Prime Numbers and their Distribution - MAT00109M
Module summary
Prime numbers are one of the most basic mathematical objects known to humankind. However, studying them utilises an unusually wide variety of methods and proofs, many of which are surprisingly deep. This module will culminate in a proof of the Prime Number Theorem, the highlight of 19th Century mathematics, requiring knowledge of the complex-analytic properties of the Riemann zeta function.
Related modules
Additional information
M-level cannot be taken if H-level was taken
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 2 2025-26 |
Module aims
Prime numbers are one of the most basic mathematical objects known to humankind. However, studying them utilises an unusually wide variety of methods and proofs, many of which are surprisingly deep. This module will culminate in a proof of the Prime Number Theorem, the highlight of 19th Century mathematics, requiring knowledge of the complex-analytic properties of the Riemann zeta function.
Module learning outcomes
By the end of the module, students should be able to:
-
Demonstrate facility with the unusually wide variety of methods and proofs which appear in number theory.
-
Apply complex analytic techniques to deduce number theoretic results.
-
Derive some of the analytic properties of the Riemann zeta function
-
Utilise the relationship between zeros of the Riemann zeta function and properties of prime numbers, specifically in the context of the proof of the Prime Number Theorem.
-
Demonstrate the ability to reason in a rigorous, precise and logical manner.
Module content
-
Multiplicative functions and average orders
-
Analytic properties of Dirichlet series and Euler products, especially the Riemann zeta function.
-
Zeros of the Riemann zeta function.
-
Proof of the Prime Number Theorem.
- [M-level only] Dirichlet L-functions and primes in arithmetic progressions.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Module feedback
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
Indicative reading
-
Apostol, Introduction to Analytic Number Theory, (Springer-Verlag New York)
-
Miller and Takloo-Bighash, An Invitation to Modern Number Theory, (Princeton University Press) (S 2.81 MIL)
-
Edwards, Riemann's Zeta Function (Dover Publications) (S 7.36 EDW)
-
Iwaniec and Kowalski, Analytic number theory (American Mathematical Society) (S 2.81 IWA)
- Montgomery and Vaughan, Multiplicative number theory I : classical theory (Cambridge University Press) (S 2.814 MON)