A classical idea for studying the behaviour of complicated functions, like the Riemann zeta function ζ(s), is to investigate averages of them. For example, the integrals over T≤t≤2T of various powers of ζ(1/2+it), sometimes multiplied by some other cleverly chosen function, have been investigated extensively to deduce upper and lower bounds for the maximum size of ζ(1/2+it). More recently, Fyodorov and Keating have proposed the investigation of much shorter integrals over T≤t≤T+1. This turns out to lead to interesting connections between various issues in number theory, analysis, mathematical physics and probability, such as branching random walk and multiplicative chaos. I will try to explain some of these connections, ideas from the proofs, and what they tell us about the zeta function.
