A twisted additive divisor problem

Abstract

We give asymptotics for shifted convolutions of the form $$\sum _{n<X}\frac{{\sigma }_{2u}(n,\chi ){\sigma }_{2v}(n+k,\psi )}{n^{u+v}}$$ for nonzero complex numbers $u,v$ and nontrivial Dirichlet characters $\chi ,\psi$. We use the technique of "automorphic regularization" to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.