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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.

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Terry Tao has a blog post on this here

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.

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    $\begingroup$ The Chowla conjecture asserts uniform equidistribution for sign patterns of the Liouville function, but for the Mobius function the distribution is a bit more complicated due to local factors (e.g., Mobius vanishing at every multiple of 4). The result of MRT that you mention has now been extended to $k \leq 4$ (with the correct logarithmic density): see Corollary 1.10 of arxiv.org/abs/1708.02610 . $\endgroup$
    – Terry Tao
    Commented Jan 9, 2020 at 16:53

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