Does the dirichlet inverse of a series with finite abscissa of convergence also has a finite abscissa of convergence?

Question

I would like to know if the Dirichlet inverse $L(s,g)$ of a series $L(s,f)$ ($f(1)\neq 0$) with finite abscissa of convergence also has a finite abscissa of convergence? Is there a specific criteria to ensure it is the case?

Answer 1

No.

For instance, suppose $f(1) = 1$ and $f(2) = -1$, while $f(n) = 0$ for all $n \geq 3$. Then $F(s) = \sum f(n) n^{-s} = 1 - 2^{-s}$, which is absolutely convergent everywhere.

But it's Dirichlet inverse is $G(s) = \frac{1}{1 - 2^{-s}}$, which only converges for $\Re s > 0$.