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15 August 2019 The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
Terence Tao, Joni Teräväinen
Duke Math. J. 168(11): 1977-2027 (15 August 2019). DOI: 10.1215/00127094-2019-0002

Abstract

Let g0,,gk:ND be 1-bounded multiplicative functions, and let h0,,hkZ be shifts. We consider correlation sequences f:NZ of the form f(a):=m1logωmxm/ωmnxmg0(n+ah0)gk(n+ahk)n, where 1ωmxm are numbers going to infinity as m and is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences f are the uniform limit of periodic sequences fi. Furthermore, if the multiplicative function g0gk “weakly pretends” to be a Dirichlet character χ, the periodic functions fi can be chosen to be χ-isotypic in the sense that fi(ab)=fi(a)χ(b) whenever b is coprime to the periods of fi and χ, while if g0gk does not weakly pretend to be any Dirichlet character, then f must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.

Citation

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Terence Tao. Joni Teräväinen. "The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures." Duke Math. J. 168 (11) 1977 - 2027, 15 August 2019. https://doi.org/10.1215/00127094-2019-0002

Information

Received: 8 August 2017; Revised: 14 December 2018; Published: 15 August 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07114912
MathSciNet: MR3992031
Digital Object Identifier: 10.1215/00127094-2019-0002

Subjects:
Primary: 11N37
Secondary: 37A45

Keywords: Chowla conjecture , Elliott conjecture , multiple recurrence , multiplicative functions

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 11 • 15 August 2019
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