MATH 8440: Analytic and Combinatorial Number Theory

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: 11:00 AM W, 3:00 PM Th

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:

Identify and state foundational theorems in analytic number theory
Articulate the major components of the proofs of these theorems
Apply the methods to solve non-routine problems at the level needed to gain a foothold in the research literature

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.

Course materials and due dates
Assignment	Due date	File
Homework #3	October 16	Download PDF
Homework #2	September 25	Download PDF
Homework #1	~~September 2~~ September 4	Download PDF

Course record

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

Dates and topics covered in MATH 8440 during Fall 2026
Date	Topic
October 5	The constant in Mertens' product theorem. The guesstimates of Legendre and Gauss for counts of primes.
October 2	Mertens' sum theorem. Mertens' product theorem.
September 30	Montgomery--Wagon's heuristic for the PNT, continued. How not to prove the PNT.
September 28	Start of discussion of Montgomery--Wagon's heuristic for the PNT.
September 21	Proof of Bertrand's postulate in the following asymptotic form: For each fixed $c_{3} < \frac{1}{3} \log 2$ and all large $x$ , the interval $(\frac{x}{2}, x]$ contains more than $\frac{c_{3} x}{\log x}$ primes. Another heuristic for the prime number theorem: If $\frac{π (x)}{x}$ “decreases steadily” to $0$ , then $π (x) \sim \frac{x}{\log x}$ , as $x \to \infty$ .
September 18	$Proof of Chebyshev's upper and lower bounds for π (x) .$
September 16	Analysis of the size of $⌊ x ⌋!$ . Prime factorization of $⌊ x ⌋!$ . Deduction that $\sum_{p^{k} \leq x}^{} \log p ⌊ \frac{x}{p^{k}} ⌋ = x \log x - x + O (\log x)$ . Reformulation in terms of $ψ (x) + ψ (\frac{x}{2}) + ψ (\frac{x}{3}) + \dots$ , where $ψ (t) = \sum_{p^{k} \leq t}^{} \log p$ .
September 14	One reason you might guess that $π (x) \sim \frac{x}{\log x}$ , as $x \to \infty$ . $p$ -adic valuations, $p$ -adic absolute values, and the product formula. Euclid's proof from the point of view of the product formula.
September 11	Completion of discussion of $2^{n} + k$ . Pseudoprimes and Carmichael numbers.
September 9	Some correspondence between Goldbach and Euler. If $f (x) \in Z [x]$ is nonconstant, then $f (2^{n})$ is composite for some positive integer $n$ . A polynomial in 26 variables whose positive range is precisely the set of prime numbers. Bunyakovsky's conjecture. Iwaniec's theorem on $n^{2} + 1$ . Start of discussion of primes of the form $2^{n} + k$ .
September 4	Proof there are infinitely many primes through understanding the maximal ideals of $\mathbb{Z}[x]$ . Washington's proof via Dedekind domains and commutative algebra. Goldbach's theorem that there are no (one-variable) polynomials that only represent primes.
September 2	The estimate $Ψ (x, 3) = \frac{1}{2 \log 2 \log 3} {(\log x)}^{2} + O (\log x)$ , via counting lattice points. Generalizing to $Ψ (x, p_{k})$ and the deduction that there are infinitely many primes. Discussion of maximal ideals in $Z [x]$ .
August 31	More on formal Dirichlet series. Möbius inversion as inverting $ζ (s)$ . Why we define $μ$ the way we do.
August 28	The Möbius function. Detecting 1 with Möbius and the Möbius inversion formula. An uncertainty principle for the Möbius function. Start of discussion of formal Dirichlet series.
August 26	Infinitely many primes from $\frac{{ζ (2)}^{2}}{ζ (4)} = \frac{5}{2}$ . A positive proportion of numbers are squarefree. A proof of Erdős that $π (N) \geq \frac{\log N}{\log 4}$ for all positive integers $N$ . Erdős's proof that $\sum_{p} \frac{1}{p}$ diverges.
August 24	Proof that $\sum_{p \leq x} \frac{1}{p} > \log \log x - 1$ . Upper bounds on $\sum_{p \leq x} \frac{1}{p}$ (first steps). Euler's solution of the Basel problem. Hacks's proof that there are infinitely many primes.
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 $\sum_{p} \frac{1}{p} = \log \log \infty$ .
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

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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.

Well-being Resources
Student Care and Outreach
University Health Center
Counseling and Psychiatric Services; CAPS 24/7 crisis support: 706‑542‑2273
Health Promotion and the Fontaine Center
Accessibility and Testing

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.