Fundamentals in Number Theory

Spring 2022, Jinan

Schedule:

Tuesday 08:00-09:50 & Thursday 08:00-09:50.

Classroom: 

Zhixin Building B124.

Prerequisites:

The first semester courses in real and complex variables, and the basic course in number theory (i.e. the first year course Basic Algebra and Geometry, see Part I of [FY]).

Syllabus:

The course is an introductory course in 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 distribution of primes and applications,

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

   6. Primitive characters, character sums, Polya–Vinogradov inequality,

   7. Sums of squares,

   8. Basic Algebraic Number Theory,

   9. Diophantine approximation, continued fractions, and the transcendence of e.

If we still have time, then we may include  

   10. Modular forms and theta series

   11. Equidistribution modulo one

Bibliography

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

   [L]. W.J. LeVeque, Fundamentals of Number Theory.

   [P]. 潘承洞，《数论基础》.

   [FY]. 冯克勤、余红兵，《整数与多项式》 第一部分.

Suggested reading:

   [A]. T. Apostol, Introduction to Analytic Number Theory.

   [MV]. H. Montgomery and R. Vaughan, Multiplicative Number Theory I. Classical Theory. [Chapters 4 & 9]

   [IK]. H. Iwaniec and E. Kowalski, Analytic Number Theory. [Chapters 1-4]

Attendance of lectures is mandatory!

Homeworks:

This will be an important part of the course. 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 (due Thursday March 3): [L] §1.1 Problems 2 & 4;   §6.3 Problems 2, 3 & 6;   §6.2 Problems 4;  §6.1 Problems 2 & 6.

Homework 2 (due Thursday March 10): [P] 第二章习题 4, 5, 6, 7, 11, 12

Homework 3 (due Thursday March 17): [L] §6.4 Problems 3(b) & 6;   §6.10 Problem 5;   §6.11 Problems 2, 3, 4, 5

Homework 4 (due Thursday March 24): [L] §6.6 Problems 4, 5, & 7;   §6.7 Problems 1 & 5

Homework 5 (due Thursday April 7): [L] §6.9 Problems 4, 6, & 7;   §6.11 Problem 13;   [P] 第三章习题 13, 14, 19

Homework 6 (due Thursday April 21): [L] §3.4 Problems 2, 6, 7, 8, & 14;   §4.3 Problems 7 & 8;   §5.4 Problems 2 & 3

Homework 7 (due Thursday May 5):    Homework_7.pdf

Homework 8 (due Thursday May 19):  Homework_8.pdf

Homework 9 (due Thursday May 26):  [L] §7.3 Problems 4 & 6;   §7.5 Problems 1 & 3;   §8.2 Problems 3, 6, 11 & 14.   (Choose 4 of them.)

Homework 10 (due Thursday June 26):  [L] §8.3 Problem 4;   §8.4 Problem 4;   §9.1 Problem 3;   §9.2 Problem 3.

Lecture notes:

Lecture note 1.pdf	Introduction	02.22
Lecture note 2.pdf	arithmetic functions	02.24,   03.01,   03.03
Lecture note 3.pdf	order of magnitude of AFs	03.08,   03.10,   03.15
Lecture note 4.pdf	the distribution of primes	03.17,   03.22,   03.29,   03.31,   04.07
[Apostol, Sections 3.8 & 5.7-5.9]	congruences and primitive roots	04.12
[LeVeque, Chapter 4]	primitive roots and indices	04,14
[Montgomery--Vaughan, Chapter 4]	Dirichlet characters	04.19,   04.21
[Iwaniec--Kowalski, Section 2.3]	Dirichlet's theorem on primes in APs	04.26,   05.03
[Montgomery--Vaughan, Chapter 9]	character sums	05.03,   05.05,   05.08
[LeVeque, Chapters 7-9]		05.12, ..., 06.09

Exercise classes:   03.24,   04.26,   05.26

Midterm exam:   04.19 or 04.21 ??   (Canceled, because of the outbreak)

Final exam:   06.20 -- 07.01

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

Teaching assistant: Liangxun Li, email: lxli@mail.sdu.edu.cn

Course homepage:  https://faculty.sdu.edu.cn/brhuang/zh_CN/zdylm/1454369/list/index.htm