A proof for the generalizated Brun's theorem about the summability of the reciprocal of twin primes

Question

Please, can anyone prove or indicate a paper or book where it is proved a generalization for the Brun's theorem? That is: the sequence $(1/p_n)$ is summable, where $p_{n+1}-p_{n}=k$, (for some $k\geq2$ even), and $p_n$ denotes the $n$th prime number.

Answer 1

The summability of reciprocals of twin primes follows from the estimate and partial summation:

Let $N_2(x)$ be the number of prime numbers $p\leq x$ such that $p$ and $p+2$ both are primes. Then $$ N_2(x) \ll \frac x{\log^2 x}. $$

An analogue of the above is

Let $r$ be an even positive integer. Let $N_r(x)$ be the number of prime numbers $p\leq x $ such that $p$ and $p+r$ both are primes. Then $$ N_r(x) \ll_r \frac x{\log^2 x}. $$

A reference for this result is Corollary 3.14 in Montgomery & Vaughan 'Multiplicative Number Theory I' Classical Theory.

Therefore, the convergence of the sum: $$ \sum_{p, p+r \ \mathrm{prime}} \frac1p $$ follows in the same way by applying partial summation.