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

Abstract

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin’s form: . 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 π(x) by .