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: $$\pi (x)=\mathrm{li}(x)+\mathcal{O}(x{e}^{-c\sqrt{\log x}})$$ 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 $\mathcal{O}(\sqrt{x}/\log x)$.