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arXiv · 2105.05317

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

Zihao Liu
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Abstract

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: $$ π(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ 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 $π(x)$ by $\mathcal O(\sqrt x/\log x)$.

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Zihao Liu. 2021-10-31. A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function. https://doi.org/10.1088/1742-6596%2F2287%2F1%2F012008

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