Producto de dos series absolutamente convergentes de Dirichlet

Question

Nos have$$(f * g)(n) = \sum_{d \mid n} f(d)g(n/d).$$How do I see that if the two Dirichlet series$$F(s) = \sum_{n =1}^\infty f(n)n^{-s},\text{ }G(s) = \sum_{n=1}^\infty g(n)n^{-s}$$converge absolutely for $\text{Re} (s) > \sigma_0$, then in the same half-plane, the following equations hold:$$F(s)G(s) = \left( \sum_{n=1}^\infty f(n)n^{-s}\right)\left( \sum_{n=1}^\infty g(n)n^{-s}\right) = \sum_{n=1}^\infty (f * g)(n)n^{-s}?$$

Answer 1

$$ \sum_{n\geq 1}\frac{(f*g)(n)} {n ^ s} = \sum_{n\geq 1} \frac {1} {n ^ s} \sum_ {d\mid n}f(d)g(n/d) = \sum_{k\geq 1} \sum_{d\geq 1}\frac1{(kd)^s}f(d)g(k) \;. $$