Lethbridge Number Theory and Combinatorics Seminar: Harald Andrés Helfgott

We will discuss a graph that encodes the divisibility properties of integers by primes. We 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 our result with Matomaki-Radziwill. For instance: for λ the Liouville function (that is, the completely multiplicative function with λ(p)=−1 for every prime), 1logx∑n≤xλ(n)λ(n+1)n=O(1loglogx‾‾‾‾‾‾‾‾√), which is stronger than well-known results by Tao and Tao-Teravainen. 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=loglogN+O(loglogN‾‾‾‾‾‾‾‾√).