On the asymptotic expansion of the sum of the first n primes

Nilotpal Kanti Sinha

On the asymptotic expansion of the sum of the first n primes

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arxiv 1011.1667 v2 pith:3B3R5M23 submitted 2010-11-07 math.NT

Nilotpal Kanti Sinha This is my paper · ORCID

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keywords asymptoticfirstprimesexpansionboundconjecturesderiveddusart

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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
math.NT 2025-05 conditional novelty 5.0 of 10

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.

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