MATH 540: Topics in Number Theory

MATH 540, Section 201: Topics in Number Theory

Announcements

Lectures will continue at the usual time and place through Friday, April 20.

On Monday, March 12, class will be moved to MATX 1118.

There will be no class on Friday, March 2.

There will be no class on Friday, February 2.

Course information

When: Mondays, Wednesdays, and Fridays, 2:00–2:50 PM
Where: MATX 1102 (Math Annex building)
Prerequisites: MATH 539 or equivalent
Course web page: http://www.math.ubc.ca/~gerg/teaching/540-Winter2007

Instructor: Prof. Greg Martin
Office: MATH 212 (Mathematics Building)
Email address: [an error occurred while processing this directive]
Phone number: (604) 822-4371
Office hours: Wednesdays 3–4 PM and Fridays 11 AM–noon, or drop in

Course description: This is a topics course in number theory entitled “Distribution of prime numbers and zeros of Dirichlet L-functions”. The twin themes of the course are to understand as well as possible the distribution of the zeros of Dirichlet L-functions (including the Riemann zeta-function), and then to use this knowledge to derive results on the distribution of prime numbers, with particular attention to their distribution within arithmetic progressions.

Topics will include:

The explicit formula for the number of primes in a finite arithmetic progression
The vertical distribution of zeros of Dirichlet L-functions; the function N(T,χ)
Zero-free regions for Dirichlet L-functions, both classical and more modern
February 5–9: Siegel zeros
February 12–16: Clumps of zeros and the Deuring–Heilbronn phenomenon
February 26–March 2: The large sieve and the Bombieri–Vinogradov Theorem
March 5–9: The horizontal distribution of zeros of Dirichlet L-functions: zero-density theorems
March 12–16: Linnik's Theorem on the least prime in an arithmetic progression
March 19–23: Zeros on the critical line
March 26–30: Primes in short intervals; irregularities of distribution (the Maier matrix method)
April 2–4: Error terms in lattice-point problems
April 11–13: Oscillations of error terms, Littlewood's results
April 16–20: Limiting distributions of almost-periodic functions and prime number races

Evaluation: Each student will be expected to give two presentations, each of length 40-50 minutes. To accommodate the student presentations, we might continue meeting for a week or two after classes formally end. Students who wish to substitute a written report for one of the two presentations can discuss it with the instructor.

Last modified Wednesday, 11-Apr-2007 12:47:19 PDT