Bounds on prime gaps as a consequence of the divergence of the series of reciprocal primes

2017/06/27 by Azevedo, Douglas
#11A41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1706.09058

Abstract

In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.

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