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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.

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    $\begingroup$ If I understand your question correctly, you are asking the following. The sum of the reciprocal of primes is divergent but the sum of the reciprocal of twin primes is convergent (as shown by Brun). Your question is there any gap $d$ greater than 2 (2 is for twin primes) between primes for which the sum of the reciprocal primes differing by $d$ is convergent. Is this understanding correct? $\endgroup$ – Nilotpal Kanti Sinha May 21 '13 at 5:30
  • $\begingroup$ @Nilotpal: yes (more or less), is there any gap $d$ greater than or equal $4$ ($d=2$ is for twin primes) between primes for which the sum of the reciprocal primes differing by d is convergent. (Your English is smoother than mine). Thank you. $\endgroup$ – Włodzimierz Holsztyński May 21 '13 at 5:41
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It turns out that $S_d$ is finite for every $d$. See my answer to this very similar question.

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The following heuristic study suggests that for any gap $d \ge 2$ the sum of the reciprocal of primes having a gap of $d$ should be convergent.

Let $L(r,x)$ be the number of conjectured occurrences of gaps of size $2r$ between successive prime $\le x$. We have

$$ L(r,x) = \int_{2}^{x} \sum_{}{}\frac{(-1)^k A(r,k)}{(\log t)^{k+1}} dt $$

The coefficients $A(r,k)$ as well the details of the above formula are explained in the following link:

http://mac6.ma.psu.edu/primes/

The above study shows that till the point $1.7427435732 * 10^{35}$ the gap of 6 is most frequent. Since we sum of the reciprocal of twin primes are convergent and they are asymptotically more denser than primes with any other gaps, we expect that the sum of the reciprocal of primes with a gap $d > 2$ to be convergent as well.

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  • $\begingroup$ @Nilotpal, it's an interesting numerical study. To obtain the respective theorems about prime gaps must be very hard though, one would guess. $\endgroup$ – Włodzimierz Holsztyński May 25 '13 at 3:52

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