Minicourse on multiplicative functions
Introductory Workshop: Analytic Number Theory February 06, 2017  February 10, 2017
Location: SLMath: Eisenbud Auditorium
Multiplicative functions
prime numbers
smooth numbers
Chowla's Conjecture
MiniCourse On Multiplicative Functions
The minicourse will be an introduction to the theory of general multiplicative functions and in particular to the theorem of MatomakiRadziwill 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 GranvilleSoundararajan 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 MatomakiRadziwill 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.
Matomaki Radzwill Notes

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MiniCourse On Multiplicative Functions
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