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Unit information: Analytic Number Theory in 2015/16

Please note: you are viewing unit and programme information for a past academic year. Please see the current academic year for up to date information.

Unit name Analytic Number Theory
Unit code MATHM0007
Credit points 20
Level of study M/7
Teaching block(s) Teaching Block 2 (weeks 13 - 24)
Unit director Professor. Tim Browning
Open unit status Not open

Complex Function Theory (Math 33000), Number Theory and Group Theory (Math 11511)



School/department School of Mathematics
Faculty Faculty of Science

Description including Unit Aims

Unit aims

To gain an understanding and appreciation of analytic number theory, and some of its most important achievements. To be able to apply the techniques of complex analysis to study a range of specific problems in number theory.

General Description of the Unit

The study of prime numbers is one of the most ancient and beautiful topics in mathematics. After reviewing some basic results in elementary number theory and the theory of Dirichlet characters and L-functions, the main aim of this lecture course will be to show how the power of complex analysis can be used to shed light on irregularities in the sequence of primes. Significant attention will be paid to developing the theory of the Riemann zeta function. The course will build up to a proof of the Prime Number Theorem and a description of the Riemann Hypothesis, arguably the most important unsolved problem in modern mathematics.

Relation to Other Units

This is one of three Level 6 and Level 7 units which develop number theory in various directions. The others are Number Theory and Algebraic Number Theory.

Further information is available on the School of Mathematics website:

Intended Learning Outcomes

Learning Objectives

To gain an understanding and appreciation of Analytic Number Theory and some of its important applications. To be able to use the theory in specific examples.

Transferable Skills

Using an abstract framework to better understand how to attack a concrete problem.

Teaching Information

Lectures and exercises.

Assessment Information

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

  1. T. M. Apostol, Introduction to analytic number theory. Springer, 1976.
  2. J. Brüdern, Einführung in die analytische Zahlentheorie, Springer, 1995.
  3. H. Davenport, Multiplicative Number Theory, third edition, Springer 2000
  4. H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press 2007
  5. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995.