Let $\lambda$ denote the Liouville function. 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 $X \to \infty$. This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$. 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 $$\sum_{ h_1,\dots,h_k \leq H} \left|\sum_{ 1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k)\right| = o(H^kX)$$ as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed. Related to this, we give the exponential sum estimate $$ \int_0^X \left|\sum_{ x \leq n \leq x+H} \lambda(n) e(\alpha n)\right| dx = o( HX )$$ as $X \to \infty$ uniformly for all $\alpha \in \mathbb{ R}$, with $H$ as before. 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.
Categories: math.NT