Topics to be covered:

• Dirichlet's Theorem on primes in progressions, Dirichlet L-functions
• Riemann's zeta-function
• the Prime polynomial theorem and function field analogues
• The Prime Number Theorem - a survey
• The Riemann Hypothesis and its consequences
• Sieve methods
• Binary quadratic forms and Dirichlet's class number formula
• Class numbers of imaginary quadratic fields
• GRH and its applications

Schedule:

Prerequisites:

I will assume knowledge of the first semester courses in real and complex variables, and the basic course in number theory.

Suggested reading:

• H. Davenport, Multiplicative Number Theory.
• T. Apostol, Introduction to Analytic Number Theory.

Lecture notes

• Lecture 1: Elementary theory of primes
• Lecture 2: Mean values of arithmetic functions
• Lecture 3: The Hardy Ramanujan theorem, more on arithmetic functions.
• Lecture 4: The prime polynomial theorem.
• Lecture 5: Characters, L-functions and Dirichlet's theorem theorem.
• Lecture 6: Non-vanishing theorems.
• Lecture 7: The Prime Polynomial Theorem for Arithmetic Progressions.
• Lecture 8: Ankeny's theorem and the least non-residue.
• Lecture 9: Binary quadratic forms and reduction theory.
• Lecture 10: The class number one problem.
• Lecture 11: Imaginary quadratic fields.
• Lecture 12: Euclidean fields and mean values of class numbers.
• Lecture 13: The analytic continuation and functional equation of Riemann's zeta function.

Homework assignments

• Assignment 1, due date: March 9, 2022.
• Assignment 2, due date: March 23, 2022.
• Assignment 3, due date: March 30, 2022.
• Assignment 4, due date: April 13, 2022.
• Assignment 5, due date: April 27, 2022.
• Assignment 6, due date: May 11, 2022.
• Assignment 7, due date: May 18, 2022
• Final Assignment, due date: June 26, 2022