The problem: I am concerned of getting a generalisation of the Wiener-Ikehara theorem for Dirichlet series which are analytic in the plane $\{s\in\mathbb{C}:\sigma>1\}$ and extend analytically over the line $\sigma=1,$ except for $1$, where they have a pole of order $k$ (not of order $1$ necessarily). I proved a generalisation in the following way. I added the extra assumption that the quotient
$$G(s)=\frac{F(s)}{{\zeta(s)}^k}$$
is represented by a Dirichlet series of non-negative coefficients for $\sigma>1$. Then, by induction on $k$, using the classical Wiener-Ikehara theorem (for a simple pole) for the initial induction step and applying the induction hypothesis on
$$H(s)=G(s){\zeta(s)}^{k-1},$$
I proved my assertion (the details are a bit technical with Abel summation). Although, I want to get a version of the above claim where the coefficients of $G$ might be negative.
The question: What kind of an assuption should I add to $G$ (for instance for its coefficients), in order to get a similar proof in this case?
Thanks in advance for your help!