Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum_{n>0}a_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That way such a "twin prime transform" of the Riemann zeta function would give for $ s=1 $ the so called Brun constant.

What would thus be the abscissa of convergence for such a series assuming the original one is $ 1 $?

  • $\begingroup$ If the sequence $a_n$ is supported on twin primes, this transform does not change the function and its abscissa. If it is supported on non primes, the transform converges for all $s$. $\endgroup$ – Ofir Gorodetsky Mar 6 '18 at 15:57

It is conjectured that there are $\gg x/\log^2 x$ twin primes up to $x$. If this is the case, then the abscissa of convergence for $\sum_p p^{-s}$, the sum taken over twin primes, is equal to $1$. If there are much fewer twin primes up to $x$ (but still infinite in total), the abscissa of convergence could be anywhere between $0$ and $1$.

  • $\begingroup$ Thank you for your answer. Would this still hold for any increasing sequence of integers containing $ \gg_{\varepsilon}x^{1-\varepsilon} $ elements up to $ x $ for every $ \varepsilon>0 $? $\endgroup$ – Sylvain JULIEN Mar 6 '18 at 16:39
  • 2
    $\begingroup$ @SylvainJULIEN: Yes, but you can also check this yourself, it is a straightforward exercise in analysis. See also Theorem 1.3 in Montgomery-Vaughan: Multiplicative number theory I. $\endgroup$ – GH from MO Mar 6 '18 at 17:28

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