We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.)

For instance: for lambda the Liouville function (that is, the completely multiplicative function with lambda(p) = -1 for every prime), (1/\log x) sum_{n≤x} lambda(n) lambda(n+1)/n = O(1/sqrt(log log x)), which is stronger than well-known results by Tao and Tao-Teravainen.

We also manage to prove, for example, that lambda(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Omega(n)=k, for any "popular" value of k (that is, k = log log N + O(sqrt(log log N)) for n≤N).

We shall also discuss a recent generalization by C. Pilatte, who has succeeded in proving that a graph with edges that are rough integers, rather than primes, also has a strong local expander property almost everywhere, following the same strategy. As a result, he has obtained a bound with O(1/(log x)^c) instead of O(1/sqrt(log log x)) in the above, as well as other improvements in consequences across the board.