Multiple sums with the Möbius function

ABSTRACT We establish nontrivial bounds for bilinear sums involving the Möbius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as Möbius-function analogues of the ternary Goldbach problem. By contrast, the binary versions of our results remain out of reach, much like the binary Goldbach problem. Nevertheless, we make partial progress in this direction by restricting the range of the third variable as far as possible.