Yesterday when reading Landau's Elementary Number Theory, I came across Brun's theorem which states that the reciprocals of twin primes add up to a finite sum (the series converges), and this made me wonder if the same is true for Sophie Germain primes (the reciprocals add up to a finite sum). But I cannot Google out anything. Can anyone give an answer or point me to some references? Thanks!
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The same sieve methods of Brun used to show the result on the twin primes can be used to estimate that, for any fixed $k$ and $a$ coprime positive integers with different parity, the number of primes $p < x$ for which $kp+a$ is prime is bounded above by $C x/(\log x)^2$ for some constant $C$. This implies that the sum of the reciprocals of that primes is convergent.
In fact, it is expected that the number of Sophie Germain primes less than $x$ is asymptotically equal to the number of twin primes less than $x$.
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$\begingroup$ Thank you very much! May I ask for some spelling out of your 2nd paragraph or some references such as survey articles? $\endgroup$ Oct 4, 2019 at 13:01
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$\begingroup$ Xarles is saying it works the same for $p,p+2$ both primes and $p,2p+1$ both primes. Estimating $O(x/\log^2 x)$ for the number of twin primes is the Brun sieve, you'll find a lot of resources about it. The random model for the primes applies the same way for the twin primes constant and the Sophie Germain primes constant, check here for an elementary way to derive it $\endgroup$– reunsOct 4, 2019 at 14:20