An alternative proof of the Dirichlet prime number theorem

TL;DR

This paper presents an alternative elementary proof of Dirichlet's theorem on arithmetic progressions, providing new insights and estimations for the prime counting function in specific cases.

Contribution

It introduces a novel proof approach for Dirichlet's theorem based on prior results, differing from Selberg's method.

Findings

New elementary proof of Dirichlet's theorem

Estimates for the prime counting function in special cases

Enhanced understanding of prime distribution in arithmetic progressions

Abstract

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative proof of it based on a previous result of us. Also we get an estimation of the prime counting function in the special cases.