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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 $A(a_1,...,a_n)=\{n|(\mu(n+1),...,\mu(n+k))=(a_1,...,a_n)\}$.

Question 1: 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 $p_1,p_2,...,p_k$, 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.

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 $\Omega_k$?

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.

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  • $\begingroup$ The image will definitely not be uniformly distributed in $\{0,1,-1\}^k$, because that would imply (for example) that $\frac23$ of the integers are squarefree, whereas we know that the correct proportion is $\frac6{\pi^2}$. $\endgroup$
    – Greg Martin
    Commented Nov 24, 2017 at 8:42
  • $\begingroup$ @GregMartin, It seems you are right... I know the entropy of Mobius flow is $\frac{\pi^2}{6}$ but unfortunately I do not know how to deduce from this result to that the correct proportion is $\frac{6}{\pi^2}$ and I do not know why the uniformly distribute will lead to the destiny of square free integers will be $\frac{2}{3}$ either, I was wondering a more precise explanation or do you have some reference? Appreciate! $\endgroup$
    – Hu xiyu
    Commented Nov 24, 2017 at 8:48
  • $\begingroup$ For some tuples, your set $A$ will be empty: for example, $A(1,1,1,1)=\varnothing$, since we can't have four consecutive squarefree numbers. Regarding the $\frac{2}{3}$, note that for $k=1$, if we had uniform distribution in $\{-1,0,1\}$, then we'd expect $\pm 1$ to appear with probability $\frac{2}{3}$ (which, as Greg mentions, is not the case). $\endgroup$
    – Wojowu
    Commented Nov 24, 2017 at 8:55
  • $\begingroup$ @Wojowu, thanks for your explain! I totally understand Greg's comments. Now it seems we need to use some k-admissible smaller subspace $\Omega_k$ of $\{0,1,-1\}^k$ to replace $\{0,1,-1\}^k$ and if we are In the best case, I still expect $f^{\mu}_k$ is uniformly distribute on $\Omega_k$. $\endgroup$
    – Hu xiyu
    Commented Nov 24, 2017 at 9:01
  • $\begingroup$ 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 $p_1,p_2,...,p_k$. $\endgroup$
    – Hu xiyu
    Commented Nov 24, 2017 at 9:09

1 Answer 1

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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.

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  • $\begingroup$ The $k=4$ case of Q1 is now known for the Liouville function, by combining recent results of myself and Joni Teravainen with an argument discovered independently by Sawin and by Matomaki: see Corollary 7.2 of arxiv.org/abs/1708.02610 . The $k=2$ case of Q2 does remain open, but if one uses logarithmic density instead of natural density then the $k=2,3$ cases are now known for both Liouville and Mobius, and the $k=4$ case known for Mobius; see Corollary 1.8, ibid. (This also gives the $k=4$ case of Q1 for Mobius.) $k>4$ remains out of reach by current methods. $\endgroup$
    – Terry Tao
    Commented Nov 24, 2017 at 22:05
  • $\begingroup$ @Terry Tao: Ah, thank you. The last time I heard about tuples was in the summer. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented Nov 25, 2017 at 7:49

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