Feb 06, 2017
Monday

10:30 AM  11:30 AM


Minicourse on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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
 Supplements



Feb 07, 2017
Tuesday

03:30 PM  04:30 PM


Minicourse on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements



Feb 09, 2017
Thursday

02:00 PM  03:00 PM


Minicourse on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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
 Supplements


