Lectures: Mondays, Wednesdays, and Fridays, 10:00-10:50 AM, room WMAX 216 (PIMS)
Office hours: by appointment (in person or by Skype)
Office: MATH 212 (Mathematics Building)
Email address:
Phone number: (604) 822-4371

Course description: This is a topics course in number theory, called “Analytic number theory II” or “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. The course will begin with a quick review of the prime number theorem and its analogue for arithmetic progressions.

Advertisement: There will be a conference on L-functions and their applications to number theory at the University of Calgary from May 30–June 3, 2011. Students who take this course should be well-prepared to get a lot out of that conference. Contact the instructor if you are interested in attending.

Prerequisites: Students should have had a previous course in analytic number theory (for example, MATH 539 here at UBC). The background of students should include the following elements, all of which should be present in those who succeeded in MATH 539: a strong course in elementary number theory (for example, MATH 537), a graduate course in complex analysis (for example, MATH 508), and the usual undergraduate training in analysis (for example, MATH 320).

Evaluation: Each student will deliver three lectures for the course, and write up (in LaTeX) lecture notes corresponding to another student's three lectures. The lectures will be chosen by the student in consultation with the instructor from the list below; most students will choose to deliver consecutive lectures on the same topic.

The last day of classes is April 6, 2011; however, because there are no exams, the lectures will continue into the beginning of the final exams period to accommodate as many students and topics as possible.

Dates Speaker Topic Writer Draft due Article due
Jan 10–12 Greg Organization and introduction
Jan 17–19 Everyone Four-minute talks (all topics)
Jan 21–28 Greg Review on L-functions and primes in arithmetic progressions: explicit formula, zero-free region, exceptional zeros
Jan 29–Feb 4 Greg Primes in short intervals; irregularities of distribution (the Maier matrix method)
Feb 7–11 Nick Linnik's Theorem on the least prime in an arithmetic progression Colin Feb 21 Feb 28
Feb 14–18 (no class)
Feb 21–25 Justin, Greg Zeros on the critical line Eric Mar 28 Apr 4
Feb 28–Mar 4 Tatchai The large sieve and the Bombieri–Vinogradov Theorem Nick Mar 14 Mar 21
Mar 7–11 Daniel The least quadratic nonresidue and the least primitive root modulo primes (unconditional and conditional results) Carmen Mar 21 Mar 28
Mar 14–18 Eric Analytic number theory without zeros (current work of Granville/Soundararajan)
Mar 21–25 Li Oscillations of error terms, Littlewood's results Tatchai Apr 4 Apr 11
Mar 28–30 Justin, Greg The nonvanishing of L-functions at the critical point and on the real axis
Apr 4–8 Carmen Limiting distributions of explicit formulas and prime number races Daniel Apr 18 Apr 25
Apr 11–15 Colin The Selberg class of L-functions Li Apr 25 May 2
Apr 18 Greg Horizontal distribution of zeros of Dirichlet L-functions; zero-density theorems
Apr 20 Greg Proofs of the prime number theorem that avoid the zeros of ζ(s)

References for these topics:

  • H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory: I. Classical theory (some errata have been posted online)
  • Note: draft versions of some chapters from their future sequel Multiplicative Number Theory: II. Modern developments can be found on the web.
  • H. Iwaniec and E. Kowalski, Analytic Number Theory
  • E. C. Titchmarsh (revised by D. R. Heath-Brown), The Theory of the Riemann Zeta-Function
  • The primary research literature (you can find references in the above books or by speaking with the instructor), almost all of which is searchable at MathSciNet

Possible references for fundamental analytic number theory:

  • H. Davenport, Multiplicative Number Theory
  • A. E. Ingham, The Distribution of Prime Numbers
  • 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

Use of the web: After the first day, no handouts will be distributed in class. All 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.