An elementary Tauberian proof of the Prime Number Theorem

TL;DR

This paper presents a straightforward, elementary real analysis-based proof of the Prime Number Theorem, avoiding complex analysis tools like the analytic continuation of the Riemann zeta function.

Contribution

It introduces a simple Tauberian approach that extends Wiener–Ikehara's theorem, removing the need for complex analysis in proving the Prime Number Theorem.

Findings

Provides an elementary proof of the Prime Number Theorem

Reduces the Tauberian assumption to local boundary behavior

Eliminates reliance on the analytic continuation of the zeta function

Abstract

We give a simple Tauberian proof of the Prime Number Theorem using only elementary real analysis. Hence, the analytic continuation of Riemann's zeta function $\zeta$ and its non-vanishing value on the whole line $\{z\in {\mathbb C};\,{\mathrm{Re}\,} z=1\}$ are no more required. This is achieved by showing a strong extension for Laplace transforms on the real line of Wiener--Ikehara's theorem on Dirichlet's series, where the Tauberian assumption is reduced to a local boundary behavior around the pole.