Mini-course on multiplicative functions

The mini-course will be an introduction to the theory of general multiplicative functions and in particular to the theorem of Matomaki-Radziwill on multiplicative function in short intervals. The theorem says that, for any multiplicative function $f: \mathbb{N} \to [-1, 1]$ and any $H \to \infty$ with $X \to \infty$, the average of $f$ in almost all short intervals $[x, x+H]$ with $X \leq x \leq 2X$ is close to the average of $f$ over $[X, 2X]$. In the first lecture we will cover briefly the "pretentious theory" developed by Granville-Soundararajan and a selection of some of the key theorems: Halasz's theorem, the Lipschitz behaviour of multiplicative functions, Shiu's bound, ... We will also describe some consequences of the Matomaki-Radziwill theorem. In the second lecture we will develop sufficient machinery to prove a simple case of the latter theorem for the Liouville function in intervals of length $x^{\varepsilon}$. In the third lecture we will explain the proof of the full result. Time permitting we will end by discussing some open challenges.

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