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

as a function of the parameters $a$ and $d$ , where $g_{1}, \dots, g_{k}$ are bounded multiplicative functions, $h_{1}, \dots, h_{k}$ are integer shifts, and $X$ is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all $X$ if $g_{1} \dots g_{k}$ does not (weakly) pretend to be a twisted Dirichlet character $n \mapsto χ (n) n^{it}$ , and behave asymptotically like a multiple of $d^{- it} χ (a)$ 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 $E_{n \leq X} λ (n + h_{1}) \dots λ (n + h_{k}) = o (1)$ for $k$ odd or equal to $2$ for all scales $X$ outside of a set of zero logarithmic density.

Keywords

multiplicative functions, Chowla's conjecture, Elliott's conjecture

Mathematical Subject Classification 2010

Primary: 11N37

Secondary: 37A45

Milestones

Received: 8 September 2018

Revised: 24 June 2019

Accepted: 24 July 2019

Published: 7 December 2019

Authors

Terence Tao
Department of Mathematics University of California Los Angeles Los Angeles, CA United States

Joni Teräväinen
Mathematical Institute University of Oxford United Kingdom

Vol. 13, No. 9, 2019

Terence Tao and Joni Teräväinen

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors