I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\lambda(n+2) =o(x)$, where $\lambda$ denotes the Liouville function. Does anyone have a reference/proof of this result ? Also, what would be de Polignac's conjecture in terms of $\lambda$ ?
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$\begingroup$ This is not known to be true. Tao claims this is a "variant" of the twin prime conjecture on the blog post you reference. It's possible that if one had a sufficiently strong quantitative version of the claim one could deduce the twin prime conjecture, but this certainly isn't known from a $o(x)$ bound. $\endgroup$– Mark LewkoCommented Jul 16, 2020 at 7:37
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$\begingroup$ @MarkLewko, okay, thanks. So, do you have any reference/proof that ''a sufficiently strong quantitative verson of the claim entails the twin prime conj'' ? Also, how strong should be the quantitative version ? $\endgroup$– Q_pCommented Jul 16, 2020 at 7:42
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2$\begingroup$ The variant is replacing $\lambda(n)$ with $\Lambda(n)$, where $\Lambda(n)$ denotes the von Mangoldt function, and extracting a main term. $\endgroup$– Peter HumphriesCommented Jul 16, 2020 at 12:20
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$\begingroup$ You can find some details here terrytao.wordpress.com/2011/11/21/… $\endgroup$– KrishnarjunCommented Jul 15, 2022 at 5:17
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