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Mathematica has PrimeZetaP for the prime zeta function $\sum_p \frac{1}{p^s}$ where the sum is taken over all primes.

How do I use Mathematica to make an equivalent summation that runs over all primes that are plus or minus 6 from another prime? In other words,

$$\sum_{{q,q\pm 6}\in\mathbb{P}} \frac{1}{q^s}$$

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  • $\begingroup$ Anything known about this function at all? $\endgroup$ – Michael E2 Feb 5 at 12:50
  • $\begingroup$ I think I’m kind of inventing it... But I’m just wondering how I’d code it :-) $\endgroup$ – Richard Burke-Ward Feb 5 at 13:38
  • $\begingroup$ I think you need to get an analytical expression for it (in the mathematical sense, e.g., a series, an integral, etc.). Well, you've got a series, but it's not a very convenient one. You could loop over the primes, but I suspect PrimeZetaP does not do that (maybe it does it for large values of s). Is anything known about a similar sum over twin primes? $\endgroup$ – Michael E2 Feb 5 at 13:48
  • $\begingroup$ Only that the sum converges to Brun’s constant which lies somewhere between (roughly) $1.8$ and $2.4$. $\endgroup$ – Richard Burke-Ward Feb 6 at 0:24
  • $\begingroup$ It should depend on s, shouldn't it? $\endgroup$ – Michael E2 Feb 6 at 0:53

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