Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-08T08:29:16.386Z Has data issue: false hasContentIssue false
  • English
  • Français

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: frantzikinakis@gmail.com
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: bernard.host@u-pem.fr
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Balog, A., Granville, A. and Soundararajan, K.Multiplicative functions in arithmetic progressions. Ann. Math. Québec 37 (2013), 3–30.CrossRefGoogle Scholar
[2]Daboussi, H.Fonctions multiplicatives presque périodiques B. D'après un travail commun avec Hubert Delange. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Asterisque 24–25 (1975), 321–324.Google Scholar
[3]Daboussi, H. and Delange, H.Quelques proprietes des functions multiplicatives de module au plus egal 1. C. R. Acad. Sci. Paris Ser. A 278 (1974), 657–660.Google Scholar
[4]Delange, H.Sur les fonctions arithmétiques multiplicatives. Ann. Sci. École Norm. Sup. 78 (1961), 273–304.CrossRefGoogle Scholar
[5]Delange, H.Sur les fonctions arithmétiques multiplicatives de module au plus égal à un. C.R. Acad. Sci. Paris Ser. A 27 (1972), 781–784.Google Scholar
[6]Delange, H.Sur les fonctions arithmétiques multiplicatives de module ⩽ 1. Acta Arith. 42 (1983), 121–151.CrossRefGoogle Scholar
[7]Elliott, P.Probabilistic Number Theory I (Springer-Verlag, New York, Heidelberg, Berlin, 1979).CrossRefGoogle Scholar
[8]Elliott, P.Multiplicative functions |g|⩽ 1 and their convolutions: An overview. Séminaire de théorie des nombres, Paris 1987-88. Progr. Math. 81 (1990), 63–75.CrossRefGoogle Scholar
[9]Elliott, P.On the correlation of multiplicative and the sum of additive arithmetic functions. Mem. Amer. Math. Soc. 112 (1994), no. 538.Google Scholar
[10]Erdös, P.Some unsolved problems. Michigan Math. J. 4 (1957), 291–300.CrossRefGoogle Scholar
[11]Frantzikinakis, N. and Host, B. Higher order Fourier analysis of multiplicative functions and applications. To appear in J. Amer. Math. Soc. arXiv:1403.0945.Google Scholar
[12]Gowers, T.A new proof of Szemerédi's theorem. Geom. Funct. Anal. 11 (2001), 465–588.CrossRefGoogle Scholar
[13]Granville, A. and Soundararajan, K.Decay of mean-values of multiplicative functions. Can. J. Math. 55 (2003), 1191–1230.CrossRefGoogle Scholar
[14]Granville, A. and Soundararajan, K.Multiplicative Number Theory: The Pretentious Approach. Book manuscript in preparation.Google Scholar
[15]Green, B. and Tao, T.Linear equations in the primes. Ann. of Math. 171 (2010), 1753–1850.CrossRefGoogle Scholar
[16]Green, B. and Tao, T.The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. 175 (2012), no. 2, 465–540.Google Scholar
[17]Green, B. and Tao, T.The Möbius function is strongly orthogonal to nilsequences. Ann. of Math. 175 (2012), no. 2, 541–566.CrossRefGoogle Scholar
[18]Green, B., Tao, T. and Ziegler, T.An inverse theorem for the Gowers U s+1[N]-norm. Ann. of Math. 176 (2012), no. 2, 1231–1372.CrossRefGoogle Scholar
[19]Halász, G.Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hung. 19 (1968), 365–403.CrossRefGoogle Scholar
[20]Matomäki, K. and Radziwiłł, M. Multiplicative functions in short intervals. To appear in Ann. of Math. arXiv:1501.04585.Google Scholar
[21]Matomäki, K., Radziwiłł, M. and Tao, T.. An averaged form of Chowla's conjecture. Algebr. Number Theory 9 (2015), 2167–2196.CrossRefGoogle Scholar
[22]Tenenbaum, G.Introduction la théorie analytique et probabiliste des nombres. Cours Spécialisés, no. 1 (Société Mathématique de France, Paris, 1995).Google Scholar
[23]Wirsing, E.Das asymptotische Verhalten von Summen uber multiplikative Funktionen. Math. Ann. 143 (1961), 75–102.CrossRefGoogle Scholar
[24]Wirsing, E.Elementare Beweise des Primzahlsatzes mit Restglied. II. J. Reine Angew. Math. 214/215 (1964), 1–18.CrossRefGoogle Scholar
[25]Wirsing, E.Das asymptotische Verhalten von Summen uber multiplikative Funktionen, II. Acta Math. Acad. Sci. Hung. 18 (1967), 411–467.CrossRefGoogle Scholar