Tony Haddad (Université de Montréal)
Date
Wednesday October 23, 20241:30 pm - 2:30 pm
Location
319 JEFFERY HALLMath & Stats Number Theory Seminar
Wednesday, October 23rd, 2024
Time: 1:30 p.m. Place: Jeffery Hall, Room 319
Speaker: Tony Haddad (Université de Montréal)
Title: A coupling for prime factors
Abstract: We present a coupling between a random integer $N_x$, chosen uniformly from the interval $[1, x]$, and a Poisson-Dirichlet process $(V_i)_{i \ge 1}$ satisfying $$\mathbb E\, \sum_{i \ge 1} |\log P_i- V_i\log x| \asymp 1,$$ where $N_x = P_1P_2 \cdots$ is the unique factorization of $N_x$ into primes (or ones), and the $P_i$'s are non-increasing. This resolves a conjecture posed by Arratia in 1998. We also explain how to apply the coupling to extract statistical information about divisors. This is joint work with Dimitris Koukoulopoulos.