The inequality log g(n) < li^{-1}(n) for all n > 0 is equivalent to the Riemann hypothesis.
New bounds for the sum of the first $n$ prime numbers
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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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math.NT 1years
2019 1verdicts
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The Landau function and the Riemann Hypothesis
The inequality log g(n) < li^{-1}(n) for all n > 0 is equivalent to the Riemann hypothesis.