NMAK16001U Analytic Number Theory (AnNum)

Volume 2024/2025

MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject


The prime number theorem gives an estimate for the number of primes less than a given value x. This theorem - which we will prove - is intimately related to the location of the zeroes of the famous Riemann zeta function. We shall study the analytic properties of the Riemann zeta functions as well as more general L-function. We consider primes in arithmetic progressions, zero-free regions, the famous Riemann hypothesis, the Lindelöf hypothesis, and related topics.

Learning Outcome

At the end of the course students are expected to have a thourough knowledge about results and methods in analytic number theory as described under course content.

At the end of the course students are expected to be able to 

  • Analyze and prove results presented in analytic number theory
  • Prove results similar to the ones presented in the course
  • apply the basic techniques, results and concepts of the course to concrete examples and exercises. 

At the end of the course students are expected to be able to

  • Explain and reproduce abstract concepts and results in analytic number theory
  • Come up with proofs for result at the course level
  • discuss topics from analytic number theory


Complex Analysis (KomAn) or equivalent

Academic qualifications equivalent to a BSc degree is recommended.
Weekly: 4 hours of lectures and 2 hours of exercises for 7 weeks.
  • Category
  • Hours
  • Lectures
  • 28
  • Preparation
  • 114
  • Exercises
  • 14
  • Exam
  • 50
  • Total
  • 206
7,5 ECTS
Type of assessment
Oral examination, 20 minutes (20-minute preparation time)
Exam registration requirements

To be allowed to take the oral exam the student should have at least 3 out of 4 hand-in exercises approved.

All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Same as ordinary exam.

To be eligible for the re-exam, students who did not get 3 out of 4 assignments approved during the ordinary term time can re-submit non-approved assignment. Deadline for this is two weeks before the beginning of the re-exam week.

Criteria for exam assesment

The student must in a satisfactory way demonstrate that they have mastered the learning outcome of the course.