Is or can Brun's twin prime theorem be generalized to a sum over all primes differing by less than a fixed gap?

Question

With Zhang's theorem, and further work by others, if I understand it aright, there are infinitely many primes with a gap of less than or equal to 680. So, working from the opposite direction, it would be interesting if the sum of reciprocals of primes with gaps all less than or equal to a given bound(a bound greater than 2 this time), were still finite.(By the way, I don't consider this a duplication of other Brun questions on Stack exchange.)

Answer 1

The answer is yes. The modern proofs of Brun's theorem show that for any fixed $k$, the number of primes $p\le x$ such that $p+k$ is also prime is at most $C_k x/(\log x)^2$, where $C_k$ is a constant depending on $k$. From this result, one can derive (using partial summation) the corollary that the sum of the reciprocals of these twin-prime-analogues converges for every fixed $k$. Of course, this in turn implies that the sum converges over all $k$ up to some bound such as $B=680$. Indeed, one can even take $B$ growing with $x$, for example $B=(\log x)^{0.999}$, and still have this result hold.