An elementary Tauberian proof of the Prime Number Theorem
Une preuve élémentaire du Théorème des nombres premiers
Résumé
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.
Mots clés
Tauberian theory
Laplace transform on the real axis
Slowly decreasing function
Wiener-Ikehara-Ingham's type theorem
Dirichlet's series
Analytic number theory
Prime numbers. 2020 Mathematics Subject Classification. 11A41 11M45 40A05 40E05 44A10 (primary)
Prime numbers
11A41
11M45
40A05
40E05
44A10 (primary)
11N05
30B50
42A38 (secondary)
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)