Let $d\ge 1$ be an integer, $a>0$ be a real number and let $\vec{s} := (s_0,\cdots,s_{d-1})$ where all the components are strictly bigger than one. We generalize the zeta function to higher dimensions as follows: \begin{equation} \zeta^{(d)} (\vec{s},a) := \sum\limits_{a \le l_{d-1} \le l_{d-2} \le \cdots \le l_0 < \infty} \prod\limits_{\xi=0}^{d-1} l_\xi^{-s_\xi} \end{equation} Clearly $\zeta^{(1)}(s,a) = \zeta(s,a)$. The question is can we compute the generalized function, to an arbitrary precision, using the zeta function and elementary functions only. Using the series representation of the zeta function (see Wikipedia page http://en.wikipedia.org/wiki/Hurwitz_zeta_function ) I have shown that the following expansion holds: \begin{equation} \zeta^{(2)}((s,s_1),a) = \frac{1}{s-1} \left[ \zeta(s+s_1-1,a) + \sum\limits_{q_1=1}^\infty \binom{1-s}{q_1} {\mathcal C}^{(2)}_{q_1} \zeta(s+s_1+q_1-1,a)\right] \end{equation} where \begin{equation} {\mathcal C}^{(2)}_{q_1} := \sum\limits_{m=1}^{q_1} \frac{1}{m+1}\sum\limits_{k=1}^m (-1)^k \binom{m}{k} k^{q_1} \end{equation}
Now, the question is what is the corresponding expansion for arbitrary values of $d$?