On the Hardy-Littlewood-Chowla conjecture on average

TL;DR

This paper proves that certain correlations involving the Möbius and von Mangoldt functions are negligible on average, extending previous results and generalizing to a broader class of multiplicative functions.

Contribution

It establishes new average bounds for correlations of multiplicative functions, improving previous ranges and generalizing results beyond the Möbius function.

Findings

Correlation sums are o(X) for most shifts under specified conditions

Results extend to non-pretentious multiplicative functions

Improves the range of H for which the conjecture holds

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.