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 $? 



Twin prime based Dirichlet series
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 $1$.


