Expansion, divisibility and parity
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)∑n≤xλ(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 n≤N).
For the Full abstract, please see: https://www.cs.uleth.ca/~nathanng/ntcoseminar/