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arXiv · 1803.03377

On the sum of the reciprocals of the differences between consecutive primes

Nian Hong Zhou
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Abstract

Let $p_n$ denote the $n$-th prime number, and let $d_n=p_{n+1}-p_{n}$. Under the Hardy--Littlewood prime-pair conjecture, we prove \begin{align*} \sum_{n\le X}\frac{\log^αd_n}{d_n} \sim\begin{cases} \frac{X\log\log\log X}{\log X}~\qquad\quad~ &α=-1,\\ \frac{X}{\log X}\frac{(\log\log X)^{1+α}}{1+α}\qquad &α>-1, \end{cases} \end{align*} and establish asymptotic properties for some series of $d_n$ without the Hardy--Littlewood prime-pair conjecture.

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Nian Hong Zhou. 2018-03-12. On the sum of the reciprocals of the differences between consecutive primes. https://doi.org/10.1007/s11139-018-0034-7

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