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

$$\begin{aligned} \sum _{n\le X}\frac{\log ^{\alpha }d_n}{d_n}\sim {\left\{ \begin{array}{ll} \quad \frac{X\log \log \log X}{\log X}~\qquad \quad ~ &{}\alpha =-1,\\ \frac{X}{\log X}\frac{(\log \log X)^{1+\alpha }}{1+\alpha }\qquad &{}\alpha >-1, \end{array}\right. } \end{aligned}$$

and establish asymptotic properties for some series of $d_n$ without the Hardy–Littlewood prime-pair conjecture.