Lectures: Mondays, Wednesdays, and Fridays, 10-11 AM, room 157 in the Barber Centre
Office hours: through Piazza, or by appointment
Office: MATH 212 (Mathematics Building)
Email address: use Piazza instead, or

Course description: This course covers the fundamental techniques in classical analytic number theory. The objects of study are the natural numbers; the theorems sought are statistical statements about the distribution of primes, the number of divisors of integers, and similar multiplicative questions; the techniques involve both “by hand” real analytic estimation and contour integration of meromorphic functions. The successful student will be well-equipped to understand much of the current research literature in this area.

Prerequisites: Students should have had a previous course in number theory (preferably MATH 537 here at UBC). It will be assumed that the student has had the usual undergraduate training in analysis (for example, MATH 320) and a strong course in complex analysis (preferably MATH 508). In particular, in complex analysis students should have a working knowledge of the residue theorem, logarithmic derivatives, and the argument principle. Students will also need to have a working knowledge of LaTeX, although this can be acquired along the way if necessary.

Course textbook: This course will require the book by Montgomery and Vaughan, Multiplicative Number Theory I: Classical Theory (Cambridge University Press, 2006). Please let me know if you encounter problems buying the textbook from the UBC bookstore. Here is a link to Hugh Montgomery's home page, at which you can access a list of errata for the book. If you find an error not on the list, you should email him!

Topics to be covered in this course:

  • Dirichlet series and the Mellin transform
  • Arithmetical functions and their summation and estimation
  • Prime counting functions and Chebyshev's and Mertens's estimates
  • The Riemann zeta function and its zeros
  • The prime number theorem and applications
  • Dirichlet characters and L-functions
  • The prime number theorem in arithmetic progressions

Use of the internet: We will be using Piazza for all class-related announcements and discussion. Piazza is a question-and-answer platform specifically designed to expedite answers to your questions, using the collective knowledge of your classmates and instructor. It has several features that facilitate discussion of mathematics, most notably LaTeX support. You are encouraged to answer your classmates' questions, or to brainstorm towards answers, every bit as much as you are encouraged to ask questions.

No handouts will be distributed in class. All homework assignments and any other course materials will be posted on Piazza in PDF format. I encourage you to ask questions on Piazza any time you're working towards understanding a concept; you can even do so anonymously if necessary. Part of the reason I don't have regularly scheduled office hours is that many questions can be answered through Piazza; in fact, I prefer using Piazza instead of email for questions related to the course (Piazza allows you to ask privately if necessary).

Evaluation: The course mark will be based on eight homework assignments, due approximately every five class days (a little more often than once every two weeks). Your homework will be marked on correctness, completeness, rigor, and elegance. A correct answer will not earn full marks unless it is completely justified, in a rigorous manner, and written in a logical sequence that is easy to follow and confirm. Survival tip: don't start these assignments the night before they're due! Anecdotal evidence suggests that each assignment could take as much as 15 hours or more to complete.

Homework solutions must be prepared in LaTeX and submitted in PDF format via email. I will supply LaTeX templates with each assignment. All homeworks are due before the beginning of class (9:59 AM) on the indicated days.

  • Homework #0: due Friday, January 13
  • Homework #1: due Wednesday, January 25
  • Homework #2: due Monday, February 6
  • Homework #3: due Friday, February 17
  • Homework #4: due Friday, March 9
  • Homework #5: due Monday, March 19
  • Homework #6: due Friday, March 30
  • Homework #7: due Monday, April 16

Students are allowed to consult one another concerning the homework problems, but your submitted solutions must be written by you in your own words. Students can be found guilty of plagiarism if they submit virtually identical answers to a question, or if they do not understand what they have submitted.

Because there are no exams, the lectures will continue into the beginning of the final exams period, ending on (probably) Wednesday, April 18.

Other possible references for analytic number theory:

  • H. Davenport, Multiplicative Number Theory
  • T. M. Apostol, Introduction to Analytic Number Theory
  • H. Iwaniec and E. Kowalski, Analytic Number Theory
  • P. T. Bateman and H. G. Diamond, Analytic Number Theory: An introductory course

Possible references for elementary number theory:

  • I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers