I am considering the following question related to the randomness of Mobius function $\mu(n)$:


$\mu(n)$ is defined as :


$\mu(1)=1$,


$\mu(p\_1...p\_t)=(-1)^t$, $\forall t\in N^\*$ $p\_1,...,p\_t$ are different primes,


$\mu(n)=0$ if $\exists p$ is a prime, $p^2|n$.


My question is, given $k\in N^\*, (a\_1,...,a\_n)\in \{0,1,-1\}^{n}$.
 Define set $A(a\_1,...,a\_n)=\{n|(\mu(n+1),...,\mu(n+k))=(a\_1,...,a\_n)\}$.



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> **Question 1:** Is there a nontrivial estimate for $\limsup\_{N\to\infty}\frac{|\{1,2,...,N\}\cap A(a\_1,...,a\_k)|}{N}$?
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This could be view as a weak-$L^{\infty}$ estimate and can be explain to be that the destiny of the image of $f$ could not concentrate at some singularity point in $\{0,1,-1\}^k$ which is for map $f^{\mu}\_k:\mathbb Z\to \{0,1,-1\}^{k}$ induced by $n\to (n+1,...,n+k)\to (\mu(n+1),...,\mu(n+k))$. 


Thanks for Gerd's comments, My original goal is to prove the image is uniformly distribute in $\{0,1,-1\}^k$ but it is failed to be true. Inspirit by Wojowu's comments, we say $(a\_1,...,a\_n)\in \{0,1,-1\}^k$ is k-admissible iff there do not exists local obstacle to make $A(a\_1,...,a\_k)=\emptyset$, for a local obstacle I mean if it is still an obstacle after $mod p\_1p\_2...p\_k$ for some $p\_1,p\_2,...,p\_k$, for example $A(1,1,1,1)$ have local obstacle due to there could not be 4 continuous square-free numbers (mod 2). Define $\Omega\_k\subset \{0,1,-1\}^k$ is the set combine with all k-admissible k-tuples, I expect $f\_{k}^{\mu}$ is uniformly distribute in $\Omega\_k$ if we are in the best case. 



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> **Question 2:** Define $\Omega\_k\subset \{0,1,-1\}^k$ is the set combine with all k-admissible k-tuples, Is $f\_{k}^{\mu}$ is uniformly distribute in $\Omega\_k$? 
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Anyway, after a check of my sight, I realize the thing I really need it that the "density" of $f^{\mu}\_{k}$ will not concentrate at some small area in $\{0,1,-1\}^k$.


I will appreciate to any meaningful comments and advice.



Is there some estimate numbers of the tuples come from Mobius function?
Studying the distribution of patterns of the Moebius function falls into an easy part, which deals with the distribution of zeroes, and a difficult part, which deals with the distribution of signs. Therefore it is more natural to separate these problems and ask for patterns of Liouville's $\lambda$-function. Here our knowledge is abysmal. Hildebrand showed that all the 8 possibilities for the values at three consecutive integers occur infinitely often. Matomaki, Radziwill and Tao showed that these patterns occur with positive lower density. Elsholtz and Buttkewitz proved that all sign patterns of length 4 occur on arithmetic progressions. Pintz has shown that the equation $\omega(n)=a$, $\omega(n+1)=b$ has infinitely many solutions, provided that $a$ and $b$ are not too small, and his method might give a lower bound for the number of tuples attained, but even the question whether all 16 sign patterns are attained for four consecutive numbers seems to be very difficult. 


In short, the answer to question 1 is yes for $k\leq 3$, and rather hopeless for all larger values. Question 2 might be solved for $k=2$, but I could not find a reference for it. For $k=3$ it is open. 


