An operator theoretic approach to the Prime number theorem

TL;DR

This paper introduces an operator theoretic version of a key tauberian theorem, providing a new, accessible proof of the Prime Number Theorem through operator theory techniques.

Contribution

It develops an operator theoretic framework for the Wiener-Ikehara tauberian theorem, leading to a novel proof of the Prime Number Theorem.

Findings

New operator theoretic proof of the Prime Number Theorem

Accessible approach using basic operator theory concepts

Establishes an operator version of the Wiener-Ikehara theorem

Abstract

In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory.