The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.
We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.
These lectures are at the level of beginning graduate students.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/7671