Andrew Granville: Optimization Problems for Multiplicative Functions (September 10, 2025)

To understand the “usual” behaviour of functions in multiplicative number theory, it is often useful to bound extreme behaviours, Halasz’s theorem being proto-typical. In this talk, we report on several projects on explicit bounds in more general settings, mean values of multiplicative functions whose values at primes are, for example (a) the kth roots of unity or (b) in [-m,m] for fixed m (greater then) or =1. Also, Halasz’s theorem is known for multiplicative functions that are k-bounded at primes, but the uniformity is poor, and we will see how this can be improved. Uploaded 25-09-25 This talk includes joint work with Kaisa Matomäki, K. Soundararajan, Kevin Church and Daodao Yang. For more information go here: https://www.simonsfoundation.org/event/mps-conference-on-universal-statistics-in-number-theory/