Equivalence of the logarithmically averaged Chowla and Sarnak conjectures

Abstract

Let $\lambda$ denote the Liouville function. The Chowla conjecture asserts that $$ \sum_{n \leq X} \lambda(a_1 n + b_1) \lambda(a_2 n+b_2) \dots \lambda(a_k n + b_k) = o_{X \to \infty}(X) $$ for any fixed natural numbers $a_1,a_2,\dots,a_k$ and non-negative integer $b_1,b_2,\dots,b_k$ with $a_ib_j-a_jb_i \neq 0$ for all $1 \leq i < j \leq k$, and any $X \geq 1$. This conjecture is open for $k \geq 2$. As is well known, this conjecture implies the conjecture of Sarnak that $$ \sum_{n \leq X} \lambda(n) f(n) = o_{X \to \infty}(X)$$ whenever $f : {\bf N} \to {\bf C}$ is a fixed deterministic sequence and $X \geq 1$. In this paper, we consider the weaker logarithmically averaged versions of these conjectures, namely that $$ \sum_{X/\omega \leq n \leq X} \frac{\lambda(a_1 n + b_1) \lambda(a_2 n+b_2) \dots \lambda(a_k n + b_k)}{n} = o_{\omega \to \infty}(\log \omega) $$ and $$ \sum_{X/\omega \leq n \leq X} \frac{\lambda(n) f(n)}{n} = o_{\omega \to \infty}(\log \omega)$$ under the same hypotheses on $a_1,\dots,a_k,b_1,\dots,b_k$ and $f$, and for any $2 \leq \omega \leq X$. Our main result is that these latter two conjectures are logically equivalent to each other, as well as to the "local Gowers uniformity" of the Liouville function. The main tools used here are the entropy decrement argument of the author used recently to establish the $k=2$ case of the logarithmically averaged Chowla conjecture, as well as the inverse conjecture for the Gowers norms, obtained by Green, Ziegler, and the author.