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New bounds for the sum of the first $n$ prime numbers

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arxiv 1606.06874 v2 pith:RELKGPIG submitted 2016-06-22 math.NT

Christian Axler This is my paper · ORCID
classification math.NT
keywords firstnumbersprimeasymptoticestimatesformularesultsaccurate
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abstract

In this paper we establish a general asymptotic formula for the sum of the first $n$ prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin. Further we prove a series of results concerning Mandl's inequality on the sum of the first $n$ prime numbers. We use these results to find new explicit estimates for the sum of the first $n$ prime numbers, which improve the currently best known estimates.

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  1. The Landau function and the Riemann Hypothesis

    math.NT 2019-07 unverdicted novelty 6.0 of 10

    The inequality log g(n) < li^{-1}(n) for all n > 0 is equivalent to the Riemann hypothesis.

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