Chandrasekhar Hall

Probabilistic Number Theory

J-M. Deshouillers

University of Bordeaux

The course, which is introductory, can be followed by graduate
students or more advanced persons.

The afternoon courses will be
self-contained:
new concepts will be
introduced during the morning tutorials.

The first lecture will be presented as a colloquium talk
with a slightly broader scope:
interactions between probability theory and number theory.
The other lectures
will be organized according to the following pattern:


1. Arithmetical functions: basic notions.
Probability: basic notions.


2. Distribution functions of arithmetical functions.
Weak convergence of random variables.
The Turan-Kubilius inequality.


3. Arithmetical functions and probabilistic models:
A weak version of the Erd\H{o}s-Kac Theorem;
a stronger form of the Hardy-Ramanujan Theorem.
The Erdos-Wintner theorem.


4. Additive number theory: basic notions.
Erdos approach to additive questionsvia probability theory.
A problem of Sidon.
The Erdos-Renyi model for $s$-th powers.


5. Arithmetico-probabilist models.
Atkin's approximation of sums of two squares.
An arithmetical Erdos-Renyi model.
Vu's economical bases formed with $s$-th powers.