MA 3001 Analytic Number Theory Spring 2010

The mastergradsseminar i matematikk this semester will be a course of analytic number theory. Prerequisites are introductory undergraduate courses in abstract algebra, complex analysis and linear algebra. The course will be taught in English, but the final exam will be in English or Norwegian at the student's option. Most of our effort will be spent counting the number of solutions to number theoretical problems that are too difficult to solve exactly. The methods that we shall use belong to undergraduate mathematics: Calculus, complex analysis, inequalities, linear algebra, and rudimentary group theory. With these rather simple tools, we shall prove Bertrand's Postulate that for any positive integer n there is a prime p between n and 2n, show that the average number of divisors of a large integer n is about log(n), prove that the arithmetic progression ak + b contains infinitely many primes if a and b have no common divisor and prove the famous Prime Number Theorem.

Messages

As we agreed, the exam will be held on May 19.

The format of the exam will be oral. You will each choose a topic to prepare before the exam, and begin by giving a 15 minute exposition of your topic. Afterwards I will ask two theoretical questions from the rest of the syllabus.

Here are the topics that you may choose, with the sections where the relevant material is found.

A. The method of Chebyshev and its applications. 1.1-1.2 (note the corrected proofs!)

B. The Dirichlet divisor problem. 1.5 and part of 1.3

B. The Euler product formula and zeros of the Riemann zeta function. 2.1 and part of 4.2

D. Fourier analysis on Z/qZ. 2.3

E. Dirichlet characters and primes in arithmetic progressions. 2.6

F. The Perron formula and the strategy of the method of contour integrals. 4.1 (corrected proof to appear) and part of 4.3

Each of the three candidates must choose a different topic. Please send me an email with three topics ranked like in this example: "I prefer: 1-D, 2-A or 3-F". I will reply with your highest ranked topic not already claimed by somebody else.

Obviously, you must exercise considerable judgement in order to give a good presentation overall of a topic in only 15 minutes.

Course information

Lectures

The lectures will be Tuesdays 12:15-14:00 and Wednesdays 15:15-17:00. The problem sessions will be Thursdays 15:15-16:00. All these will be held in room 922 of SBII. The office hour will be Thursdays 12:15-13:00 in room 1044 in SBII.

There will be eleven weeks of lectures, in weeks 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17.

Lecturer

Marius Overholt marius.overholt@uit.no

Textbook

I will provide the manuscript of an English-language textbook Elements of Analytic Number Theory.

New proofs of 1.1.1 og 1.2.1

Plan and Syllabus

Week 4-6: Sections 1-5 of chapter 1: Arithmetic Functions

Week 7-11: Sections 1-7 of chapter 2: Characters and Euler Products

Week 15-17: Sections 1-3 of chapter 4: The Method of Contour Integrals

Homework

I will assign four or five problems from the book each week. You may turn in work for me to correct if you want to. But homework is not obligatory, and I will solve the assigned problems in the problem session.

First homework set: Problems 1, 2 and 3 from Section 1.1.

Second homework set: Problems 1 and 2 from Section 1.2 and Problem 1 from Section 1.3.

Third homework set: Problem 5 from Section 1.2 and Problems 4 and 9 from Section 1.3 and Problem 4 from Section 1.4.

Fourth homework set: Problems 1 and 11 and 15 from Section 1.4 and Problems 4 and 6 from Section 1.5.

Fifth homework set: Problems 2 and 3 from Section 2.1 and Problems 1 and 3 from Section 2.2.

Sixth homework set: Problem 9 from Section 2.1 and Problem 7 from Section 2.2 and Problem 1 from Section 2.3.

Seventh homework set: Problems 1 and 2 from Section 2.4 and Problems 2 and 6 from Section 2.5.

Eighth homework set: Problems 4 and 7 from Section 2.6 and Problems 3 and 5 from Section 2.7.

Ninth homework set: Problems 1 and 5 and 7 from Section 4.1 and Problems 2 and 5 from Section 4.2.

Exam

The exam will be held on May 19, and will be oral.

2010-04-22, Marius Overholt