A more basic question is, what is the easiest way to deduce the prime number theorem from the asymptotic relation $$p_n \sim n \log n \ (n \to \infty),$$ where $p_n$ denotes the $n$th prime? Maybe a proof in the twin prime case would be similar?
(Pardon the notation $t_n$ for the $n$th twin prime. If you have a better notation, let me know and I will edit the question.)