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_{k=1}^\infty \frac{x^k}{k^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 :
$$\tag{1}\psi(1+x)=-\gamma-\sum_{n=1}^\infty \zeta(n+1)\;(-x)^n$$
while the reflection formula allows to get the even values of $\zeta$ directly as :
$$\tag{2}\pi\;x\;\cot(\pi\;x)=-2\sum_{n=0}^\infty \zeta(2n)\;x^{2n}$$
A generating function for the polylogarithm was obtained too :
$$\tag{3}z\,\Phi(z, 1, 1-x)=\sum_{n=0}^\infty\;\operatorname{Li}_{n+1}(z)\;x^n$$
using the Lerch zeta function $\displaystyle\;\Phi(z, s, \alpha) := \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}$.
$(1)$ may be generalized to the Hurwitz zeta function $\zeta(s,\alpha)=\Phi(1, s, \alpha)$ :
\begin{align}
\tag{4}\psi(\alpha)-\psi (\alpha+x)&=\sum_{n=1}^{\infty}\zeta(n+1,\alpha)\,(-x)^{n},\qquad |x|<|\alpha|\\
\tag{5}\frac{\Phi(1, s,\alpha) - x\,\Phi(x, s,\alpha)}{1-x}&=\sum_{n=0}^{\infty} \zeta(s,n+\alpha) \: x^n\\
\end{align}
(the last one found here )
