Lectures: Mondays, Wednesdays, and Fridays, 10-11 AM, room MATH 105 (Mathematics Building)
Office hours: by appointment
Office: MATH 212 (Mathematics Building)
Email address:
Phone number: (604) 822-4371

Course textbook: 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!

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.

Evaluation: The course mark will be based on six homework assignments, due approximately every two weeks of class. 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 20 hours or more to complete.

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, through Friday, April 23.

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 web: After the first day, no handouts will be distributed in class. All homework assignments and other course materials will be posted on this course web page. All documents will be posted in PDF format and can be read with the free Acrobat reader. You may download the free Acrobat reader at no cost. You may access the course web page on any public terminal at UBC or via your own internet connection.

All homeworks are due at the beginning of class (10:00 AM) on the indicated days. The links below lead to the most up-to-date versions of the homework (reflecting any needed corrections, for example).

Other possible references for analytic number theory:

  • H. Davenport, Multiplicative Number Theory
  • T. M. Apostol, Introduction to 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