Mean squares of quadratic twists of the Möbius function (Q2689516)

Let \(\chi_d\) be the Kronecker symbol \(\left(\frac{d}{\cdot}\right)\), and fix two non-negative, smooth functions \(\Phi(x)\) and \(W(x)\) that are compactly supported on \((0, \infty)\). In the paper under review, the authors approximate, conditionally, the weighted sum \[ S(X,Y;\Phi,W):=\sum_d\left(\sum_n\mu(n)\chi_{8d}(n)\Phi\left(\frac{n}{X}\right)\right)^2\,W\left(\frac{d}{X}\right), \] where the outer sum runs over odd and square-free integers \(d\). More precisely, assuming the generalized Riemann hypothesis, they show that \[ S(X,Y;\Phi,W)=CXY+O(X^{1/2+\varepsilon}Y^{3/2+\varepsilon}+XY^{1/2+\varepsilon}), \] providing yet a valid asymptotic formula for the wide range \(Y\ll X^{1-\varepsilon}\). The leading constant \(C\) determined explicitly on the paper.