\begin{align*} \sum_{k=2}^{\infty}\zeta\left(k\right)z^{k} & =\sum_{k=2}^{\infty}z^{k}\sum_{j=1}^{\infty}\frac{1}{j^{k}}\\ & =\sum_{j=1}^{\infty}\sum_{k=2}^{\infty}\frac{z^{k}}{j^{k}}\\ & =\sum_{j=1}^{\infty}\sum_{k=2}^{\infty}\left(\frac{z}{j}\right)^{k}\\ & =\sum_{j=1}^{\infty}\left(\frac{z}{j}\right)^{2}\frac{1}{1-\frac{z}{j}}\cmt{\because\left|z\right|<1}\\ & =z^{2}\sum_{j=1}^{\infty}\frac{1}{j\left(j-z\right)}\\ & =\left(-z\right)\cdot\left(-z\right)\sum_{j=1}^{\infty}\frac{1}{j\left(j-z\right)}\\ & =-zH_{-z}\cmt{\because H_{z}=z\sum_{k=1}^{\infty}\frac{1}{k\left(k+z\right)}}\\ & =-z\left(\psi\left(1-z\right)+\gamma\right)\cmt{\because\psi\left(z\right)=-\gamma+H_{z-1}}\\ & =-z\left(\psi\left(z\right)+\pi\tan^{-1}\left(\pi z\right)+\gamma\right)\cmt{\because\text{ディガンマ関数の相反公式}\psi\left(1-z\right)-\psi\left(z\right)=\pi\tan^{-1}\left(\pi z\right)} \end{align*}