# Fiches de cours

## Introduction to analytical number theory

#### Enseignant(s) :

Lin Yongxiao
Raju Chandra Sekhar

English

#### Summary

The aim of this course is to present the basic techniques of analytic number theory.

#### Content

This course provides an introduction to analytic number theory. After introducing the basic definitions and methods, our aim will be to prove Dirichlet's theorem on primes in arithmetic progressions and the prime number theorem.

Covered topics include:

1. Arithmetic functions: Multiplicative functions, Dirichlet convolutions
2. Asymptotic estimates: Euler's summation formula, Summation by parts, Dirichlet's hyperbola method
3. Elementary results on the distribution of prime numbers: Chebyshev's theorem, Mertens' theorems
4. Dirichlet series: Euler product, Perron's formula
5. Primes in arithmetic progressions: Dirichlet characters, Dirichlet L-functions, Proof of Dirichlet's theorem on primes in arithmetic progressions
6. The Riemann zeta function: Analytic continuation, Functional equation, Hadamard product
7. The prime number theorem: Explicit formula, Zero-free region, Proof of the prime number theorem

#### Learning Prerequisites

##### Required courses

• Analyse I, II, III
• Algèbre Linéaire I, II
• Algèbre I

#### Learning Outcomes

By the end of the course, the student must be able to:
• Analyse and solve a basic problem from analytic number theory.

#### Teaching methods

Ex cathedra lecture with exercises.

#### Expected student activities

Proactive attitude during the course and the exercise sessions, possibly with individual presentation of the solution of exercise problems.

#### Assessment methods

Written exam.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

#### Supervision

 Office hours No Assistants Yes Forum No

#### Resources

##### Bibliography

• Introduction to Analytic Number Theory, T. M. Apostol
• A Course in Arithmetic, J.-P. Serre
• Multiplicative Number Theory, H. Davenport
• Multiplicative Number Theory I. Classical Theory, H. L. Montgomery & R. C. Vaughan

#### Prerequisite for

MATH-417 Topics in Number Theory

### Dans les plans d'études

• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Introduction to analytical number theory
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11   MAA331MAA331
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand