This question was already asked earlier at MSE.

Let $M$ denote the sequence of values of $\mu(n)$, the Mobius function which begins $M = (1,-1,-1,0,-1,1,-1,0,\dotsc)$. The questions below concern patterns that may or may not arise within $M$.

Questions: (1) It's easy to find palindromes in $M$ if we are free to choose the starting point. For example, the values from $n=102$ through $n=110$ yield the length $9$ palindrome $(-1,-1,0,-1,1,-1,0,-1,-1)$. However, the existence of a palindrome of any length greater than $1$ beginning at $n=1$ seems far fetched. Can it be proved that none exists? In this regard, is it also true that for large $n$, the sum $\sum_{k=1}^{k=n} \mu(k) \mu(n+1-k)$ is small when compared to $\sum_{k=1}^{k=n} \mu(k)^2$?

(2) Although $M$ is perhaps best known for its random aspects, it does at least have some predictable features. For example, it vanishes on certain arithmetic progressions including $\mu(4n)= 0$. This shows that the string $1111$ does not appear anywhere in $M$. [All strings and substrings are consecutive]. Let's abbreviate f.s. = forbidden (sub)sequence to be any such finite string that does not appear in $M$. Of course, any extension of an f.s. such as $01111$ or $1111{-}1$ is likewise forbidden so define a minimal f.s. to be one which contains no shorter f.s. inside it. For instance, each length $4$ string drawn from $+1,-1$ is a minimal f.s. since (i) it's forbidden and (ii) one checks that all such length $3$ strings do in fact occur.

What can one say about the collection of all minimal forbidden sequences? In particular, is it possible that there are only finitely many of them?

Thanks!

This question was already asked earlier at MSE.

Let $M$ denote the sequence of values of $\mu(n)$, the Möbius function which begins $M = (1,-1,-1,0,-1,1,-1,0,\dotsc)$. The questions below concern patterns that may or may not arise within $M$.

Questions: (1) It's easy to find palindromes in $M$ if we are free to choose the starting point. For example, the values from $n=102$ through $n=110$ yield the length $9$ palindrome $(-1,-1,0,-1,1,-1,0,-1,-1)$. However, the existence of a palindrome of any length greater than $1$ beginning at $n=1$ seems far fetched. Can it be proved that none exists? In this regard, is it also true that for large $n$, the sum $\sum_{k=1}^{k=n} \mu(k) \mu(n+1-k)$ is small when compared to $\sum_{k=1}^{k=n} \mu(k)^2$?

(2) Although $M$ is perhaps best known for its random aspects, it does at least have some predictable features. For example, it vanishes on certain arithmetic progressions including $\mu(4n)= 0$. This shows that the string $1111$ does not appear anywhere in $M$. [All strings and substrings are consecutive]. Let's abbreviate f.s. = forbidden (sub)sequence to be any such finite string that does not appear in $M$. Of course, any extension of an f.s. such as $01111$ or $1111{-}1$ is likewise forbidden so define a minimal f.s. to be one which contains no shorter f.s. inside it. For instance, each length $4$ string drawn from $+1,-1$ is a minimal f.s. since (i) it's forbidden and (ii) one checks that all such length $3$ strings do in fact occur.

What can one say about the collection of all minimal forbidden sequences? In particular, is it possible that there are only finitely many of them?

Thanks!

This question was already asked earlier at MSE.

Let M denote the sequence of values of $\,$ $\mu(n)$ $\,$ the Mobius function which begins $\,$ M = (1,-1,-1,0,-1,1,-1,0,...) . The questions below concern patterns that may or may not arise within M.

Questions: (1) It's easy to find palindromes in M if we are free to choose the starting point. For example, the values from n=102 through n=110 yield the length 9 palindrome (-1,-1,0,-1,1,-1,0,-1,-1). However, the existence of a palindrome of any length beginning at n=1 seems far fetched. Can it be proved that none exists? $\;$ In this regard, is it also true that for large n, the sum $\,$ $\sum_{k=1}^{k=n}$$\mu(k)$$\mu(n+1-k)$ $\,$ is small when compared to $\,$ $\sum_{k=1}^{k=n}$$\mu(k)^2$ $\,$ ?

(2) Although M is perhaps best known for its random aspects, it does at least have some predictable features. For example, it vanishes on certain arithmetic progressions including $\mu(4n)$= 0 . This shows that the string 1111 does not appear anywhere in M . [All strings and substrings are consecutive]. Let's abbreviate f.s. = forbidden (sub)sequence to be any such finite string that does not appear in M . Of course, any extension of an f.s. such as 01111 or 1111-1 is likewise forbidden so define a minimal f.s. to be one which contains no shorter f.s. inside it. For instance, each length 4 string drawn from +1,-1 is a minimal f.s. since (i) it's forbidden and (ii) one checks that all such length 3 strings do in fact occur.

What can one say about the collection of all minimal forbidden sequences? In particular, is it possible that there are only finitely many of them?

Thanks

This question was already asked earlier at MSE.

Let $M$ denote the sequence of values of $\mu(n)$, the Mobius function which begins $M = (1,-1,-1,0,-1,1,-1,0,\dotsc)$. The questions below concern patterns that may or may not arise within $M$.

Questions: (1) It's easy to find palindromes in $M$ if we are free to choose the starting point. For example, the values from $n=102$ through $n=110$ yield the length $9$ palindrome $(-1,-1,0,-1,1,-1,0,-1,-1)$. However, the existence of a palindrome of any length greater than $1$ beginning at $n=1$ seems far fetched. Can it be proved that none exists? In this regard, is it also true that for large $n$, the sum $\sum_{k=1}^{k=n} \mu(k) \mu(n+1-k)$ is small when compared to $\sum_{k=1}^{k=n} \mu(k)^2$?

(2) Although $M$ is perhaps best known for its random aspects, it does at least have some predictable features. For example, it vanishes on certain arithmetic progressions including $\mu(4n)= 0$. This shows that the string $1111$ does not appear anywhere in $M$. [All strings and substrings are consecutive]. Let's abbreviate f.s. = forbidden (sub)sequence to be any such finite string that does not appear in $M$. Of course, any extension of an f.s. such as $01111$ or $1111{-}1$ is likewise forbidden so define a minimal f.s. to be one which contains no shorter f.s. inside it. For instance, each length $4$ string drawn from $+1,-1$ is a minimal f.s. since (i) it's forbidden and (ii) one checks that all such length $3$ strings do in fact occur.

What can one say about the collection of all minimal forbidden sequences? In particular, is it possible that there are only finitely many of them?

Thanks!