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In Terry Tao's notes on the Chowla conjecture, the said conjecture states that for any fixed integer $m>0$ and nonzero $(a_1,\ldots,a_m)\in\{0,1\}^m$, $$ \lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leq x}\mu(n+1)^{a_1}\cdots\mu(n+m)^{a_m}=0, $$ where $\mu$ denotes the Möbius function. I would like information about the rate of this conjectured convergence. Are there any widely believed conjectures along these lines?

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up vote 11 down vote accepted

I imagine the "correct" conjecture is the Möbius $s$-tuples conjecture in the form that if $s \in \mathbb{N}$, $\alpha_1, \ldots, \alpha_s \in \mathbb{N}$ with at least one $\alpha_i$ odd, and $d_1, \ldots, d_s \in \mathbb{Z}$ distinct, then for all $\varepsilon > 0$ we have that $$\sum_{n \leq x}{\mu(n + d_1)^{\alpha_1} \mu(n + d_2)^{\alpha_2} \cdots \mu(n + d_s)^{\alpha_s}} \ll_{\varepsilon} x^{1/2 + \varepsilon}$$ uniformly for all $|d_i| \leq x$. However, I don't imagine there is any good evidence for this other than the fact that the Riemann hypothesis implies the case $s = 1$. I don't think this problem has really been studied enough for there to be a widely-believed generalisation of Chowla's conjecture.

A form of this conjecture is assumed by Ng in this paper on the distribution of the Möbius function in short intervals; he assumes that this conjecture holds with "for all $\varepsilon > 0$" replaced by "for some $0 < \beta_0 < 1/2$".

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Note Carmon and Rudnick's recent paper on a function field analogue: The autocorrelation of the Mobius function and Chowla's conjecture for the rational function field. The bound they get is quite far from the analogue of $x^{1/2+\epsilon}$, and by analysing their proof, you can see it is actually wrong. But it is only an specific analogue ($\lim q\rightarrow\infty$, "$x$" a fixed degree). – Dror Speiser Jun 17 '13 at 1:59

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