Counting Divisors in the Outputs of a Binary Quadratic Form

For a fixed natural number $h$, we prove meromorphic continuation of the two-variable Dirichlet series ${\sum }_m{r}_2(m){\sigma }_w(m+h)(m+h{)}^{-s+w}$ to ${\mathbb{C}}^2$ and use this to obtain asymptotics for ${\sum }_{m^2+n^2\le X}{\sigma }_w(m^2+n^2+h)$. We approach this continuation through spectral theory. Our results are comparable to earlier work of Bykovskii, who used different methods to study the sums ${\sum }_{n^2\le X}{\sigma }_w(n^2+h)$.