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 $k,\ell \ge 1$ and distinct integers $h_2,\dots ,h_k,a_1,\dots ,a_{\ell }$, we have $$\sum _{n\le X}\mu (n+h_1)\cdots \mu (n+h_k)\Lambda (n+a_1)\cdots \Lambda (n+a_{\ell })=o(X)$$ for all except $o(H)$ values of $h_1\le H$, so long as $H\ge (\log X{)}^{\ell +\epsilon }$. 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 generalize from the Möbius function $\mu$ to arbitrary (non-pretentious) multiplicative functions.

arXiv comment17 pages

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Cited by in corpus (1)

• The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero