Calculating prime sequences 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?
 A: You should learn about Dirichlet characters. There are $\phi(q)$ (completely multiplicative) Dirichlet characters $\chi$ modulo $q$, and summing over all of them one gets the following nice relationship, for any $a$ that is relatively prime to $q$:
$$
\frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \chi(n) = \begin{cases}1, &\text{if } n\equiv a\pmod q, \\0, &\text{if } n\not\equiv a\pmod q. \end{cases}
$$
Therefore (for $1\le a\le q$ and $(a,q)=1$)
\begin{align*}
\sum_{k=0}^\infty \frac{\omega(qk+a)}{(qk+a)^s} &= \sum_{n=1}^\infty \frac{\omega(n)}{n^s} \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \chi(n) \\
&= \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \sum_{n=1}^\infty \frac{\omega(n)\chi(n)}{n^s} \\
&= \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \prod_p \frac1{1-\chi(p)p^{-s}} \sum_p \frac{\chi(p)}{p^s} \\
&= \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} L(s,\chi) \sum_p \frac{\chi(p)}{p^s},
\end{align*}
where $L(s,\chi)$ is a Dirichlet $L$-function.
A: Define a generalization of the prime zeta function, where the primes range over the congruene classes:
$$P_{a,b}(s)=\sum_{p\equiv b \text{ mod a}}\frac{1}{p^s}$$
Also define the indicator function $$\delta_{a,b}(k)=\text{1 if k mod a = b},\text{ 0 if k mod a}\ne b$$
Then for b>0,$$\sum_{k=0}^\infty\frac{\omega(ak+b)}{(ak+b)^s}=\frac{1}{a^s}\sum_{j=1}^a\sum_{k=1}^a\delta_{a,b}(kj)P_{a,k}(s)\zeta(s,\frac{j}{a})$$
Where $\zeta(s,q)$ is the hurrwitz zeta function.
This seems very straight forward,
I don't understand what it is you did with your "dirichlet characters"
