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If $p$ is a prime number, and $q$ is its twin prime, the sum of the reciprocal twin numbers is convergent and the value of the sum of the series is the Brun constant. Now, if we consider the prime numbers $x=p+\alpha$ where $\alpha$ is a constant such that $x$ is also a prime, we can consider the sum: $S=\frac{1}{\sum_{k=1}^{N}x_k}$ The question is: is it possible to show for what values of $\alpha$ the series is convergent? Thanks in advance.

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    @Riccardo.Alestra can you edit your formula or the title/text please? It just doesn't fit together...2012-07-26

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For any non-zero integer $\alpha$, the Selberg sieve can be used to show that the number of primes $p \le n$ such that $p + \alpha$ is prime is at most $C_\alpha n/(\log^2 n)$. The value of $\alpha$ does have an effect: if $\alpha$ is divisible by many small primes, then the constant in the upper bound will be higher. On the other hand if $\alpha$ is odd, there is at most one prime satisfying the condition.

Either way, it is easy to show by partial summation that the reciprocal sum converges. Brun's original proof gave a slightly weaker upper bound, but one that is still strong enough to obtain convergence.