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

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^{\alpha}d_n}{d_n} \sim\begin{cases} \frac{X\log\log\log X}{\log X}~\qquad\quad~ &\alpha=-1,\\ \frac{X}{\log X}\frac{(\log\log X)^{1+\alpha}}{1+\alpha}\qquad &\alpha>-1, \end{cases} \end{align*} and establish asymptotic properties for some series of $d_n$ without the Hardy--Littlewood prime-pair conjecture.