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 ${\sum }_{1\le n\le X}\lambda (n+h_1)\cdots \lambda (n+h_k)=o(X)$ as $X\to \infty$. This conjecture remains unproven for any $h_1,\dots ,h_k$ with $k\ge 2$. 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 $$\sum _{h_1,\dots ,h_k\le H}\left|\sum _{1\le n\le X}\lambda (n+h_1)\cdots \lambda (n+h_k)\right|=o(H^{k}X)$$ as $X\to \infty$ whenever $H=H(X)\le X$ goes to infinity as $X\to \infty$, and $k$ is fixed. Related to this, we give the exponential sum estimate $${\int }_0^X\left|\sum _{x\le n\le x+H}\lambda (n)e(\alpha n)\right|dx=o(HX)$$ 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.

32 pages; proof of Proposition A.3 corrected

References in corpus (1)

• Asymptotics for multilinear averages of multiplicative functions

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