Number theory

Spring 2016

 Lectures: T-Th 12:30 --1:50 APM B412 Office Hour: M 10:00 AM--12:00 AM APM 7230 TA information: Corey Stone cdstoneucsd edu TA Office hour: F, 2-3; Th, 5-6 APM 6331

Book
• W. J. LeVeque, Fundamentals of number theory. (The main textbook)
• H. Davenport, Multiplicative number theory. (Excellent advance book; we discuss a little bit of this book.)
Topics

I am looking forward to this class. We will have fun with various topics in number theory, and we will see how integrated mathematics is. In this course, we will study

• Distribution of prime numbers:
• Various arithmetic functions.
• Baby prime number theorem.
• Zeta and L-functions.
• Dirichlet theorem: prime numbers in an arithmetic progression.
• Prime Number Theorem and the Mobius randomness law.

Homework

• Homework are due on Tuesdays in class. You should hand in your homework in class.
• Late Homework are not accepted.
• There will be 9 problem sets. Your cumulative homework grade will be based on the best 8 of the 9.
• You can work on the problems with your classmates, but you have to write down your own version. Copying from other's solutions is not accepted and is considered cheating.
• A good portion of the exams will be based on the weekly problem sets. So it is extremely important for you to make sure that you understand each one of them.

• Your weighted score is the best of
• Homework 30%+ midterm exam I 20%+ midterm exam II 20%+ Final 30%
• Homework 30%+ The best of midterm exams 20%+ Final 50%
• This is a relatively advance course, and I am sure that all of you are motivated and hardworking students. Since most likely the size of the class would be small, I will have a chance of knowing you guys better. That can help me greatly to know how you are doing in the course. That said to determine your letter grades first I look at your weighted score and use the following table. Your letter grades can only get better from there based on your attendance and involvement in class or office hours.
•  A+ A A- B+ B B- C+ C C- 97 93 90 87 83 80 77 73 70
• Homework and midterm exams will be returned in the discussion sections.
• If you wish to have your homework or exam regraded, you must return it immediately to your TA.
• Regrade requests will not be considered once the homework or exam leaves the room.
• If you do not retrieve your homework or exam during discussion section, you must arrange to pick it up from your TA within one week after it was returned in order for any regrade request to be considered.
Further information
• There is no make-up exam.
• Keep all of your returned homework and exams. If there is any mistake in the recording of your scores, you will need the original assignment in order for us to make a change.
• No textbooks, calculators and electronic devices are allowed during exams.
Exams.

• The first exam:
• Date: May 3.
• Topics: Details of the lemmas proved in class; Outline of proofs of important theorems; Homework assignments; Exercises in your book Sections 6.1-6.8.
• Exam: Here is the first exam.
• The second exam:
• Date: May 24.
• Topics: It will be an oral exam. I will meet with you individually, and the question will be about general concepts that are discussed in class, and outline of proofs of major results proved in the lectures. And details of short proofs of some the results.
• The final exam:
• Topics: All the topics that were discussed in class, your homework assignments, and relevant examples and exercises in the practice exercises and your book.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.
• Extra help:
• To be posted later.
• Assignments.

• Due April 12: here is the first problem set.
• Due April 19: here is the second problem set.
• Due April 26: here is the third problem set.
• Due May 3: here is the fourth problem set.
• Due May 10: here is the fifth problem set.
• Due May 17: here is the sixth problem set.
• Due May 24: here is the seventh problem set.
• For your practice: here is the eighth problem set. (In Problem 3, you are supposed to prove that dual of direct sum of two finite abelian groups is isomorphic to the direct sum of the dual of these groups.)
My notes.

• Here is my note on arithmetic and multiplicative functions.
• Here is my note on cyclotomic polynomials and proof of a special case of Dirichlet theorem: there are infinitely many primes of form nk+1. We prove a little more: the n-th cyclotomic polynomial has a root mod p if and only if p is of the form nk+1.
• Here is my note on zeta function, Euler equation, Chebyshev's baby prime number theorem, certain weighted sums over primes, and proof of Bertrand hypothesis.
• Here is my note on the needed material from complex analysis.
• Here is my note on a proof of Dirichlet Theorem on primes in an arithmetic progression.
• Here is a summary of our proof of Dirichlet Theorem on primes in an arithmetic progression.
• Here is my note on the characters of finite abelian groups.
• Here is my note on Prime Number Theorem and Mobius Randomness law.