I agree with Mark Grant's comment above, since I also remember that the first proofs of the Prime number theorem given by J. Hadamard and C. J. de la Vallée Poussin, were quite long and involved: however, many mathematicians worked to simplify their proofs. Currently, you can find reasonably short proofs in (graduate) textbooks as [1], chapter 6, pp. 200-238: you may read that chapter and figure out if your students 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.