NMAK16001U Analytic Number Theory (AnNum)

Volume 2022/2023

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
Type of assessment details
Oral examination with 20 minutes 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 he/she has mastered the learning outcome of the course.