The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Abstract

Let $g_0,\dots ,g_k:\mathbb{N}\to \mathbb{D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots ,h_k\in \mathbb{Z}$ be shifts. We consider correlation sequences $f:\mathbb{N}\to \mathbb{Z}$ of the form $$f(a):=\underset{m\to \infty }{}\frac{1}{log{\omega }_m}{\sum }_{x_m/{\omega }_m\le n\le x_m}\frac{g_0(n+a{h}_0)\cdots g_k(n+a{h}_k)}{n},$$ where $1\le {\omega }_m\le x_m$ are numbers going to infinity as $m\to \infty$ and $$ is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences $f$ are the uniform limit of periodic sequences $f_i$. Furthermore, if the multiplicative function $g_0\cdots g_k$ “weakly pretends” to be a Dirichlet character $\chi$, the periodic functions $f_i$ can be chosen to be $\chi$-isotypic in the sense that $f_i\left(ab\right)=f_i\left(a\right)\chi \left(b\right)$ whenever $b$ is coprime to the periods of $f_i$ and $\chi$, while if $g_0\cdots g_k$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.