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Hi everyone

I am trying the evaluate sums of the form $$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$ for general $a_{1,1},\ldots,a_{m,m} \in \mathbb{C}$ (I probably also need to assume real part greater than $0$ to ensure convergence. If you want feel free to assume that $a_{i,j}\in \bar{\mathbb{Q}}$ and the $\{a_{i,j}\}$ for fixed $j$ constitutes an orbit under $G_{\mathbb{Q}}$, although it probably won't make any difference). Sums of this type was considered by Shintani in the seventies. He used geometric series and the integral formula for the gamma function to turn it into a multiple integral of the form $$ \int_{0}^{\infty}\ldots \int_0^{\infty}\frac{ e^{-(a_{1,1}t_1+\ldots+ a_{m,1}t_m)} }{1-e^{-(a_{1,1}t_1+\ldots+ a_{m,1}t_m)}}\ldots \frac{ e^{-(a_{m,1}t_1+\ldots+ a_{m,m}t_m)} }{1-e^{-(a_{m,1}t_1+\ldots+ a_{m,m}t_m)}}t_1^{k-1} \ldots t_m^{k-1} dt_1\ldots dt_m$$ however this doesn't seem - to me at least - to be much easier to evaluate explicitly if $m >4$. Hence I would be very interested if anyone out there knows of a trick ( perhaps a functional equation that I don't know of...) which would help me evaluate them. Any thoughts would be welcome.

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Have you considered trying residues? – Steve Huntsman Jan 31 '10 at 17:07
I guess, the considering $\frac{d}{dx} \log(1-e^{-x}) = e^{-x}/(1-e^{-x})$ doesn't really help... – Suvrit Aug 6 '13 at 5:16

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