A sketch of proof of Prime Number Theorem

Question

I'm looking for some sketch of the elementary proof of the Prime Number Theorem, which would suffice to explain someone the general mechanism of proving PNT without going into deep analytic methods etc.

Answer 1

Try this prize-winning exposition:

• A Motivated Account of an Elementary Proof of the Prime Number Theorem by Norman Levinson

For some of the history of the proof, see:

• The Elementary Proof of the Prime Number Theorem: An Historical Perspective by D. Goldfeld [pdf]

• Elementary Proof of the Prime Number Theorem by Joel Spencer and Ronald Graham

Answer 2

Don Zagier has an article "Newman's Short Proof of the Prime Number Theorem", available for download here. It is an exposition of D.J.Newman's work, and consists of a self-contained four-page proof of the Prime Number Theorem. From the introduction:

We describe the resulting proof, which has a beautifully simple structure and uses hardly anything beyond Cauchy's theorem.

I know you asked for an elementary proof $-$ which presumably rules out Cauchy's theorem $-$ but such proofs are needlessly complicated, and don't lead to any understanding of why the PNT is true.