Mathematics 539, Analytic Number Theory I, Section 101

also offered as 430C (Topics in Analysis)

Term 1, 2003, MWF 9-10

  • Instructor : David Boyd
    Office: 200 Mathematics
    Office hours: Mon 1:00-2:00 and Weds 1:00-2:00
    e-mail: boyd at math dot ubc dot ca

    URL of this web page:

  • Textbook: T. M. Apostol, Introduction to Analytic Number Theory
  • It is definitely worth purchasing the text (see below).
    There is one copy of the text on one day loan in the Mathematics Library.

  • Course description:
    The course is a first course in analytic number theory suitable for 4th year honours mathematics students and 1st year graduate students or 2nd year graduate students. We will assume that students have had a previous course in number theory (preferably 537 = 437, which was offered in 2002-2003, but even a course similar to 312 which is taught at many universities would be acceptable for a student who has the necessary background in analysis). It will be assumed that the student has had the usual undergraduate training in analysis (e.g. Math 320) and complex analysis (e.g. Math 300) to the level of the residue theorem, but the complex analysis course could be taken concurrently.
    The main goal will be to prove Dirichlet's theorem on primes in arithmetic progressions and the prime number theorem.

  • Assignments and Supplementary course material:
    There will be no paper handouts in class. Any such material will be linked to this web page and be in pdf format. To read pdf you need Adobe's free acrobat reader.
    Most of the exercises on the assignments will be taken from the text or adapted from those in the text. You are encouraged to try other exercises from the text.

  • Marking:
    The course mark will be based entirely on the assignments which will assigned at regular intervals. Be sure to do these and hand them on on time!
    Assignment #1, due Monday, September 22
    Assignment #2, due Monday, October 6
    Assignment #3, due Monday, October 20
    Assignment #4, due Monday, November 3
    Assignment #5, due Monday, November 17
    Assignment #6, due Friday, November 28

  • Course outline (chapter references to the text in parentheses):
    1. Arithmetical Functions and their estimation (2,3)
    2. The prime counting function and Chebyshev's estimates (4)
    3. Dirichlet series (11)
    4. Dirichlet characters and Dirichlet L-functions (6)
    5. Dirichlet's theorem on primes in arithmetic progressions (7)
    6. The Riemann zeta function (12)
    7. The prime number theorem (13)

  • Other material in the text:
    The Historial Introduction is well worth reading.
    We assume that the material of Chapters 1, 5, 9 and 10 is familiar to the student from an earlier course. You should read over the material of Chapter 1 before the course begins and the material of Chapters 5, 9 and 10 before we begin the discussion of Dirichlet characters and the Dirichlet L-function.
    We introduce Dirichlet series in a formal way earlier than is done in the text since this makes some of the manipulations of the earlier chapters more transparent. But the serious use of analytic functions does not begin until we begin to study the Riemann zeta function and Dirichlet L-functions.

  • References:
    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
    K. Chandraekharan, Introduction to Analytic Number Theory
    H. Davenport, Multiplicative Number Theory
    L. K. Hua, Introduction to Number Theory