Are there known any series that converge for $\Re(s)>0$ or at least for $\Re(s)>\frac{1}{2}$ that are equal to $\frac{f(s)}{\zeta (s)}$, where $\zeta (s)$ is Riemann zeta function and $f(s)$ is arbitrary function?
Examples that I already know: $$\tag{1}\sum _{n=1}^{\infty } \frac{\mu (n)}{n^s}=\frac{1}{\zeta (s)}$$ $$\tag{2}\sum _{n=1}^{\infty } \frac{\lambda (n)}{n^s}=\frac{\zeta (2 s)}{\zeta (s)}$$ $$\tag{3}\sum _{n=1}^{\infty } \frac{\phi (n)}{n^s}=\frac{\zeta (s-1)}{\zeta (s)}$$ where $\mu (n)$ is Möbius function, $\lambda (n)$ is Liouville function and $\phi (n)$ is Euler's totient function.
$(1)$ and $(2)$ seem to converge for $\Re(s)>\frac{1}{2}$ but proof of it is still an open problem, as far as I know. $(3)$ converge only for $\Re(s)>2$.
I mean there are plenty representations of $\zeta (s)$ in numerator as series that converge in whole critical strip but I can not find any for $\zeta (s)$ in denominator. Do you know any other examples of such series?
EDIT: Of course I meant to converge in critical strip except poles caused by nontrivial zeros of $\zeta(s)$.