# Expansion, Divisibility and Parity

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/logx)∑n≤xlambda(n)λ(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=loglogN+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/(logx)c)

instead of O(1/sqrtloglogx) in the above, as well as other improvements in consequences across the board.