Estimates for π(x) for large values of x and Ramanujan's prime counting inequality

2017/03/07 by Christian Axler, Axler, Christian
Mathematics · #11A41 (Secondary) #11N05 (Primary) #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1703.02407

openalex publication_date 2017/03/07 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In this paper we use refined approximations for Chebyshev's ϑ-function to establish new explicit estimates for the prime counting function π(x), which improve the current best estimates for large values of x. As an application we find an upper bound for the number H₀ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every x ≥ H₀.

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