Undergraduate Seminar: Glimpses of Modern Number Theory

**▲ Instructor: **Guanghua Ji

**▲ Syllabus:** **1, Riemann hypothesis and prime numbers**

It starts with the basic properties of prime numbers and the prime number theorem. A list of open questions about primes and related results are mentioned. We will continue to study of the Riemann zeta function, including the Euler product, analytic continuation and functional equation, zeros, Hadamard’s product formula, the function N(T) and the zero-free region of Riemann zeta-function. These lead to a proof of the prime number theorem.

Lecture note: Heath-Brown, Prime number theory and the Riemann zeta-function

Other Reference Books

** 2, Introduction to modular forms **

Modular forms play a central role in modern and classical number theory. Most

prominently, they play a key role in Wiles' proof of Fermat's Last Theorem. The goal of this course is to give an elementary introduction to the theory of modular forms. There is no prerequisite.

Textbook: J. P. Serre, A course in arthematic, GTM7, Chapter VII.

Other Reference Books

If time permits, we will also consider the following topics.

** 3, Introduction to elliptic curves**

Textbook: W. Stein, Elementary Number Theory and Elliptic Curves, Springer, 2009.

Other Reference Books

**▲ Grading:** Based on homework. No exam.

**▲** **Classroom:** New math building, B120

**▲** **Time:** 6:00pm--7:50pm, Wed.

ALL ARE WELCOME!