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

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arxiv 2111.10001 v5 pith:JLGPLQ3G submitted 2021-11-19 math.NT

Michaela Cully-Hugill , Daniel R. Johnston This is my paper · ORCID
classification math.NT
keywords errorexplicittermformulamangoldtriemann--vonapplicationsconsecutive
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abstract

We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramar\'e (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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  1. Effective short intervals containing primes

    math.NT 2025-08 conditional novelty 5.0 of 10

    For every n ≥ 106 and every x ≥ 1, there is always a prime in the interval [x, x + x^(1−1/n)], making several classical prime-gap results fully explicit.

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