University of Georgia · Department of Mathematics

MATH 8440: Topics in Analytic and Combinatorial Number Theory

Fall 2026

Instructor
Paul Pollack
Email
pollack@uga.edu
Class meetings
Monday, Wednesday, and Friday, 1:15–2:10
Class location
Boyd 640
Office
Boyd 406
Office hours
To be announced

Overview

Bulletin description

Topics in combinatorial and analytic number theory, such as sieve methods, probabilistic models of prime numbers, the distribution of arithmetic functions, the circle method, additive number theory, and transcendence methods.

Topical outline

This version of MATH 8440 is an introduction to analytic methods in number theory, with an emphasis on concrete, easily described problems. Possible topics include:

  • Elementary results on the distribution of prime numbers, from Euclid to Euler to Chebyshev to Mertens
  • Dirichlet’s theorem on prime numbers in arithmetic progressions
  • Elementary sieve methods and their applications, including Brun’s theorem on twin primes and Schnirelmann’s theorem toward Goldbach’s conjecture
  • Character sums and the distribution of power residues
  • The distribution of arithmetic functions, with particular attention to the number of prime divisors, number of divisors, and sum of divisors
  • The prime number theorem

Prerequisites

Prerequisites are minimal. Familiarity with algebra and analysis, both real and complex, at the advanced undergraduate level is assumed. Prior acquaintance with elementary number theory will be helpful but is not essential.

Timeline

The last time this course was offered, it proceeded roughly as follows: elementary theory of primes (two weeks); the weak forms of the prime number theorem proved by Chebyshev and Mertens (two weeks); Dirichlet’s theorem on primes in arithmetic progressions and applications (three weeks); an introduction to sieve methods (four weeks); and Waring’s problem (four weeks).

But this time will surely be different!

Learning outcomes

By the end of the course, students should be able to:

Textbook

Cover of Not Always Buried Deep by Paul Pollack

The required main reference is the instructor’s book:

Not Always Buried Deep: A Second Course in Elementary Number Theory (American Mathematical Society, 2009).

No purchase is required. Thanks to the kind permission of the American Mathematical Society, the book can be downloaded free of charge.

Course materials

Homework assignments, handouts, and other course materials will be posted here, usually as PDF files.

No course materials have been posted yet.

Course record

The table below will be updated after class meetings. Newest entries will appear first.

Date Topic
August 21 Arithmetic functions. Multiplicative functions, and examples. Euler's factorization theorem. Application: There are infinitely many primes. In fact, the sum of reciprocals of the primes diverges. Euler's claim that ∑ p 1 p = log ⁡ log ⁡ ∞ .
August 19 Furstenberg's proof. Coprime integer sequences. Goldbach's proof. Prime divisors of Fermat numbers. Infinitude of primes congruent to 1 mod powers of 2. A folklore proof with Mersenne numbers.
August 17 Go over syllabus. Euclid's proof. Stieltjes's version. A lesser-known proof of Euclid's lemma. Braun's proof.

Homework, exams, and grading

There are no exams in this class. Your grade is based entirely on homework. If you decide to concentrate in number theory or arithmetic geometry, the material in this course may well form part of your oral exams, and I encourage you—implore you!—to take the homework seriously.

You should expect homework assignments roughly every two weeks, on average. Each problem will carry a point value. To receive an A in the class, you should complete at least roughly half of the assigned problems. Students who attend class but do not turn in homework will receive a B at the end of the term.

In general, late homework is not accepted, but I am happy to discuss individual circumstances on a case-by-case basis.

Attendance is required. Missing more than four classes may result in automatic withdrawal.

Policies and student support

UGA well-being resources

UGA Well-being Resources promote student success by cultivating a culture that supports a more active, healthy, and engaged student community.

Anyone needing assistance is encouraged to contact Student Care and Outreach in the Division of Student Affairs at 706‑542‑8479. Student Care and Outreach helps students navigate difficult circumstances by connecting them with the most appropriate resources or services. They also administer the Embark@UGA program, which supports students experiencing, or who have experienced, homelessness, foster care, or housing insecurity.

UGA provides clinical and non-clinical options to support student well-being and mental health—any time, any place. Whether on campus or studying from home or abroad, UGA Well-being Resources are here to help.

Additional information, including free digital well-being resources, is available through the UGA app and the UGA Well-being Resources website.

Academic honesty and AI statement

All UGA students are bound by the Honor Code:

“I will be academically honest in all of my academic work and will not tolerate academic dishonesty of others.”

A Culture of Honesty contains the University’s policy and procedures for handling cases of suspected dishonesty.

What that means for this class: Everything you turn in has to reflect your own understanding of the material. Were you interrogated alone in a room with nothing but your wits and your answer sheet, you should be able to explain what you wrote and why you wrote it. But how you come to that understanding is left up to you. I strongly encourage you to come to me with questions and discuss the material with your classmates. You are also free to consult textbooks and, yes, AI tools.

Syllabus changes

This syllabus is a general plan for the course. Deviations announced to the class by the instructor may be necessary.