Complex Summation

Algebra Level 3

Let \(S =\) {\(a+bi: a, b \in \mathbb{z}, 1 \leq |a|,|b|\leq 2011\).}.

Find \( \frac{\sum\nolimits_{z \in S} \frac{1}{z^{4}-1}}{\sum\nolimits_{z \in S} \frac{1}{z^{2}-1}}\).

This is not original.

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