Mean values of divisors twisted by quadratic characters

In 2000, J. B. Conrey, D. W. Farmer, and K. Soundararajan found an asymptotic formula for all $X,Y>0$ of the following character sum: $$\sum_{m \leq X, n \leq Y} \Big (\frac {m}{n} \Big ).$$
It is relatively easy to obtain such an asymptotic formula when $X, Y$ are far apart in size. However, things are much subtler when $X$ and $Y$ are of comparable size. In this talk, we consider an analogue case by establishing the mean values of divisors twisted by quadratic characters. Other similar results will also be discussed.