The Prime Number Theorem via Karamata's proof of the Landau-Ingham Tauberian theorem
arXiv:1311.2491
Abstract
This is an exposition, in 12 pages including all prerequisites and a generalization, of Karamata's little known elementary proof of the Landau-Ingham Tauberian theorem, a result in real analysis from which the Prime Number Theorem follows in a few lines. We believe this to be the most transparent of the elementary proofs of the PNT.
Latex2e, 17 pages, no figures. Proof of Theorem 2.4 has been improved, otherwise minor changes. To appear in L'Enseignement Mathematique