On the Hardy–Littlewood–Chowla conjecture on average

Abstract

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 $k,\ell \ge 1$ and distinct integers $h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $ , we have: $$ \begin{align*}\sum_{n\leq X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)\end{align*} $$ for all except $o(H)$ values of $h_1\leq H$ , so long as $H\geq (\log X)^{\ell +\varepsilon }$ . This improves on the range $H\ge (\log X)^{\psi (X)}$ , $\psi (X)\to \infty $ , obtained in previous work of the first author. Our results also generalise from the Möbius function $\mu $ to arbitrary (non-pretentious) multiplicative functions.