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Séminaire Bourbaki

# Adam HARPER — The Riemann zeta function in short intervals

Institut Henri Poincaré
Amphithéâtre Hermite
11 rue Pierre-et-Marie-Curie, 75005 Paris

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 Tt2T 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 TtT+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.