I am currently struggling to understand Section 11.9 in Tom Apostol's Introduction to Analytic Number Theory, where he talks about taking the logarithm of a Dirichlet series.
The idea is that we have some Dirichlet series $F(s) = \sum_{n=1}^\infty f_nn^{-s}$ with $f(1)\neq 0$ and
- $F(s)$ is absolutely convergent
- $F(s) \neq 0$
for any $s$ with $\mathfrak{R}(s)>\sigma_0$.
Apostol says that we can then always find some Dirichlet series $G(s)$ that is also absolutely convergent for $\mathfrak{R}(s)>\sigma_0$ and fulfils $e^{G(s)} = F(s)$ for any such $s$, and that $$G(s) = \ln f(1) + \sum_{n=2}^\infty \frac{(f' * f^{-1})(n)}{\ln n} n^{-s}$$ where $f^{-1}$ is the Dirichlet inverse of $f$ and $f'(n) = f(n) \ln n$, i.e. they correspond to the multiplicative inverse and the negative derivative of $F$.
However, if I am not missing anything here, I think all of this only works if $\sum_{n=1}^\infty f'(n) n^{-s}$ and $\sum_{n=1}^\infty f^{-1}(n) n^{-s}$ converge absolutely, but I don't see why they should, in general.