I agree with [Mark Grant's comment](https://mathoverflow.net/questions/306630/teaching-prime-number-theorem-in-a-complex-analysis-class-for-physicists#comment762392_306630) above, 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 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](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0220903), [Zbl 0145.29901](https://www.zbmath.org/?q=an%3A0145.29901).