Le th\'eor\`eme Taub\'erien pour la transformation de Laplace complexe et th\'eor\`eme des nombres premiers

TL;DR

This paper presents a new proof of a Tauberian theorem related to complex Laplace transforms, providing a simplified approach to the prime number theorem, which describes the asymptotic distribution of prime numbers.

Contribution

It introduces a novel proof of the Tauberian theorem for complex Laplace transforms, offering a simpler derivation of the prime number theorem.

Findings

New proof of the Tauberian theorem for complex Laplace transforms

Simplified derivation of the prime number theorem

Enhanced understanding of the connection between Laplace transforms and prime distribution

Abstract

This is a new proof of the tauberian theorem of complex Laplace transformation for getting a simple proof of the prime numbers theorem $\pi(x) \sim \frac{x}{\log(x)}$ for a largest real $x$.