Section: Course Materials | C3.8 Analytic Number Theory (2025-26)

Lecturer: Profile: Ben Green

Course information

General prerequisites:

Basic ideas of complex analysis. Elementary number theory. Some familiarity with Fourier series will be helpful but not essential.

Course term: Michaelmas

Course lecture information: 16 lectures

Course weight: 1

Course level: M

Assessment type: Written Examination

Course overview:

The aim of this course is to study the prime numbers using the famous Riemann \(\zeta\)-function. In particular, we will study the connection between the primes and the zeros of the \(\zeta\)-function. We will state the Riemann hypothesis, perhaps the most famous unsolved problem in mathematics, and examine its implication for the distribution of primes. We will prove the prime number theorem, which states that the number of primes less than \(X\) is asymptotic to \(X\log X\).

Learning outcomes:

In addition to the highlights mentioned above, students will gain experience with different types of Fourier transform and with the use of complex analysis.

Course synopsis:

Introductory material on primes. Arithmetic functions: Möbius function, Euler's \(\phi\)-function, the divisor function, the \(\sigma\)-function. Multiplicativity. Dirichlet series and Euler products. The von Mangoldt function.
The Riemann \(\zeta\)-function for \(Re (s) > 1\). Euler's proof of the infinitude of primes. \(\zeta\) and the von Mongoldt function.
Schwarz functions on \(\mathbf{R}\), \(\mathbf{Z}\), \(\mathbf{R}/\mathbf{Z}\) and their Fourier transforms. *Inversion formulas and uniqueness*. The Poisson summation formula. The meromorphic continuation and functional equation of the \(\zeta\)-function. Poles and zeros of \(\zeta\) and statement of the Riemann hypothesis. Basic estimates for \(\zeta\).
The classical zero-free region. Proof of the prime number theorem. Implications of the Riemann hypothesis for the distribution of primes.

Section outline

- Select activity Lecture notes
  
  Lecture notes File
- Select activity Sheet 0
  
  Sheet 0 File
  
  This sheet gives some practice/revision with basic analytic estimates and the O() notation. It will not be marked and there will be no class on it but I am happy to go over questions after lectures. At some point during the term I will make solutions available.
- Select activity Sheet 1
  
  Sheet 1 Assignment
  
  This is sheet 1 for the first class. It covers material up to the end of Section 2 of the notes.
  
  The sheet is available as sheet1.pdf (without solutions), or as sheet1-AC.pdf which has solutions to Sections A and C. Full solutions will be made available after the classes.
- Select activity Sheet 2
  
  Sheet 2 Assignment
  
  Sheet 2 covers material up to the end of Section 3 of the notes and reinforces some of the earlier material. As usual, it is initially available in two forms: with no solutions, and with solutions to Sections A and C.
- Select activity Sheet 3
  
  Sheet 3 Assignment
  
  Sheet 3 covers material up to the end of Section 7 of the notes.
- Select activity Sheet 4
  
  Sheet 4 Assignment
  
  Sheet 4 covers material from the whole course but with an emphasis on the later sections.

Main content blocks

Section outline