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

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 conventional methods, new approach uses contour-integration analyze Riemann's counting function $J(x)$, which only differs from $\pi(x)$ by $\mathcal O(\sqrt x/\log x)$.