Vol. 13, No. 9, 2019

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The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

Terence Tao and Joni Teräväinen

Vol. 13 (2019), No. 9, 2103–2150

We study the asymptotic behaviour of higher order correlations

EnXdg1(n + ah1)gk(n + ahk)

as a function of the parameters a and d, where g1,,gk are bounded multiplicative functions, h1,,hk are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g1gk does not (weakly) pretend to be a twisted Dirichlet character nχ(n)nit, and behave asymptotically like a multiple of d itχ(a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t = 0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture EnXλ(n + h1)λ(n + hk) = o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density.

multiplicative functions, Chowla's conjecture, Elliott's conjecture
Mathematical Subject Classification 2010
Primary: 11N37
Secondary: 37A45
Received: 8 September 2018
Revised: 24 June 2019
Accepted: 24 July 2019
Published: 7 December 2019
Terence Tao
Department of Mathematics
University of California Los Angeles
Los Angeles, CA
United States
Joni Teräväinen
Mathematical Institute
University of Oxford
United Kingdom