Title of the course: Probabilistic Methods in Number Theory: From Basics to the Erdős-Kac Theorem
Instructor: Dr. Arturo Jaramillo
Institution: The Center of Research in Mathematics (CIMAT)
Dates: 21-27 July 2025
Prerequisites: Calculus 1. Some knowledge of basic probability would be really helpful.
Level: Undergraduate, graduate
Abstract: This minicourse offers an introduction to analytic and probabilistic number theory, starting with foundational ideas and culminating in a proof of the Erdős-Kac Theorem. We begin by discussing elementary properties of prime numbers and elementary asymptotic estimations, including results like Mertens’ theorems. The course then addresses the interplay between number theory and probability, emphasizing the relation between multiplicities of uniform samples in 0,…,n and the geometric distribution. The final sessions focus on a probabilistic proof of the Erdős-Kac Theorem, drawing inspiration from Stein’s method. By the end of the course, participants will gain insight into the probabilistic nature of additive functions, such as the number of distinct prime factors, and appreciate their convergence properties through the lens of central limit theorems.
Language: English