Asymptotics for multilinear averages of multiplicative functions

Nikos Frantzikinakis (1) , Bernard Host (2)

1 Department of Mathematics, University of Crete
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Mots clés: Multiplicative functions; multilinear averages; correlations; Hal\'asz; Wirsing.

Domaines: Systèmes dynamiques [math.DS]

Nikos Frantzikinakis, Bernard Host. Asymptotics for multilinear averages of multiplicative functions. Mathematical Proceedings of the Cambridge Philosophical Society, 2016, 161 (1), pp.87-101. ⟨10.1017/S0305004116000116⟩. ⟨hal-01267079⟩