One of my friend (who is working in mathematics) was asking the following question. Let us take Liouville [λ(n) function](http://en.wikipedia.org/wiki/Liouville_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 ?



Fluctuations of Liouville function
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](https://www.renyi.hu/conferences/ll70/program/).


