Mean squares of quadratic twists of the M\"obius function

In this paper, we evaluate asymptotically the sum $$\sum_{d \leq X} \left( \sum_{n \leq Y} \mu(n)\leg {8d}{n} \right)^2,$$ where $\leg {8d}{n}$ is the Kronecker symbol and $d$ runs over positive, odd, square-free integers.