On the moments of averages of quadratic twists of the M\"obius function

Abstract

We consider the moment of quadratic twists of the M\"obius function of the form \[ S_k(X,Y) = \sum_{d\leq X} \left( \sum_{n\leq Y} \left(\frac{8d}{n}\right) \mu(n)\right)^k, \] where $\left(\frac{8d}{\cdot}\right)$ is the Kronecker symbol and $d$ runs over positive, odd and square-free integers. We give unconditional results for their asymptotic behaviors.