MODERN PRIME NUMBER THEORY
We will discuss important recent results concerning prime numbers. The highlights will include (1) Huxley’s theorem on the nonexistence of large gaps between prime numbers, (2) the theorem of Goldston-Pintz-Yildirim on the existence of small gaps between prime numbers, (3) the Agrawal-Kayal-Saxena algorithm for recognizing prime numbers in polynomial time, and (4) Linnik’s theorem on the least prime in arithmetic progressions. You will learn modern techniques of analytic number theory such as mean value theorems for Dirichlet polynomials, density results for the zeroes of the Riemann zeta function, and important ideas inspired by the Hardy-Littlewood method.
By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.
Elective live courses: regular homework, and presentation or final