Asymptotics for multilinear averages of multiplicative functions

TL;DR

This paper extends Halász's classical result to multilinear averages of multiplicative functions, establishing asymptotics and convergence properties, and verifies a two-dimensional variant of Elliott's conjecture.

Contribution

It introduces a new asymptotic analysis for multilinear averages of multiplicative functions, confirming a two-dimensional version of Elliott's conjecture.

Findings

Derived asymptotics for multilinear averages

Proved convergence results for multilinear expressions

Verified a two-dimensional variant of Elliott's conjecture

Abstract

A celebrated result of Hal\'asz describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalizes a well known convergence result of Wirsing. The key ingredients are a recent structural result for bounded multiplicative functions proved by the authors and the mean value theorem of Hal\'asz.