I am interested in a sum of the shape $$ \sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\ \mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2). $$ I figured it must have been considered before, but I have not been able to find a reference for it. Any information would be appreciated. Thank you very much.

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    $\begingroup$ The usual technique of writing $\mu^2(x_1x_2) = \sum_{d^2\mid x_1x_2} \mu(d)$ and reversing the order of summation should be a good start here; the new inner sum simply counts integers in an arithmetic progression and thus we have as good an error term as we could hope for (if $B$ is not too small in terms of $H$). $\endgroup$ – Greg Martin Feb 7 at 18:25

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