A prime $p$ is called a [Sophie Germain prime](https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes) if $2p+1$ is also prime:
[OEIS A005384](https://oeis.org/A005384).
Whether there are an infinite number of such primes is unsolved.
My question is:



> 
> If there are an infinite number of Germain primes,
> is the sum of the reciprocals of these primes known to converge, or diverge?
> 
> 
> 


Of course if there are only a finite number of Germain primes, the sum is finite.
And a lower bound on any infinite sum can be calculated. But it is conceivable
that it is known that the sum either converges or is a finite sum.
And maybe even an upperbound is known?




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(My connection to this topic is via this question:
["Why are this operator's primes the Sophie Germain primes?"](https://mathoverflow.net/q/48638/6094).)



Sum of reciprocals of Sophie Germain primes
Here is a general result. For a sequence of nonnegative numbers $\{a\_n\}$, let $A(x) = \sum\_{n \leq x} a\_n$. For example, if $S \subset \mathbf Z^+$ and we set $a\_n = 1$ when $n\in S$ and $a\_n = 0$ when $n \not\in S$, then $A(x)$ is the number of elements of $S$ that are $\leq x$.


Exercise: If $A(x) = O(x/(\log x)^r)$ for a positive integer $r$ and all $x \geq 2$, then $\sum\_{n \leq x} a\_n/n$ converges as $x \to \infty$ if $r \geq 2$ and $\sum\_{n \leq x} a\_n/n = O(\log \log x)$ for $r = 1$.


Example: if $f\_1(T), \ldots, f\_r(T)$ are polynomials with integer coefficients that fit the hypotheses of the Bateman-Horn conjecture (twin primes are $f\_1(T) = T$ and $f\_2(T) = T+2$, while Sophie Germain primes are $f\_1(T) = T$ and $f\_2(T) = 2T+1$), then Bateman and Stemmler showed $60$ years ago that the number of $n \leq x$ such that $f\_1(n), \ldots, f\_r(n)$ are all prime is $O(x/(\log x)^r)$, where the $O$-constant depends on $f\_1, \ldots, f\_r$.
Therefore if above we take $S$ to be the $n \in \mathbf Z^+$ such that $f\_1(n), \ldots, f\_r(n)$ are all prime and define $a\_n$ to be $1$ or $0$ according to $n \in S$ or $n \not\in S$, then the exercise above says the sum of all $1/n$ for $n \in S$ converges if $r \geq 2$.


So for any sequence of pairs of primes $p$ and $ap+b$ that are expected to occur infinitely often ($p$ and $p+2$, or $p$ and $2p+1$, or $\ldots$), the sum of $1/p$ for such primes converges.


That the sum of the reciprocals of the twin primes converges indicates that this summation is the wrong thing to be looking at. We want a strategy to prove the infinitude of twin primes, and that suggests a better sum. The Bateman-Horn conjecture predicts the number of $n \leq x$ such that $f\_1(n), \ldots, f\_r(n)$ are all prime is asymptotic to $Cx/(\log x)^r$ where $C$ is a positive constant depending on $f\_1, \ldots, f\_r$, and if $A(x) \sim cx/(\log x)^r$ as $x \to \infty$ for some $c > 0$ then $\sum\_{n \leq x} a\_n(\log n)^{r-1}/n \sim c\log\log x$. Therefore we expect (but have never proved) that the sum of $(\log p)/p$ over prime $p \leq x$ such that $p$ and $p+2$ are prime should grow like $c\log\log x$ for some constant $c > 0$, and a similar asymptotic estimate (for a different constant $c$) should hold for the sum of $(\log p)/p$ over all prime $p \leq x$ such that $p$ and $2p+1$ are prime.


