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. Booker |

Open unit status | Not open |

Pre-requisites |
MATH33000 Complex Function Theory |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**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.

**Unit Description**

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 two units which develop number theory in various directions. The other is Number Theory.

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.

Lectures and exercises.

90% Examination

10% Coursework

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.

**Recommended**

- Tom M. Apostol,
*Introduction to Analytic Number Theory*, Springer, 1976 - Harold Davenport,
*Multiplicative Number Theory*, Third edition, Springer, 2000 - Hugh L. Montgomery and Robert C. Vaughan,
*Multiplicative Number Theory: I. Classical Theory*, Cambridge University Press, 2007 - Gérald Tenenbaum,
*Introduction to Analytic and Probabilistic Number Theory*, Cambridge University Press, 1995