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I've been playing around with the product of polylogarithms, and have come across the following triple sum:

$$ \sum_{n=1}^\infty \sum_{m=1}^{\infty} \sum_{v=1}^{\infty} \frac{z^{m+n+v}\omega^{m+2v}}{(mnv)^{3}}$$

Where $\omega=e^{\frac{2\pi i}{3}}$. I would like to be able to additively decompose this sum without dealing with objects like

$$\omega\sum_{m,n,v\in\mathbb{N}, m+2v \equiv 1 \mod 3} \frac{z^{m+n+v}}{(mnv)^{3}} $$

Because the modulo in the index sort of limits my ability to manipulate what's going on. I would like to define a new index $a$ where $m+2v=3a+1$, but at the moment it looks like I'm going to have to do a ton of algebra to eliminate the $m$ and the $v$. Is there a straightforward way to make the substitution which doesn't turn into manipulating power series of the indices themselves?

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Have you considered splitting the sum into parts, one for each residue class (e.g. of $m$)? Anyways, the Iverson bracket is extraordinarily useful for complicated summation tasks, once you get used to it. (see –  Hurkyl Jul 20 '12 at 0:43

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