Let $p^*_n$ be the $n$ th element of a subset of primes such that $p^*_{n+1}>p^*_n$ and $p^*_n < O((n+2) ln((n+2))^3)$. Define $f(z)$ as the analytic continuation of $\prod_{n>0} (1+\dfrac{1}{p^{*^z}_n-1})$.
The analytic continuation is to the largest possible domain. If there is no natural boundary then this is the entire complex plane.
Conjecture 1 : The analytic continuation to the entire complex plane Always exists.
Conjecture 2 : If $f(1)$ is a pole then it is the only pole in the complex plane $C$ of $f(z)$.
It is easy to show for $Re(z) > 1$ but is the conjecture true ??
Notice $f(1)$ is not Always a pole see :
Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?