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 $${\mathbb{E}}_{n\le X/d}{g}_1(n+a{h}_1)\cdots g_k(n+a{h}_k)$$ 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 $n\mapsto \chi (n)n^{it}$, and behave asymptotically like a multiple of $d^{-it}\chi (a)$ 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 ${\mathbb{E}}_{n\le X}\lambda (n+h_1)\cdots \lambda (n+h_k)=o(1)$ for $k$ odd or equal to $2$ for all scales $X$ outside of a set of zero logarithmic density.

48 pages, no figures. Referee comments incorporated

References in corpus (4)

• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• Odd order cases of the logarithmically averaged Chowla conjecture
• Nilsequences and Multiple Correlations along Subsequences
• On binary correlations of multiplicative functions

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• Expansion, divisibility and parity
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• On the local Fourier uniformity problem for small sets