The famous Polignac conjecture posits that the $n$-th prime gap $g\_{n}:=p\_{n+1}-p\_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta\_{Pol}:=s\mapsto\sum\_{n>0}(ng\_{n}/2)^{-s}}$ has an abscissa of convergence $\sigma\_{Pol}$ less or equal to $1$.


Is this abscissa of convergence known unconditionally or under some widely believed conjecture other than Polignac's?



Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?
Well, the abscissa of convergence is $1$, though I see no way to deduce anything about it from the Polignac conjecture. $\sigma\_P \le 1$ is obvious from $|\zeta\_P(s)| \le \zeta(\Re s)$, while for $\sigma\_P \ge 1$ it is enough to prove that $\zeta\_P(1) = \infty$:


$$\zeta\_P(1) \ge \sum\_{k=1}^\infty \frac{1}{2^{k+1}}\sum\_{n = 2^k}^{2^{k+1}}\frac{1}{g\_n} \gg \sum\_{k = 1}^\infty \frac{1}{2^k} \frac{2^k}{k} = \infty,$$


where in the second step we used that $\sum\_{n = 2^k}^{2^{k+1}} g\_n \le p\_{2^{k+1}} \ll 2^k \log(2^k)$ and, say, AM-HM inequality.


