On the Hardy-Littlewood-Chowla conjecture on average

Abstract

There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any $k,\ell\ge1$ and distinct integers $h_2,\ldots,h_k,a_1,\ldots,a_\ell$, we have $$\sum_{n\leq X}μ(n+h_1)\cdots μ(n+h_k)Λ(n+a_1)\cdotsΛ(n+a_{\ell})=o(X)$$ for all except $o(H)$ values of $h_1\leq H$, so long as $H\geq (\log X)^{\ell+ε}$. This improves on the range $H\ge (\log X)^{ψ(X)}$, $ψ(X)\to\infty$, obtained in previous work of the first author. Our results also generalize from the Möbius function $μ$ to arbitrary (non-pretentious) multiplicative functions.