Math 105

Topics in Number Theory

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

### News and current assignment

Due to one last presentation that is not quite ready, we will meet the x-period on Tuesday, Nov. 19. This is from noon to 12:50 pm in 004 Kemeny.

There are no further written assignments.

Notes from week 1 are here. Please alert me to typos, inaccuracies, etc.

Notes from week 2 are here. Please alert me to typos, inaccuracies, etc.

Notes from week 3 are here. Please alert me to typos, inaccuracies, etc.

Notes from week 4, are here. Please alert me to typos, inaccuracies, etc.

Notes from week 5, are here. Please alert me to typos, inaccuracies, etc.

Notes from week 6, are here. Please alert me to typos, inaccuracies, etc.

Notes from week 7, are here. Please alert me to typos, inaccuracies, etc.

Notes from week 8, are here. Please alert me to typos, inaccuracies, etc.

Notes from week 9, are here. Please alert me to typos, inaccuracies, etc.

For further reading on the final topics in the notes, you might look at the talk slides: http://www.math.uga.edu/~pollack/NTS0812.pdf

Our text is "Introduction to analytic number theory" by Tom M. Apostol, published by Springer. This book is at Wheelock Books. It is available on amazon as well (today, 9/16/13, it is about \$43 for the hardcover edition, but the price goes up and down).

We meet MWF at 11:15, the room is 004 Kemeny.

### Abstract

Introduction to analytic number theory

Analytic number theory uses tools from analysis, both real and complex, to study the distribution of special numbers (such as primes) within the integers and to study the distribution of number-theoretic functions. An early triumph in the field is the theorem of Dirichlet that in any residue class a (mod b), where a, b are relatively prime, there are infinitely many primes. The text we'll be using is fairly gentle and is in fact in the Springer series Undergraduate Texts in Mathematics. Always optimistic, I hope we can get through most of the book, with more thoroughness in some areas than others. Some of the highlights will be Dirichlet's theorem mentioned above, Gauss sums (with applications to the law of quadratic reciprocity and the Polya-Vinogradov inequality), and the prime number theorem.

Prerequisites: An undergrad number theory course, as well as some abstract algebra. I'll be happy to try and fill in gaps for motivated students.

Grading: There will be weekly written assignments, plus some class presentations. There will be no formal exams. Enrolled graduate students who have been admitted to candidacy will be excused from the written assignments.

### Classes

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

We may meet several of the x-hours, but this will always be announced in advance.

There will be no classes October 16 and 25.

Students are encouraged to attend the Math Department colloquia (Thursdays at 4 pm), especially those featuring a number theorist:
Sept. 26, Jim Brown, Clemson U.
Oct. 31, Yitang Zhang, UNH
Nov. 14, Adriana Salerno, Bates C.
These three should also be giving number theory seminars. The seminar will meet at 2:30 pm on Thursdays in 343 Kemeny, and there will be other speakers than these as well. You are encouraged to attend.

### Staff

Instructor:
Carl Pomerance -- 339 Kemeny / Tel. 6-2635
Office hours: Tuesday and Thursday 9:00 AM--9:55 AM and by arrangement at other times.
There will be no regularly scheduled office hours the week of Oct. 14 nor on Oct. 24.

See above.

### 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

Assignment #1 (due at 11:15 am, Monday 9/23):

(1) Let S be a countable set of positive real numbers such that the number of members of S lying in (0,x] is O(x) for x starting at 2. Show that the sum of reciprocals of members of S lying in this interval is O(log x). (Hint: Use partial summation, also called Abel summation. This goes for the next problem as well. You would have to generalize the identity in the notes and book somewhat. There is another approach to these two problems that's more elementary. For example, if an is the n-th term of the set, then the number of terms up to this number is of course n, but the assumption in the problem says that it is at most can...)

(2) (Continuation) Now suppose that for x starting at 2, the number of members of S in the interval is O(x/(log x)2). Show that the sum of the reciprocals of the members of S is finite.

Do problems 15, 16, 17, 28, 30 in Chapter 1 of the text. (Hints: On number 28, note that in general, gcd(a,b) =gcd(a-b,b). On 30, here's a way not to do the problem, but the neat way is somewhat similar: Using that there's a prime p larger than n/2 and less than or equal to n, we see that p must divide the denominator of the harmonic sum to n when reduced to lowest terms, so the harmonic sum cannot be an integer. The reason you should not use this proof is that we haven't proved the deal about the existence of p.)

Assignment #2 (due at 11:15 am, Monday 9/30):
Do problems 2, 3, and 5 in Chapter 2, plus these supplementary problems. (Note that one of these problems was edited on 9/26.)

Assignment #3 (due at 11:15 am, Monday 10/7):
Problem 12 in Chapter 2 (hint: are both functions multiplicative?)
Also, these problems. (Posted the evening of 10/1 and with a new problem on the morning of 10/2.)

Assignment #4 (due at 11:15 am, Monday 10/14):
Do these problems. (A small edit was made on 10/9.)

Assignment #5 due Monday, October 21 at 11:15 AM: Ch. 6, #14, 15, 17. (In 17b it is to be assumed that f is not identically 0.)

Assignment #6 due Monday, Oct. 28, 2013 at 11:15 AM: Do these problems.

Your assignment for Monday, Nov. 4 is to pick a topic for a class presentation, tell me about it, and have either a meeting with me scheduled or the actual presentation scheduled. For students past quals, this is voluntary.

Assignment for Monday, Nov. 11: Do these problems.

### Exams

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