Overview

Probabilistic number theory was born in a famous collaboration of Erdös and Kac. Nowadays the use of classical probabilistic techniques in number theory is abundant. However, up until recently, there had not been much room for modern deep techniques of probability theory. During the past few years though this has changed notably, with the use of random fragmentation processes, Poisson branching processes, Stein’s method and martingales playing a key role in advancements on number-theoretic problems as diverse as Pratt trees, random multiplicative functions and factorization algorithms.

Multiplicative number theory is a broad field which arose from the study of prime numbers and is currently one of the most vibrant areas of number theory. This part of the workshop will focus mainly on the theory of pretentious multiplicative functions, pioneered by Granville and Soundararajan. This theory serves as an alternative approach to several classical problems, such as the distribution of prime numbers in arithmetic progressions, and is now as strong as the classical methods both qualitatively and quantitatively. Its most famous application is arguably proving the Arithmetic Quantum Unique Ergodicity conjecture by Holowinsky and Soundararajan, which involved demonstrating weak subconvexity for a general family of L-functions. In general, the theory of L-functions and the more elementary theory of multiplicative functions have several connections. Part of the workshop will be dedicated to highlighting these links.

Lecture series will be given by Carl Pomerance (Dartmouth), K. Soundararajan (Stanford) and Gerald Tenenbaum (Nancy).