Math 53. Analytic Number Theory

Spring 2013


Notes on the proof of the Prime Number Theorem.

Notes on Complex Analysis:
Part 1
Part 2.

Notes on the sum of the reciprocals of squares and fourth powers, etcetera.

Notes on Dirichlet characters and another proof of Dirichlet's theorem.

Notes on Abel Summation.

Homework 9.

Homework 8.

Homework 7.

Homework 6.

Homework 5.

Homework 4.

Homework 3.

Homework 2.

Homework 1.


Not Always Buried Deep: A Second Course in Elementary Number Theory by Paul Pollack.

The textbook is mandatory.

Topics we will cover

Possibly the most important result in Mathematics in the 19th century was the proof of the Prime Number Theorem, a theorem that not even the mighty Gauss could prove. In this class, we will overview prime number theory. We will prove that there are infinitely many primes in the arithmetic progression a, a+b, a+2b, ... (where a and b are coprime), known as Dirichlet's theorem. We will prove the important Chebyshev and Mertens estimates. We will give an elementary proof of the aforementioned prime number theorem. Outside of prime number theory, we will cover sieve theory and use it to attack the famous Goldbach problem, the twin prime conjecture and other popular unsolved number theory problems. In particular, we might get to cover the important Goldston-Pintz-Yildirim theorem proven in the last decade. Other number theory curiosities will be covered along the way, such as theorems on perfect numbers, amicable numbers and theorems on the distribution of important arithmetical functions.


The course grade will be based on:
Homework 25%,
Presentations 25%,
Midterm (take-home) 25%,
Final Exam (take-home) 25%.


There will be written homework due roughly every week. The most recent homework will be posted in the announcements and a copy of all homeworks can be accessed here. Collaboration in the homework is permitted, but you are not allowed to copy someone else's work. The solutions must be written individually. You have to mention on your problem set the names of the students that you worked with.


For each written homework, the student must select a problem (and solution) to present in class. Each student will present a different problem, so I suggest the students to team up to decide which student presents which problem.


There will be one midterm and one final exam. Both will be take home exams. Dates to be announced later. On the midterm and the final exam you must work on the problems on your own. No collaboration permitted in the exams.

Accommodations Statement

If you believe that you need accommodations for a disability, please contact Leslie Hempling in the Office of Student Disability Services, located in Parrish 130, or e-mail to set up an appointment to discuss your needs and the process for requesting accommodations. Leslie Hempling is responsible for reviewing and approving disability-related accommodation requests and, as appropriate, she will issue students with documented disabilities an Accommodation Authorization Letter. Since accommodations may require early planning and are not retroactive, please contact her as soon as possible. For details about the Student Disabilities Service and the accomodations process, visit

You are also welcome to contact me privately to discuss your academic needs. However, all disability-related accommodations must be arranged through Leslie Hempling in the Office Of Student Disability Services.

Last modified on May 1, 2013.