Naser Talebizadeh (Fall 2017):

Description: The aim of the course is to study sieves and their applications in analytic number theory. This course is for graduate students interested in number theory in a broad sense. Historically the sieve was a tool to solve problems about prime numbers, such as the Goldbach conjecture or the twin prime conjecture. We will start with basic ideas of sieve theory, such as the sieve of Eratosthenes, Brun's combinatorial sieve, Selberg's upper bound sieve, and the large sieve. My lectures are based on Heath-Brown's lecture notes; available on arxive \url{https://arxiv.org/abs/math/0209360}.

Grading: There will be periodic homework assignments which are mandatory for undergraduate students. The grade of undergraduate students is based on the assignments and a take home exam or a presentation at class. I'll ask graduate student to present a lecture that will be assigned to them at the second week of the course.

Office hours: TBA

Textbook and lecture notes:

1)Heath-Brown's lecture notes \url{https://arxiv.org/abs/math/0209360}

2)The comprehensive book of Iwaniec and J.Friedlander ``Opera de Cribro'' AMS Colloquium Publications, vol 57, is sufficient and recommended for casual reading. I will not follow the book completely or precisely.

Selected topics are: The Eratosthenes sieve - The Brun combinatorial sieve - The Selberg sieve - The Bombieri sieve - The parity phenomena - Producing primes by sieve - Small gaps between primes - Primes represented by polynomials -Zillions of applications