Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T03:56:07.737Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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

References

Ayoub, R., ‘On Rademacher’s extension of the Goldbach–Vinogradoff theorem’, Trans. Amer. Math. Soc. 74 (1953), 482–491.Google Scholar
de Bruijn, N. G., ‘On the number of positive integers ⩽x and free of prime factors > y ’, Nederl. Acad. Wetensch. Proc. Ser. A 54 (1951), 50–60.Google Scholar
Buttkewitz, Y. and Elsholtz, C., ‘Patterns and complexity of multiplicative functions’, J. Lond. Math. Soc. (2) 84(3) (2011), 578–594.Google Scholar
Chowla, S., The Riemann Hypothesis and Hilbert’s Tenth Problem (Gordon and Breach, New York, 1965).Google Scholar
Frantzinakis, N. and Host, B., ‘Asymptotics for multilinear averages of multiplicative functions’, Math. Proc. Cambridge Philos. Soc. 161(1) (2015), 87–101.CrossRefGoogle Scholar
Green, B. and Tao, T., ‘Linear equations in primes’, Ann. of Math. (2) 171(3) (2010), 1753–1850.Google Scholar
Green, B. and Tao, T., ‘The Möbius function is strongly orthogonal to nilsequences’, Ann. of Math. (2) 175(2) (2012), 541–566.Google Scholar
Green, B., Tao, T. and Ziegler, T., ‘An inverse theorem for the Gowers U s+1[N]-norm’, Ann. of Math. (2) 176(2) (2012), 1231–1372.Google Scholar
Halberstam, H. and Richert, H.-E., Sieve Methods (Academic Press, London, 1974).Google Scholar
Harman, G., Pintz, J. and Wolke, D., ‘A note on the Möbius and Liouville functions’, Studia Sci. Math. Hungar. 20(1–4) (1985), 295–299.Google Scholar
Hildebrand, A., ‘On consecutive values of the Liouville function’, Enseign. Math. (2) 32(3–4) (1986), 219–226.Google Scholar
Matomäki, K. and Radziwiłł, M., ‘Multiplicative functions in short intervals’, Ann. of Math. (2) 183 (2016), 1015–1065.Google Scholar
Matomäki, K., Radziwiłł, M. and Tao, T., ‘An averaged form of Chowla’s conjecture’, Algebra Number Theory 9(9) (2015), 2167–2196.Google Scholar
Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III (Princeton University Press, Princeton, NJ, 1993), With the assistance of Timothy S. Murphy.Google Scholar
Tao, T., ‘The logarithmically averaged Chowla and Elliott conjectures for two-point correlations’, Preprint, 2015, arXiv:1509.05422.Google Scholar