Simple analytic proofs of some versions of the abstract Prime Number Theorem

Abstract

On exactly one typewritten page, Ernst Witt sketched in the 60th of the last century a proof of the Prime Number Theorem, even in a generalized abstract situation. A closer inspection reveals a deep correlation to several other publications before and around this time. We elaborate his outline in more detail (without trying to find the "historical truth" of priority) and demonstrate that only some part of his approach can be combined with the quite recent very elegant method of Newman and Korevaar to yield a really simple analytic proof of a version of the abstract Prime Number Theorem that is close to Beurling's original one.