This is an answer to the initial question (asking for a generating function of $\zeta(s)$) :
- $−\ln(1−x)$ is the generating function of $\ \frac 1n$.
- $-\frac{\ln(1−x)}{1-x}$ is the generating function of the harmonic number $\ H_n=\sum_{k=1}^n\frac 1k$
The generating function of the generalized harmonic number $\ H_{n,s}:=\sum_{k=1}^n\frac 1{k^s}\ $ is given by :
$$\frac{\operatorname{Li}_s(x)}{1-x}$$
with $\operatorname{Li}_s$ the polylogarithm.
Should you simply want $\ \displaystyle\sum_{n=1}^\infty \frac{x^n}{n^s}=\operatorname{Li}_s(x)\ $ then dot dot's answer is right of course !
Note that a generating function for $\zeta(n)$ is known as the digamma function :
$$\psi(1+x)=-\gamma-\sum_{n=1}^\infty \zeta(n+1)\;(-x)^n$$
while the reflection formula allows to get the even values of $\zeta$ directly as :
$$\pi\;x\;\cot(\pi\;x)=-2\sum_{n=0}^\infty \zeta(2n)\;x^{2n}$$
A generating function for the polylogarithm was obtained too :
$$z\,\Phi(z, 1, 1-x)=\sum_{n=0}^\infty\;\operatorname{Li}_{n+1}(z)\;x^n$$
using the Lerch zeta function $\displaystyle\;\Phi(z, s, \alpha) := \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}$.
