Multiple sums with the M\"obius function

TL;DR

This paper develops bounds for bilinear sums involving the M"obius function over solutions to various equations, advancing understanding of the M"obius function's behavior in additive problems similar to Goldbach's conjecture.

Contribution

It introduces new bounds for bilinear sums with the M"obius function over broad classes of equations, extending analogies with the Goldbach problem.

Findings

Established nontrivial bounds for bilinear sums involving the M"obius function.

Provided M"obius-function analogues of the ternary Goldbach problem.

Made partial progress on binary cases by restricting variable ranges.

Abstract

We establish nontrivial bounds for bilinear sums involving the M\"obius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as M\"obius-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.