# Lethbridge Number Theory and Combinatorics Seminar: Harald Andrés Helfgott

## Topic

Expansion, divisibility and parity

## Details

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‾‾‾‾‾‾‾‾√).

## Additional Information

**Time**: 9.30am Pacific/ 10.30am Mountain

**Online Via Zoom**:

https://uleth.zoom.us/meeting/register/tJcpc--rjwsGtE7inVJcDzcjleYhbzaso14

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

This is a Past Event

Event Type

**Scientific, Seminar**

Date

**April 3, 2023**

Time

**-**

Location

University of Lethbridge