w(n)=v(n) is the number of distinct prime factors of n. i have found Its Dirichlet Series and Abscissa of Convergence from Apostol's book.
Proof. Let $a_{n}$ indicate whether $n$ is prime. For $\sigma>1$
\begin{aligned}
\zeta(s) \sum_{p} \frac{1}{p^{s}} &=\left(\sum_{n=1}^{\infty} \frac{1}{n^{s}}\right)\left(\sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}\right) \\
&=\sum_{n=1}^{\infty} \frac{1}{n^{s}} \sum_{d | n} a_{n} \\
&=\sum_{n=1}^{\infty} \frac{1}{n^{s}} \sum_{p | n} 1 \\
&=\sum_{n=1}^{\infty} \frac{\nu(n)}{n^{s}}
\end{aligned}
When I search this arithmetic function , i also read from Titshmarch and Montgomery's Books; $\sum_{n=1}^{\infty} \omega(n) n^{-s}=\zeta(s) \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log \zeta(n s)$ =$\zeta(s) \sum_{p} \frac{1}{p^{s}}$
So; My little 2 questions are How can we prove that equality?
And i am also curious about abscissa of Absolute convergence. What about it? You can also give some good referrences (books,article,hints,the books solutions, pdf-examples etc) to me about the Dirichlet series and their Analytic properties. Thanks for your answers.