Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

2021/09/06 by Daniel R. Johnston, Johnston, Daniel R.
Mathematics · #11Y70 (Primary) 11M26 (Secondary) #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2109.02249

openalex publication_date 2021/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using a recent verification of the Riemann hypothesis up to height 3⋅ 10¹², we provide strong estimates on π(x) and other prime counting functions for finite ranges of x. In particular, we get that |π(x)-li(x)|<√(x)log x/8π for 2657≤ x≤ 1.101⋅ 10²⁶. We also provide weaker bounds that hold for a wider range of x, and an application to an inequality of Ramanujan.

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