Math 539: Analytic Number Theory

This course is a PIMS Network Course and is also open to the public

Spring Term 2027

General Information

Office: MATX 1112, 604-827-3031
Email: lior@math.ubc.ca; please include the course number in the subject line, if applicable.
Office hours (AY 2025-2026): by appointment (I'm on sabbatical)

Classes: TTh 9-10:30 at ESB 4127 and on Zoom – see schedule below.
Syllabus (PDF)
(Rough) lecture notes (PDF)
There is a Piazza discussion board; contact the instructor for the sign-up link.
Here's an English translation of Riemann's 1859 paper.

We shall count (that is, estimate the number of) integer and prime number solutions to equations. We will being with combinatorial (“elementary”) methods, continue to Fourier analysis, and finally use zeta-function (contour integration) techniques. The main pre-requisites are Elementary Number Theory and Real and Complex analysis (say at the level of UBC MATH 537, 320, and 508, respectively). We will use some basic ideas from ring theory and finite abelian groups, and will develop all the Fourier analysis we will use.

References

H. Davenport, Multiplicative Number Theory.
H. Montgomery and R. Vaughn, Multiplicative Number Theory I: Classical Theory.
H. Iwaniec and E. Kowalski, Analytic Number Theory.

Problem Sets

Problem sets are due at 9:00 Vancouver time (start of class) on the date indicated. Please submit typeset solutions as PDF files, either on Canvas or by email. In the second case the subject line must be: "MATH 539 PSxx submission" where xx is the number of the problem set.
- The source files for the problem sets rely on my macro file. Feel free to use it for your submissions as well.
We will figure out how to distribute solutions.

Lecture-by-Lecture information

Warning: the following information is tentative and subject to change at any time

Chapter	Week	Date	Material
Elementary Counting	1	T 5/1	Introduction Arithmetic functions
	1	Þ 7/1	Averages of arithmetic functions
	2	T 12/1	Divisor switching Counting primes
Fourier methods	2	T 7/1	$ℤ / N ℤ$ Roth's Theorem
	3	Þ 19/1	(continued)
	3	Þ 21/1	${(ℤ / N ℤ)}^{\times}$ Dirichlet characters
	4	T 26/1	Primitive Characters Dirichlet's Theorem
	4	Þ 28/1	(continued) Gauss's sum
	5	T 2/2	$ℝ / ℤ$ Poisson Sum
	5	Þ 4/2	Pólya--Vinogradov
	6	T 9/2	$ℝ$ Fourier inversion
The Prime Number Theorem	6	Þ 11/2	Mellin transform Zetafunction counting
	Feb 17-21		UBC Winter break
	7	T 23/2	Multiplicative smoothing
	7	Þ 25/2
	8	T 2/3
	8	Þ 4/3
	9	T 16/3
	9	Þ 18/3
	10	T 23/3
	10	Þ 25/3
Further Topics	11	T 31/3
	11	Þ 1/4
	12	T 6/4
	12	Þ 8/4

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Last modified Friday June 12, 2026