On the error term in the explicit formula of Riemann--von Mangoldt

2021/11/19 by Michaela Cully-Hugill, Daniel R. Johnston, Cully-Hugill, Michaela +1 · 3 citations
Mathematics · #11M26 (Primary) 11N05 #11Y70 (Secondary) #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2111.10001

openalex publication_date 2021/11/19 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We provide an explicit O(xlog x/T) error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.

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• Transcendency of variants of Mills' constant
• Effective short intervals containing primes
• An explicit form of Ingham's zero density estimate

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