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)\}$ .

Is there a nontrivial estimate for $ \limsup_{N\to\infty}\frac{|\{1,2,…,N\}\cap A(a_1,…,a_k)|}{N}$ ?

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$ . 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.

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.