Ruixiang Zhang (Fall 2019):

Description: Possible topics I will do include:

1. Dirichlet's Theorem on primes in progressions

2. Dirichlet L-functions, class numbers

3. Analytic theory: analytic continuation, functional equations and the Prime Number Theorem

4. Non-vanishing of L(1, \chi_d), Siegel's theorem and its ineffectivity

5. The Bombieri-Vinogradov theorem

6. Elementary Sieve

7. Vinogradov's Theorem and his bilinear sieve method

8. Primality testing

9. The Burgess bound

10. Elementary counting in finite fields

Grading: Grading will be based on a few homework assignments.

Textbook: *Multiplicative Number Theory* by Harold Davenport

I will cover most of the material in the textbook. You are expected to finish reading the book on your own by the end of the semester.

Note that you should take Math 623 before this course, and it is required that you feel comfortable with what is in the first 5 chapters of Stein-Shakarchi's book *Complex Analysis *before taking this course.

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 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