A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function. Issue 1 (1st June 2022)

• Record Type: Journal Article
• Authors: Liu, Zihao
• Is Part Of: Journal of physics. Volume 2287:Issue 1(2022)
• Publication Date: 2022-06-01
• Subjects: Physics -- Congresses; 530.5
• DOI: 10.1088/1742-6596/2287/1/012008
• Languages: English
• ISSNs: 1742-6588

Abstract

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 ) = li ( x ) + O ( x e − c 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 O ( x log log x / log x ) .