# 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