Sharper bounds for the error term in the Prime Number Theorem

2022/06/25 by Andrew Fiori, Fiori, Andrew, Habiba Kadiri +2
Computer Science · Mathematics · #11A05 #11M06 #11M26 #11N25 #11N56 #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.12557

openalex publication_date 2022/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function ψ(x) to bounds of the same type on both θ(x) and π(x). This follows up our previous work on ψ(x) in \citeFKS, and prove that | π(x) - Li(x) | ≤ 9.2211 x√(log(x)) exp ( -0.8476 √(log(x)) ) for all x≥ 2. Additionally, we are able to obtain the best numeric bounds for x on a very large interval (all x up to exp(1.8⋅10⁹)).

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