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 $\sum_{ n^2 \leq X} \sigma_w(n^2 + h)$.
Categories: math.NT