For a fixed positive integer $j$, consider the arithmetical function :
$$\vartheta _{j}(k+1)=\left\{\begin{matrix} 1 \;\;, & k+1=j^{l}\;\;(l=1,2,3...)\\ 0 \;\;, & \text{otherwise} \end{matrix}\right.$$ We construct the Dirichlet series : $$\sum_{m=0}^{\infty}\left | G_{m+1} \right |\sum_{k=0}^{m}(-1)^{k}\binom{m}{k}\frac{\vartheta _{j}(k+1)}{(k+1)^{s}}$$ Where $|G_{k}|$ is the kth Gregory coefficient, and $s\in\mathbb{C}$. Does this series converge, and if yes, what is its abscissa of convergence ?