Asymptotics for multilinear averages of multiplicative functions

Abstract

A celebrated result of Halász 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 Halász.

14 pages. Referee's comments incorporated. To appear in the Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (1)

• An averaged form of Chowla's conjecture

Cited by in corpus (9)

• An averaged form of Chowla's conjecture
• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• Linear correlations of multiplicative functions
• The logarithmically averaged Chowla and Elliott conjectures for two-point correlations
• Sign patterns of the Liouville and Möbius functions
• On Furstenberg systems of aperiodic multiplicative functions of Matomaki, Radziwill and Tao
• Effective Asymptotic Formulae for Multilinear Averages of Multiplicative Functions
• Partition regularity of Pythagorean pairs
• On averages of completely multiplicative functions over co-prime integer pairs