Maxim Gerspach: Dirichlet series, the Riemann zeta function and random multiplicative functions

Abstract

In this survey-type talk I want to give an overview of the types of probabilistic and analytic tools and ideas that go into the heuristic and rigorous study of the Riemann zeta function and related quantities. I will explain certain models that are used in order to get a good heuristic understanding of the typical and extremal behaviour of the zeta function, and how to analyze these models rigorously by relating them to probabilistic objects such as branching random walks and multiplicative chaos, as well as applying analytic techniques in the so-called Hardy space of Dirichlet series.