Geodesic
On the Hardy–Littlewood–Chowla conjecture on average
Forum of Mathematics, Sigma, Tome 10 (2022)

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
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     author = {Jared Duker Lichtman and Joni Ter\"av\"ainen},
     title = {On the {Hardy{\textendash}Littlewood{\textendash}Chowla} conjecture on average},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2022},
     doi = {10.1017/fms.2022.54},
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     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.54/}
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Jared Duker Lichtman; Joni Teräväinen. On the Hardy–Littlewood–Chowla conjecture on average. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.54

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