An averaged form of Chowla's conjecture

Abstract

<p> Let $\lambda$ denote the Liouville function.&nbsp; A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k) = o(X)$ as $X \to \infty$.&nbsp; This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$.&nbsp; In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely<br /> $$ \sum_{h_1,\dots,h_k \leq H} \left|\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k)\right| = o(H^kX)$$<br /> as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed.&nbsp; Related to this, we give the exponential sum estimate<br /> $$ \int_0^X \left|\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)\right| dx = o( HX )$$<br /> as $X \to \infty$ uniformly for all $\alpha \in \R$, with $H$ as before.&nbsp; Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $\frac{\log\log H}{\log H}$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.<br /> &nbsp;</p>