An averaged form of Chowla's conjecture

Abstract

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

$$\sum _{1\le n\le X}\lambda \left(n+h_1\right)\cdots \lambda \left(n+h_k\right)=o\left(X\right)$$

as $X\to \infty$. This conjecture remains unproven for any $h_1,\dots ,h_k$ with $k\ge 2$. Using the recent results of Matomäki and Radziwiłł on mean values of multiplicative functions in short intervals, combined with an argument of Kátai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely

$$\sum _{h_1,\dots ,h_k\le H}\left|\sum _{1\le n\le X}\lambda \left(n+h_1\right)\cdots \lambda \left(n+h_k\right)\right|=o\left(H^{k}X\right)$$

as $X\to \infty$, whenever $H=H\left(X\right)\le X$ goes to infinity as $X\to \infty$ and $k$ is fixed. Related to this, we give the exponential sum estimate

$${\int }_0^X\left|\sum _{x\le n\le x+H}\lambda \left(n\right)e\left(\alpha n\right)\right|dx=o\left(H“X\right)$$

as $X\to \infty$ uniformly for all $\alpha \in ℝ$, with $H$ as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $loglogH∕logH$) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.