I once had an independent study reading Ayoub’s book on analytic number theory. In that book, I remember the Prime Number Theorem being a hard slog. It turns out that D. J. Newman published a short 6-page proof in 1980. The proof requires complex analysis at the level of an undergraduate course.
This message by Joe Shipman to the Foundations of Mathematics mailing list contains a link to a write-up of the proof by Zagier.