An elementary Tauberian proof of the Prime Number Theorem

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 $\{z\in \mathbb{C};\mathrm{R}\mathrm{e}z=1\}$ 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.