# Coursebooks 2017-2018

## L-functions and random matrices

Michel Philippe

English

#### Remarque

pas donné en 2017-18

#### Summary

This year, the course will discuss the analytic theory of L-functions. These objects of analytic nature (the epitome is Riemann's zeta function) encode deep arithmetic properties of number theoretic objects. We will discuss various modern methods to study them either individually or in families.

#### Content

I. Examples and basic properties.

II. Weil's explicit formula, zero free regions and prime number type theorems.

III. The distribution of zeros of L-functions: the Katz-Sarnak philosophy.

IV. L-functions inside the critical strip

- zero density estimates

- The mollification method

- the resonnance method.

- the amplification method.

#### Learning Prerequisites

##### Recommended courses

Analysis I II III IV, Algebra I II; Introduction to analytic number theory.

##### Important concepts to start the course

Some knowledge (and interest) for number theory along with advanced cerebral activity may be indispensable to start the course

#### Learning Outcomes

By the end of the course, the student must be able to:
• Demonstrate a mastery of the basic techniques studied in the course and during the exercise sessions.

#### Teaching methods

Ex cathedra lectures.

The exercise sessions will be devoted to the understanding of the notions developed in the course and their extensions to more general situations.

#### Expected student activities

A very active participation to the exercise sessions will be expected.

#### Assessment methods

Oral exam based on the material developed during the course and the exercise sessions.

#### Resources

##### Bibliography

E. Kowalski, H. Iwaniec, Analytic Number Theory.

J.-P. Serre, A Course in Arithmetic.

#### Prerequisite for

solving the Riemann hypothesis and winning the 1M USD Millenium prizes of the Clay Mathematics Institute

### In the programs

• Mathematics for teaching, 2017-2018, Master semester 2
• Semester
Spring
• Exam form
Oral
• Credits
5
• Subject examined
L-functions and random matrices
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Mathematics for teaching, 2017-2018, Master semester 4
• Semester
Spring
• Exam form
Oral
• Credits
5
• Subject examined
L-functions and random matrices
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

### Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Under construction
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German