I remember reading about a conjecture one night a while ago, but I can’t seem to find anything about it anymore. I have forgotten it’s name, but I believe the conjecture went as follows:
Suppose $\lambda(n)$ is the liouville function, i.e. $\lambda(n)=(-1)^{\Omega(n)}$, where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. Then the correlation of $\lambda(n)$ and $\lambda(n+1)$ is zero.
I remember getting the impression that it was a pretty important conjecture in number theory (or was a notable consequence of an important conjecture) as it intuitively means “prime numbers don’t care about your additive structure.” However, despite remembering it being important, I can’t find any mention of it anymore.
If anyone knows the name of this conjecture, please let me know.
Chowla's conjecture states that the sum $$ \sum_{n\leq x} \lambda(n) \lambda(n+k_1) \cdots \lambda(n+k_s)= o(x). $$ for all $s\geq 1.$
Here is the corrected link to Terry Tao's blogpost.