Lectures: Tuesdays and Thursdays, 11:00 AM–12:20 PM, room MATH 225 (Mathematics Building)
Office hours: right after class, by appointment, or drop in
Office: room MATH 212 (Mathematics Building)
Email address: gerg@math.ubc.ca

Lecture notes:

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 power series, poles and the residue theorem, logarithmic derivatives and the argument principle, and analytic continuation. Students with incomplete mastery of these topics are welcome—they should just expect to brush up on these background topics as needed when we encounter them.

Course textbook: The recommended textbook for this course is the book by Montgomery and Vaughan, Multiplicative Number Theory I: Classical Theory (Cambridge University Press, 2006). Here is a link to Hugh Montgomery's home page, at which you can access a list of errata for the book.

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
  • (if time permits) Dirichlet characters and Dirichlet's theorem on primes in arithmetic progressions

Other possible references for analytic number theory:

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

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

Evaluation: The course mark will be based on attendance, particularly of group work done in class about three times a month. There will be no official homework or final exams, though I will post lists of suggested problems to work on every couple of weeks. Please notify me in advance (by email) if you ever have to miss a class.

Group work schedule:

Suggested problems:

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