I agree with Mark Grant, since I also remember that the first proofs of the Prime number theorem given by [J. Hadamard](https://en.wikipedia.org/wiki/Jacques_Hadamard) and [C. J. de la Vallée Poussin](https://en.wikipedia.org/wiki/Charles_Jean_de_la_Vall%C3%A9e_Poussin), were quite long and involved: however, many mathematicians worked to simplify their proofs. Currently you can find proofs it in (graduate) textbooks as the one of Veech (1967) (ch. 6, pp. 200-238): you can read that chapter and figure out if your student will be able to attend fruitfully a lecture dealing with an abridged version of it. On my side, I remark that there are many interesting tools developed/introduced for the proof, for example the Tauberian theory wich is an interesting topic per se.

[1] Veech, W. A. (1967), *A second course in complex analysis*, New York-Amsterdam: W.A. Benjamin, Inc., pp. IX+246, MR0220903, Zbl 0145.29901