Topics in number theory
Summary
This year's topic is "Advanced Analytic Number Theory": this is a continuation of the course MATH-313 "Introduction to Analytic Number Theory". We will cover primes in arithmetic progressions, the Landau-Siegel zero, the Bombieri-Vinogradov Theorem and Vinogradov's three primes theorem (itp).
Content
This year, we will continue the course "Introduction to Analytic Number Theory" with more advanced topics:
- Primes in arithmetic progressions : the Hadamard/de la Vallee-Poussin zero-free regions for Dirichlet L-functions.
-The Landau-Siegel zero and the Siegel-Walfisz Theorem.
-Primes in large arithmetic progressions: the large Sieve and the Bombieri-Vinogradov Theorem.
-Ternary additive problems: introduction to the circle method and Vinogradov's Three Primes Theorem:
- Every sufficiently large odd integer is the sum of three prime numbers.
Keywords
Primes numbers
Arithmetic progressions
L-functions and zero-free regions
The large Sieve
The circle method
Learning Prerequisites
Required courses
Analysis III & IV
Introduction to Analytic Number Theory.
Recommended courses
Important concepts to start the course
-Good knowledge of analysis in particular Fourier theory and theory of the complex variable.
Learning Outcomes
By the end of the course, the student must be able to:
- Synthesize the analytic aspects of the theory of numbers
- Solve advanced problems in analytic number theory
Transversal skills
- Access and evaluate appropriate sources of information.
- Make an oral presentation.
- Demonstrate the capacity for critical thinking
Teaching methods
Ex-Cathedra Course
Expected student activities
We expect a proactive attitude during the courses and the exercises sessions (possibly with individual presentation of the solution of various problems).
Assessment methods
Oral 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 |
| Others | a moodle with ressources for the course will be maintained |
Resources
Bibliography
Davenport: Multiplicative Number Theory
Iwaniec-Kowalski: Analytic Number Theory
Ressources en bibliothèque
Prerequisite for
Current research in number theory
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Topics in number theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Topics in number theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Topics in number theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks