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

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 $n\mapsto \chi \left(n\right)n^{it}$, 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.