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

TL;DR

This paper analyzes the asymptotic behavior of correlations of multiplicative functions at various scales, providing structural results that support cases of the Chowla and Elliott conjectures, especially for odd and small even cases.

Contribution

It extends previous work by characterizing the asymptotic behavior of higher order correlations of multiplicative functions for almost all scales, linking them to twisted Dirichlet characters.

Findings

Correlations vanish for most scales unless functions mimic twisted Dirichlet characters.

Establishes the $k$-point Chowla conjecture for odd $k$ and for $k=2$ at almost all scales.

Provides structural understanding of multiplicative functions' correlations across scales.

Abstract

We study the asymptotic behaviour of higher order correlations $$ \mathbb{E}_{n \leq X/d} g_1(n+ah_1) \cdots g_k(n+ah_k)$$ 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 \leq 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.