Let $\mu$ be the Mobius function from $\mathbb{N}$ to $\{-1, 0, 1\}$. It is well known for the frequency of $-1, 1$, and $0$ for the sequence $(\mu(1), \mu(2), \mu(2), \dots, )$. 


For any $k\in \mathbb{N}$, it is natural to ask what is the frequency of any given block of $k$-digits in $\{-1, 0, 1\}^{k}$ . I do not know whether this is known in the literature. 


Any comments and remarks will be appreciated. 



frequence of block of digits in Mobius sequence
Terry Tao has a blog post on this [here](https://terrytao.wordpress.com/2015/09/06/sign-patterns-of-the-mobius-and-liouville-functions/)


The Chowla conjecture asserts that all $k-$ blocks are equidistributed.


Matomaki, Radziwill and Tao (MRT) have shown that each of the sign patterns in 
$\{-1,0,+1\}^k$ 
is attained by the Möbius function for a set ${n}$ of positive lower 
density for $k\leq 4.$ 
What this means is that for all $(a\_1,\ldots, a\_k)\in \{-1,0,+1\}^k,$ if $k\leq 4,$ 
 there is a subset $I$ of $\mathbb{N}$ such that there is some $\varepsilon>0,$ 
with $$\lim\_{N\rightarrow \infty} \frac{I \cap \{1,2,\ldots,N\}}{N}>\varepsilon,$$
and
$$
\{(\mu(n),\mu(n+k-1)=(a\_1,\ldots,a\_k):\forall n \in I\}, 
$$


**Edit:** Edited to update the value of $k$ for which the MRT result holds.


