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arXiv · 2111.08912

On the Hardy-Littlewood-Chowla conjecture on average

Jared Duker Lichtman·Joni Teräväinen
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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 $$\sum_{n\leq X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)$$ for all except $o(H)$ values of $h_1\leq H$, so long as $H\geq (\log X)^{\ell+\epsilon}$. 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 M\"obius function $\mu$ to arbitrary (non-pretentious) multiplicative functions.

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Jared Duker Lichtman, Joni Teräväinen. 2021-11-17. On the Hardy-Littlewood-Chowla conjecture on average. https://doi.org/10.1017/fms.2022.54

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