One of my friend (who is working in mathematics) was asking the following question. Let us take Liouville λ(n) function.

et S={ λ(1), λ(2), λ(3), ..... } . Then every finite length (say l) subsequence of S occurs infinitely many times. In other words every finite block of $\pm$ signs occurs in this sequence infinitely often i.e. for any given numbers $\epsilon_i = \pm 1 ( 1 \le i \le l)$, there are infinitely many integers $n>= 1$, such that $f(n+i)=\epsilon_i (1 \le i \le l )$.

I can prove it for $l=1,2. l=1$ is the trivial case.

For $l=2$, the case for ${+1,-1}$ or ${-1,+1}$ follows trivially from $l=1$ case.

For ${+1,+1}$ or ${-1,-1}$ case: Let us take a odd positive number n. Let us also choose f(n)=+1. This is trivially accomplished by taking n as a product of even number of primes. if n+1 or n-1 has f(n)=+1 then we get a pair with {+1,+1}. Else, f(n+1)=f(n-1)= -1. Hence f(n+1/2)=f(n-1/2)=1 and this gives us a pair {+1,+1}. This proves the case for {+1,+1}. Similarly it may be proved for {-1,-1}.

What is known about $k \ge 3$. Is it something already proved ?


Hildebrand (On consecutive values of the Liouville function, Enseign. Math. (2) 32 (1986), 219–226) proved the conjecture for $l=3$, i.e. all 8 combinations $\pm 1,\pm 1,\pm 1$ occur infinitely often in the Liouville sequence. Christian Elsholtz proved very recently that all 16 combinations $\pm 1,\pm 1,\pm 1,\pm 1$ occur along arithmetic progressions infinitely often. I believe this is the state of the art.

Added. My response above is almost 7 years old. For the current state of the art, I recommend watching Tao's recent lecture at the Building Bridges II. conference.


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