Some explicit estimates for the error term in the prime number theorem

2022/04/05 by Daniel R. Johnston, Andrew Yang, Johnston, Daniel R. +1 · 3 citations
Mathematics · #11N05 #11Y70 (Primary) 11M26 (Secondary) #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.01980

openalex publication_date 2022/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By combining and improving recent techniques and results, we provide explicit estimates for the error terms |π(x)-li(x)|, |θ(x)-x| and |ψ(x)-x| appearing in the prime number theorem. For example, we show for all x≥ 2 that |ψ(x)-x|≤ 9.39x(log x)^1.515exp(-0.8274√(log x)). Our estimates rely heavily on explicit zero-free regions and zero-density estimates for the Riemann zeta-function, and improve on existing bounds for prime-counting functions for large values of x.

Cited by

• An effective Bombieri-Vinogradov error term for sifting problems
• An explicit form of Ingham's zero density estimate
• Refinements for primes in short arithmetic progressions

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