Asymptotic expression for the $n$th prime number

Question

Quoting from the Wikipedia article:

As a consequence of the prime number theorem, one gets an asymptotic expression for the $ n $th prime number, denoted by $ p_n $: $$ p_n \sim n \log n.$$

Can you explain how we get this approximate expression? I understand the number of prime numbers less than or equal to an integer $ N $, denoted as $ \pi(N) $ is approximately: $$ \pi(N) \sim \frac{N}{\log N}. $$

If the $ n $th prime number is approximately $ n \log n $, it implies that there are approximately $ n $ prime numbers less than or equal to $ n \log n $, i.e., $$ \pi(n \log n) \sim n. $$ But substituting $ N $ with $ n \log n $ in the formula of $\pi(N)$ we get $$ \pi(N) \sim \frac{N}{\log N} = \frac{n \log n}{\log (n \log n)} \not\sim n. $$ What am I doing wrong? How can I show that the statement quoted from Wikipedia is correct?

Answer 1

$$\log(n\log n)=\log n+\log\log n=(1+o(1))\log n$$ therefore $$\frac{n\log n}{\log(n\log n)}=\frac{n\log n}{(1+o(1))\log n} =\frac{n}{(1+o(1))}\sim n.$$

Answer 2

It seems you are very new to prime number theory hence I am not sure if you understand the little $o$ notations. Hence I present @Lord Shark's answer in layman's terms.

$$ \log (n\log n) = \log n + \log\log n $$

But $\log\log n$ is exponentially smaller compared to $\log n$ i.e. it can be ignore compared to $\log n$. Hence as an approximation, we have

$$ \log (n\log n) \sim \log n $$

With this approximation

$$ \pi (n\log n) \sim \frac{n\log n}{\log (n\log n)} \sim \frac{n\log n}{\log n} \sim n $$

The smallest values of $x$ for which $\pi(x) = n$ is $x = p_n$ hence we have

$$ \pi (p_n) = n \sim \pi(n\log n) $$

or $p_n \sim n\log n$.