We will discuss a graph that encodes the divisibility properties of integers by primes.
This graph is shown to have a strong
local expander property almost everywhere. We then obtain
several consequences in number theory, beyond the traditional
parity barrier. For instance: for Λ the Liouville function (that is, the completely
multiplicative function with Λ(p) = -1 for
every prime p),
(1/log x) Σn ≤ x (Λ(n) Λ(n + 1)) / n
= O(1/(log log x)1/2), which is stronger than a well-known result by Tao.
We also manage to prove, for example, that Λ(n+1) averages to 0
at almost all scales when n is restricted to have a specific
number of prime divisors Ω(n) = k, for any "popular" value of k
(that is, k = log log N + O((log log N)1/2) for n ≤ N).