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Analytic Number Theory - MAT00051M

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  • Department: Mathematics
  • Module co-ordinator: Dr. Christopher Hughes
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2021-22

Related modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Pre-requisite knowledge for MSc students: good knowledge of calculus; first courses in both real and complex analysis.

Module will run

Occurrence Teaching period
A Autumn Term 2021-22

Module aims

  • To show how analytic properties of the Riemann zeta function lead to results on prime numbers.

  • To understand properties of Dirichlet L-functions, and their relation to primes in arithmetic progressions.

Module learning outcomes

At the end of the module you should be able to:

  • To understand how complex analytic techniques enable number theoretic results to be deduced.

  • To know some of the analytic properties of the Riemann zeta function

  • Appreciate the connection between zeros of the Riemann zeta function and properties of prime numbers.

  • To know the prime number theorem and the relation between its error term to the location of zeros of the zeta function.

  • To understand Dirichlet characters and analytic properties of Dirichlet L-functions

  • To know Dirichlet’s theorem on primes in arithmetic progressions

Module content



  • Arithmetic functions, partial summation, Dirichlet series and Euler products.
  • The Riemann zeta function, its functional equation and analytic continuation. Evaluation at even integers.
  • Zeros of the zeta function, and the Riemann Hypothesis.
  • The Prime Number Theorem.
  • Dirichlet characters and Gauss sums.
  • Dirichlet L-functions, their analytic continuation and functional equations.
  • Primes in arithmetic progressions.


Task Length % of module mark
Online Exam -less than 24hrs (Centrally scheduled)
Analytic Number Theory
2 hours 100

Special assessment rules



Task Length % of module mark
Online Exam -less than 24hrs (Centrally scheduled)
Analytic Number Theory
2 hours 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)
  • Mezzadri and Snaith (eds), Recent Perspectives in Random Matrix Theory and Number Theory, LMS Lecture Notes Series, 322. (S 2.81 MEZ)

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.