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Consider the sum $$\sum_{\substack{n,m\leq N\\ n, m \equiv a \bmod q\\n \not= m}}\mu(n)\mu(m).$$ What are some of the best known estimates regarding this sum? For the case where we allow $n=m$, I am aware of the following preprint of Nunes. But what the sum above? Here, $\mu$ is the Möbius function.

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    $\begingroup$ Isn't this sum $( \sum_{ { n \leq N , n \equiv a\operatorname{ mod }q }} \mu(n) )^2 - \sum_{ { n \leq N ,n \equiv a \operatorname{ mod }q }} \mu(n)^2$? I would estimate the two terms separately. $\endgroup$
    – Will Sawin
    Apr 28, 2021 at 15:08
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    $\begingroup$ I just realised that this is not a clever question. For $q\leq (\log N)^A$, GH from MO points out that the first sum in your comment is estimated by Siegel-Walfisz (mathoverflow.net/questions/390774/…). So, I am guessing that without conditions on $q$, there aren't many good estimates of the sum in the question available? $\endgroup$
    – user147650
    Apr 28, 2021 at 15:14
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    $\begingroup$ Not that I am aware of (except under GRH and in the function field model). $\endgroup$
    – Will Sawin
    Apr 28, 2021 at 15:24
  • $\begingroup$ No problem, thanks. $\endgroup$
    – user147650
    Apr 28, 2021 at 19:17

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