# Twin prime based Dirichlet series

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 $1$?

• 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 $s$. – 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 $1$.
• 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 $\varepsilon>0$? – Sylvain JULIEN Mar 6 '18 at 16:39