Expansion, divisibility and parity

Speaker: Harald Andrés Helfgott

Date: Mon, Apr 3, 2023

Location: Online, PIMS, University of Lethbridge

Conference: Lethbridge Number Theory and Combinatorics Seminar

Subject: Mathematics

Class: Scientific

Abstract:

Harald Andrés Helfgott University of Göttingen, Germany, and Institut de Mathématiques de Jussieu, France)

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), (1/logx)nxλ(n)λ(n+1)/n=O(1/(loglogx)), 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)) for nN).

For the Full abstract, please see: https://www.cs.uleth.ca/~nathanng/ntcoseminar/