Special Number Theory Seminar

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\leq 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.