[In Terry Tao's notes on the Chowla conjecture](http://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-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?



Is there a stronger (but widely believed) version of the Chowla conjecture?
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](http://www.cs.uleth.ca/~nathanng/RESEARCH/mobiusshort.pdf) 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 $0 < \beta\_0 < 1/2$".


