#### 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
##### Abstract

We study the asymptotic behaviour of higher order correlations

${\mathbb{E}}_{n\le X∕d}{g}_{1}\left(n+a{h}_{1}\right)\cdots {g}_{k}\left(n+a{h}_{k}\right)$

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 ${g}_{1}\cdots {g}_{k}$ does not (weakly) pretend to be a twisted Dirichlet character $n↦\chi \left(n\right){n}^{it}$, and behave asymptotically like a multiple of ${d}^{-it}\chi \left(a\right)$ 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 ${\mathbb{E}}_{n\le X}\lambda \left(n+{h}_{1}\right)\cdots \lambda \left(n+{h}_{k}\right)=o\left(1\right)$ 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
Primary: 11N37
Secondary: 37A45