Coursebooks

Introduction to the analytical number theory

MATH-313

Lecturer(s) :

Lin Yongxiao
Topacogullari Berke

Language:

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

Learning Outcomes

By the end of the course, the student must be able to:

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

Prerequisite for

MATH-417 Topics in Number Theory

In the programs

  • Mathematics, 2019-2020, Bachelor semester 5
    • Semester
      Fall
    • Exam form
      Written
    • Credits
      5
    • Subject examined
      Introduction to the analytical number theory
    • 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 MAA331MAA331
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Lecture
Exercise, TP
Project, other

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