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Proposed by Sam Vandervelde, St. Lawrence University, Canton, NY.
First we let \mathbb{Q}^* = \{ \frac{a}{b} \ |  a,b\in \mathbb{Z} ,\ a \neq 0,\ b> 0 and (a,b)=1 \}. In other words, \mathbb{Q}^* is the set of all nonzero rational numbers written in lowest terms. Find, with proof, the value of

\displaystyle\sum_{\frac{a}{b} \in \mathbb{Q}^*}{\frac{1}{(ab)^2}}.

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