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 \(\sum_{m^2 + n^2 \leq X} \sigma_w(m^2 + n^2 + h)\). We approach this continuation through spectral theory. Our results are comparable to earlier work of Bykovskii, who used different methods to study the sums \(\sum_{n^2 \leq X} \sigma_w(n^2 + h)\).