Fundamentals in Number Theory

Spring 2020/21, Jinan

Schedule: Monday 10:10-12:00, Wednesday 10:10-12:00.

Classroom: Zhixin Building B119.

Prerequisites:

I will assume knowledge of the first semester courses in real and complex variables, and the basic course in number theory.

Syllabus:

The course is an introductory course in (analytic) number theory. The topics include

    1. The Euclidean algorithm, unique factorization, greatest common divisor, linear Diophantine equations, congruences, Chinese Remainder Theorem, Legendre symbol, quadratic reciprocity law, primitive roots and indices

    2. Arithmetic functions

    3. The order and average order of magnitude of arithmetic functions

    4. The Prime Number Theorem and its applications

    5. Dirichlet characters, Dirichlet's theorem on primes in arithmetic progressions 

    6. Primitive characters, character sums, Polya–Vinogradov inequality, and Burgess' bounds

    7. Sums of squares

If we still have time, then we may include    (In fact, we won't have time. 06.07)

    8. Modular forms and theta series

    9. Equidistribution modulo one

    10. Diophantine approximation, Liouville's theorem on rational approximations to algebraic numbers, Thue's equation

Bibliography

Any introductory book on number theory will be useful. For example, see:

    1. W.J. LeVeque, Fundamentals of Number Theory.

    2. 潘承洞，《数论基础》.

Suggested reading:

    3. T. Apostol, Introduction to Analytic Number Theory.

    4. H. Montgomery and R. Vaughan, Multiplicative Number Theory I. Classical Theory.

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

Attendance of lectures is mandatory!

Homeworks:

This will be an important part of the course. In order to be eligible to take the final exam, at least 50% of the assignments have to be turned in on the week of their due date. 10% of the final grade will be determined from the homework scores, which will be obtained as the average grade of a certain number of assignments.

Homework_1.pdf   due Wednesday, March 17, 2021

Homework_2.pdf   due Monday, March 29, 2021

Homework_3.pdf   due Wednesday, April 14, 2021

Homework_4.pdf   due Wednesday, April 28, 2021

Homework_5.pdf   due Monday, May 17, 2021

Homework_6.pdf   due Monday, May 31, 2021

Lecture notes:

Lecture note 1.pdf  03.01

Lecture note 2.pdf  03.03, 03.08, 03.10

Lecture note 3.pdf  03.10, 03.15, 03.17, 03.22

Lecture note 4.pdf  03.24, 03.29, 03.31, 04.12, 04.14

[Apostol, Sections 3.8 & 5.7-5.9]   04.19

[LeVeque, Chapter 4]   04.21

[Montgomery and Vaughan, Chapter 4] & [Iwaniec and Kowalski, Section 2.3]   04.26, 04.28, 05.08

[Montgomery and Vaughan, Chapter 9]   05.10, 05.12, 05.19, 05.24, 05.26, 05.31, 06.02, 06.07

[LeVeque, Chapter 7]   06.09

Exercise classes:   04.07,   05.17,  06.16 

Contact: Bingrong Huang, brhuang@sdu.edu.cn, Office : Mingde Building C701

Teaching assistant: Shenghao Hua,  huashenghao@vip.qq.com

Course homepage:  https://faculty.sdu.edu.cn/brhuang/zh_CN/zdylm/1322195/list/index.htm