Math 105

Elementary Analytic Number Theory

Instructor: Carl Pomerance (carl.pomerance at dartmouth.edu)

### News

Homework due Wednesday, May 27:
Supplementary problems: posted here.
Chapter 7, numbers 15, 16, 17.

### Abstract

This course is an introduction to elementary analytic number theory. The primary theme is the prime number theorem and the role of elementary methods.

We shall follow the book of Paul Pollack, ``Not always buried deep", in particular Chapters 3, 4, 6, and 7.

1. Chapter 3, Elementary prime number theory, II
Highlights consist of estimates of Chebyshev on counting primes, including a proof of Bertrand's Postulate: ``Chebyshev said it, I'll say it again, there's always a prime between n and 2n. This chapter also contains the theorem of Mertens, plus many challenging exercises.

2. Chapter 4, Primes in aritmetic progressions
This chapter introduces group characters and applies them to prove the theorem of Dirichlet that for each ``legal" residue class there are infinitely many primes.

3. Chapter 6, Sieve methods
This chapter introduces some combinatorial ideas for attacking some notorious problems such as the twin prime conjecture and Goldbach's conjecture.

4. Chapter 7, An elementary proof of the prime number theorem
The title says it all.

### Classes

Room: 004 Kemeny
Lectures: Monday-Wednesday-Friday 11:15 am--12:20 pm (11 hour)
X-hour: Tuesday 12:00 noon - 12:50pm

We generally will not use the x-period, but it would be very good for students to attend the number theory seminar. This usually meets on Thursdays at 2:30 pm in Room 343 Kemeny, for 1 hour, though there will be a few weeks when it meets on Tuesdays instead of Thursdays.

There will be no class on Friday, April 17 and no class on Friday, April 24, nor office hours on the previous Thursdays.

### Staff

Instructor:
Carl Pomerance -- 339 Kemeny / Tel. 6-2635
Office hours: Tuesday, Wednesday, Thursday 9:00 AM--9:55 AM and by arrangement at other times.

### Textbook

Paul Pollack
Not always buried deep: a second course in elementary number theory

### Homework

Homework is due at the start of the class period on the due date.
Homework will be generally due once per week on Mondays.
Assignments will be posted on this website, with extra problems and/or comments added as the week progresses.

### Past assignments

Homework due Wednesday, May 20:
Supplementary problems: posted here.
Chapter 6, numbers 18a, 20, 21, 22a, 25, 30.
(Please double-check the book website so that you are doing the correct problems; numbering has changed.)

Homework due Monday, May 11:
Chapter 6, problems 2, 7 (second part).
[Note that a new draft of the book was uploaded this week, which changes somewhat the numbering of the problems.]
Supplementary problems are posted here.

Homework due Monday, May 4:
posted here.

Homework due Monday, April 27:
Chapter 4, numbers 10 to 14.

Homework due April 20.
Chapter 4, numbers 2, 3, 4.
Here are three more problems.

Homework due Monday, April 13:
Chapter 3, problems 13, 14, 15, 19, 22, 23.
The assignment is complete.

Homework due Monday, April 6.
Chapter 3, problems 1, 5, 6, 7, 8.
Here are two more problems.
The assignment is complete.

### Exams

There will be no exams, but students will be expected to make some presentations to the class.