An elementary Tauberian proof of the Prime Number Theorem

2024/04/19 by Philippe Angot, Angot, Philippe
Mathematics · #11A41 #11M45 #11N05 #30B50 #40A05 #40E05 #42A38 (secondary) #44A10 (primary) #Advanced Mathematical Theories #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2404.13019

openalex publication_date 2024/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ζ and its non-vanishing value on the whole line \z∈ \mathbb C; 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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