**PART I** (Initial version)

Let $P$ be the set of all primes $2\ 3\ \ldots$. Let

$$P_d\ \ :=\ \ \{\ p\in P\ :\ \exists_{q\in P}\ \ 0 < |p-q|\le d\ \}$$

and

$$S_d\ :=\ \sum_{p\in P_d}\ \frac 1p$$

for every real $d>0$. Thus $d\mapsto S_d$ is non-decreasing, $S_2 < \infty$, and $\lim_{d\rightarrow\infty} S_d = \infty$. What else is known about $S_d$ ? For which values of $d$ the sum $S_d$ is finite?

**PART II** (additional)

Let $d\ m\ n$ be positive integers. Is it true that

$$ (m < n)\quad\Rightarrow\quad \left( S_{m+d}-S_m\ \ge\ S_{n+d}-S_n \right)$$

**?** - It seems (to me) that it should be true.