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VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

Part of: Ergodic theory Multiplicative number theory

Published online by Cambridge University Press:  26 September 2019

TERENCE TAO  and
JONI TERÄVÄINEN
TERENCE TAO
Affiliation:
Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles CA 90095, USA; tao@math.ucla.edu
JONI TERÄVÄINEN
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, UK; joni.teravainen@maths.ox.ac.uk

Abstract

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We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern $n+1\in A$, $n+2\in A$, $n+3\in A$ is positive as long as $A$ has density greater than $\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $\frac{1}{3}$, below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$, $P^{+}(n+1),P^{+}(n+2)$ of three consecutive integers. Second, we show that the tuple $(\unicode[STIX]{x1D714}(n+1),\unicode[STIX]{x1D714}(n+2),\unicode[STIX]{x1D714}(n+3))~(\text{mod}~3)$ takes all the $27$ possible patterns in $(\mathbb{Z}/3\mathbb{Z})^{3}$ with positive lower density, with $\unicode[STIX]{x1D714}(n)$ being the number of distinct prime divisors. We also prove a theorem concerning longer patterns $n+i\in A_{i}$, $i=1,\ldots ,k$ in approximately multiplicative sets $A_{i}$ having large enough densities, generalizing some results of Hildebrand on his ‘stable sets conjecture’. Finally, we consider the sign patterns of the Liouville function $\unicode[STIX]{x1D706}$ and show that there are at least $24$ patterns of length $5$ that occur with positive upper density. In all the proofs, we make extensive use of recent ideas concerning correlations of multiplicative functions.

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 37A45: Relations with number theory and harmonic analysis
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e33
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) 2019

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