Instructor: Dimitris Koukoulopoulos |
Class dates: From October 2 to December 21, 2012 |
This is an introductory course to sieve methods. We will start with a study of the so-called combinatorial sieve, which will include Brun's pure sieve and the beta-sieve, leading to the fundamental lemma of sieve methods. Several applications will be given, such as the Brun-Titchmarsh inequality and bounds for the number of primes that are values of a given polynomial. Then we will introduce Selberg's approach to sieving. We shall also discuss the optimality of Selberg's sieve and the parity problem of sieve methods. Next, we will study prime solutions to linear equations and Cramer's model of the integers and, subsequently, the revolutionary work of Goldston, Pintz and Yildirim on small gaps among primes. The course will then shift to the study of bilinear sums and the large sieve, as well as one of its main applications: the Bombieri-Vinogradov theorem. We will conclude with three additional topics, provided that time permits: smooth numbers and large gaps between primes, the linear sieve and Maier's matrix method, and the affine sieve.
Your grade will be based entirely on the number of points you accumulate on the following assignments:
You are welcome to email me electronic versions of your homework.
There is no textbook for this course. The course will be mainly based on the following notes which will be updated as the course progresses.
Notes (Chapters 1-9, Appendices A-C. Last update: December 12, 2012. Added Chapter 9; this is essentially the final version of the notes, modulo small future corrections.)
Other main references include (note that both books have been reserved in the math library of UdeM):