Random multiplicative functions and their distribution

by Seth Hardy (University of Warwick, Royaume-Uni)

Monday Mar 24, 2025, 3:30 PM → 4:30 PM Europe/Paris

Salle Pierre Grisvard (IHP - Bâtiment Borel)

A central question in analytic number theory is to understand the size of the partial sums of the Möbius function. This motivated the 1944 paper of Wintner where he introduced the concept of a random multiplicative function: a probabilistic model for the Möbius function. In recent years, it has been uncovered that there is an intimate connection between random multiplicative functions and the theory of Gaussian Multiplicative Chaos, an area of probability theory introduced by Kahane in the 1980's. These ideas have been pioneered by Harper, who, in a recent pre-print, uses this (highly probabilistic) connection to show that character sums typically display better than square-root cancellation. We will survey selected results and discuss recent research on the distribution of partial sums of random multiplicative functions (which one can imagine as relating to character sums) when restricted to integers with a large prime factor.