Instructor: Cezar Lupu (BIMSA)

Coaches: Dongsheng Wu 吴冬生 (BIMSA)、 Tianyu Zhang 张天羽 (Ohio State U)

Location: A120

Course description:

In this course, we explore the connections between two important fields of mathematics such as analysis and number theory. First, we introduce the necessary analytic tools such as series of real numbers, derivatives and integrals, sequences and series of functions. Moreover, we study arithmetic functions, but from an analytic perspective, dwelling more on their asymptotic expansion. Also, we study the distribution of prime numbers via elementary methods. One of the key figures for this endeavor is the celebrated Riemann zeta function (for real argument) which can be regarded as an important tool in connection with prime numbers.

Here is an outline.

• What is the connection between analysis and number theory? The big picture.
• Infinite sequences and series of real numbers. Convergence and examples. Convergence criteria.
• Derivatives and integrals of functions on the real line and their properties.
• Sequences and series of real functions. Pointwise and uniform convergence.
• Asymptotics and summation formulas.
• Arithmetic functions. Elementary properties and asymptotic estimates.
• Prime numbers and their properties and distribution.
• The Riemann zeta function for real argument and its special values.

Syllabus could be downloaded here.

Homework 1 (due July 24th)
Homework 2 (due July 29th)
Homework 3 (due August 4th)
Homework 4 (due August 8th)
Research Projects