I am curious to know if this function was in the literature, and mainly if it is possible define an analytic extension out of $\Re s>0$.
Let $N_n$ the $n$th primorial number, and $s=\sigma+it$ the complex variable. Then it is easy, using absolute convergence, to know that
$$\rho(s)=\sum_{n=1}^\infty\frac{1}{N_n^s}\tag{1}$$ is analytic for $\Re s>0$ since converges absolutely for $\sigma>0$. One knows that has a pole at $s=0$. Also if $s=k+i\cdot0$ is a negative integer our series diverges.
I call $(1)$ the Primorial zeta function. But I don't know if the surname zeta is suitable.
Question. Was in the literature our function $(1)$? Do you think that is it possible to define an analytic continuation, out of our pole $s=0$, to a small strip at the left of the origin? If $(1)$ was in the literature, please refer the literature answering this question as a reference request, then I am going to search and read those statements. Many thanks.
I searched in Google pseudo z functions, primorials.
I know that if one can to find, for example, an integral representation for a function that extends the domain of definition of our complex function, then one can define an analytic extension from it.