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Asymptotics for multilinear averages of multiplicative functions

Published online by Cambridge University Press:  04 March 2016

NIKOS FRANTZIKINAKIS  and
BERNARD HOST
NIKOS FRANTZIKINAKIS
Affiliation:
University of Crete, Department of Mathematics, Voutes University Campus, Heraklion 71003, Greece. e-mail: [email protected]
BERNARD HOST
Affiliation:
Université Paris-Est Marne-la-Vallée, Laboratoire d'analyse et de Mathématiques Appliquées, UMR CNRS 8050, 5 Bd Descartes, 77454 Marne la Vallée Cedex, France. e-mail: [email protected]
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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 generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 87 - 101
Copyright
Copyright © Cambridge Philosophical Society 2016 

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