(joint work with M. Radziwill) We will discuss a graph that encodes the divisibility properties of integers by primes. We show that 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), (1/log x) ∑n ≤ x λ(n) λ(n+1)/n = O(1/√(log log x)), 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 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)) for n ≤ N. We will give a quick overview of the combinatorial ideas behind the proof.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
