A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function

TL;DR

This paper introduces a new analytic contour-integration method to prove the prime number theorem directly via Riemann's prime counting function, offering an alternative to traditional asymptotic expansions.

Contribution

It presents a novel proof of the prime number theorem using contour integration on Riemann's prime counting function, diverging from conventional methods.

Findings

Proves the prime number theorem in de la Vallée Poussin's form

Develops a new analytic approach using contour integration

Shows Riemann's prime counting function closely approximates π(x)

Abstract

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann's prime counting function $J(x)$, which only differs from $\pi(x)$ by $\mathcal O(\sqrt x/\log x)$.