  Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real
Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of
binary forms defined over $\mathbb{Z}$ has been the focus of extensive
investigations in recent years, with spectacular applications to Manin's
conjecture for Ch\^atelet surfaces. We introduce a far-reaching generalization
of this problem, in particular replacing $\chi$ by Jacobi symbols with both
arguments having varying size, possibly tending to infinity. The main results
of this paper provide asymptotic estimates and lower bounds of the expected
order of magnitude for the corresponding averages. All of this is performed
over arbitrary number fields by adapting a technique of Daniel specific to
$1\ast 1$. This is the first time that divisor sums over values of binary forms
are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our
work is a key step in the proof, given in subsequent work, of the lower bound
predicted by Manin's conjecture for all del Pezzo surfaces over all number
fields, under mild assumptions on the Picard number.
