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arXiv · 2404.13019

An elementary Tauberian proof of the Prime Number Theorem

Philippe Angot
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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.

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Philippe Angot. 2024-04-19. An elementary Tauberian proof of the Prime Number Theorem. https://arxiv.org/abs/2404.13019

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