The prime number theorem and the nth prime This is a much clearer restatement of an earlier question.
In section 1.8 of Hardy & Wright, An Introduction to the Theory of Numbers, it is proved that the function inverse to $ x ⁄ \log⁡ x$  is asymptotic to 
$x \log⁡ x$. “From this remark we infer,” they say, that:
(*) The prime number theorem, $\pi(x)\sim x⁄ \log ⁡x$ , is equivalent to the theorem $p_n \sim n \log ⁡n$, where $p_n$ denotes the $n^{\text{th}}$ prime.
That the theorems are equivalent is easy to prove by a different method, as in Apostol's Introduction to Analytic Number Theory, Theorem 4.5. But how does the equivalence follow from H & W’s “remark”? As they say in section 1.5, since $\pi(p_n ) = n$, “$\pi(x)$, as function of $x$, and $p_n$, as function of $n$, are inverse functions”; but the inverses of asymptotic functions are not usually themselves asymptotic to one another. Would someone please explain how H & W mean for us to deduce (*)?
 A: While I can't speak directly for Hardy and Wright, I think the following is a plausible explanation, based on a copy of the fourth edition of H&W.  Just prior to claiming the equivalence of $\pi(x) \sim x/\log x$ and $p_n \sim n \log n$, they spell out the argument that the inverse function of $x/\log x$ is asymptotic to $x \log x$.  For completeness, here is the brief argument, very mildly paraphrased:

If $y = x / \log x$ then $\log y = \log x - \log \log x$.  Since $\log\log x = o(\log x)$ we have $\log y \sim \log x$ and thus $x = y \log x \sim y \log y$.

The point here is that this argument illustrates a "moral": the key observation is that once we establish that $x$ and $y$ are not too far apart (that is $\log x \sim \log y$) then we can justify shifting between them and that this allows us to (asymptotically) invert functions which do not have nice inverses.  Imagine now $\pi(x)$ in the place of $y$, not in an exact copy of the above proof, but a modified version with this moral intact:

If $y \sim x / \log x$ then $\log y = \log x - \log \log x + o(1)$.  Since $\log\log x = o(\log x)$ and $o(1) = o(\log x)$ we have $\log y \sim \log x$ and thus $x \sim y \log x \sim y \log y$.

This establishes Theorem 8 in a way that strongly echoes the preceding discussion (and without making any general claim of asymptotics of inverse functions).  Likewise, the same argument goes through for $\asymp$ instead of $\sim$, with $O(1)$ replacing $o(1)$.
I do agree there is sloppiness in saying that this inference follows "from the remark" and not in a manner akin to the remark.  It's also interesting that they take the trouble to write out a proof of Theorem 9 ($p_n \asymp n \log n$) from Theorem 7, but refer to Theorem 8 ($p_n \sim n \log n$) as a trivial consequence of Theorem 6 (the antecedent theorems in both cases being the corresponding estimate on $\pi(x)$).  I'm inclined to chalk this up to human fallibility :).
A: If the authors assume the reader has had a course in analysis, I would say that their assumption of this to be straightforward is a fair one.
At integer $x$ there would be about $n=x/\log x$ [define $n = \lceil x/\log x\rceil$ and "about $n$" is $n(1 \pm o(1)]$ ] primes so $p_n$ would be $x(1\pm o(1))$ [by the Prime Number Theorem, otherwise there would be too many or too few primes by the integer $p_n$]. So $p_n$ is $x(1\pm o(1)) = (1\pm o(1))(\log x)(x/\log x) =(1\pm o(1))n \log x$.
I would then trust that the reader would be able to see that $\log x = (1+o(1)\log n$. 
Nothing too slick here, just basic analysis.
