Let $\omega(k)$ be the prime omega function, it counts how many distinct prime factors k has.
The dirichlet series for $\omega(k)$ can be written as,$\sum_{k=1}^\infty\frac{\omega(k)}{k^s}=\prod_{p}\frac{1}{1-p^{-s}}*\sum_{p}\frac{1}{p^s}=\zeta(s)*P(s)$ I know I cant re-write $\sum_{k=0}^\infty\frac{\omega(ak+b)}{(ak+b)^s}=\prod_{p\equiv\text{b mod a} }\frac{1}{1-p^{-s}}*\sum_{p\equiv\text{b mod a}}\frac{1}{p^s}$ But can I re-write it, as somthing similar?