Topics in Number Theory
Instructor: Carl Pomerance (carl.pomerance at dartmouth.edu)
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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.
Introduction to analytic number theory
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.
Room: 004 Kemeny?
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:
Homework is due at the start
of the class period on the due date.
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):
Assignment #3 (due at 11:15 am, Monday 10/7):
Assignment #4 (due at 11:15 am, Monday 10/14):
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.
There will be no exams, but homework will be graded and students will be expected to make some presentations to the class.
Grades will be based on homework, class participation, and oral presentations. (Graduate students who are working towards their dissertations will be judged on a separate standard.)
Collaboration on homework is definitely allowed and even encouraged. However, it is tempting to think that you understand something that was figured out by your friend. When you hand in a solution, you should know it well enough that you could explain it to others. Please name others you worked with when handing in homework papers. Merely copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code, even if attribution is made. You should understand what you turn in.
I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.
The Student Disabilities Coordinator, Nancy Pompian, can be reached at 6-2014 if you have any questions. Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.