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SIGN PATTERNS OF THE LIOUVILLE AND MÖBIUS FUNCTIONS

Part of: Multiplicative number theory

Published online by Cambridge University Press:  21 June 2016

KAISA MATOMÄKI ,
MAKSYM RADZIWIŁŁ  and
TERENCE TAO
KAISA MATOMÄKI
Affiliation:
Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland; ksmato@utu.fi
MAKSYM RADZIWIŁŁ
Affiliation:
Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA; maksym.radziwill@gmail.com
TERENCE TAO
Affiliation:
Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095, USA; tao@math.ucla.edu

Abstract

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Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand’s result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $({\it\mu}(n),{\it\mu}(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints 11N37: Asymptotic results on arithmetic functions 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e14
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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