For the averaged-prime sequence, analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures are proved from known explicit estimates.
On the asymptotic expansion of the sum of the first n primes
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abstract
An asymptotic formula for the sum of the first n primes is derived. This result improves the previous results of P. Dusart. Using this asymptotic expansion, we prove the conjectures of R. Mandl and G. Robin on the upper and the lower bound of the sum of the first n primes respectively.
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On the arithmetic average of the first $n$ primes
For the averaged-prime sequence, analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures are proved from known explicit estimates.