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## ECMM741 - Analytic Number Theory (2018)

MODULE TITLE | Analytic Number Theory | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECMM741 | MODULE CONVENER | Dr Julio Andrade (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 |

Number of Students Taking Module (anticipated) | 16 |
---|

The study of properties of the integer numbers, in particular of prime numbers, is one of the most ancient topics in mathematics. The study of the prime numbers and their distribution is also considered to be one of the most beautiful topics in mathematics. Analytic number theory is the area of mathematics that uses methods from mathematical analysis to solve problems about the integers. After reviewing some basic concepts in elementary number theory, the main aim of this lecture course is to show how powerful mathematical analysis is, in particular, complex analysis, in the study of the distribution of prime numbers.

Prerequisite module: ECM3703

The aim of this course is to introduce you to the theory of prime numbers, showing how the irregularities in this sequence of integer numbers can be tamed by the power of complex analysis. The main aim of the course is to present a proof of the Prime Number Theorem which is the corner-stone of prime number theory. We also will discuss the Riemann Hypothesis, which is arguably the most important unsolved problem in modern mathematics.

From this module, you will acquire a working knowledge of the main concepts of analytic number theory, together with some appreciation of modern results and techniques, and some recent research in the area.

On successful completion of this module ** you should be able to**:

**Module Specific Skills and Knowledge**

2. apply the techniques of complex analysis and analytic number theory to solve a range of seen and unseen problems;

**Discipline Specific Skills and Knowledge**

6. apply a range of techniques from the module with precision and clarity.

**Personal and Key Transferable / Employment Skills and Knowledge**

8. communicate your work professionally, and using correct mathematical notation.

1. Fundamental Theorem of Arithmetic and Some Foundations on Prime Numbers

- Divisibility

- Prime Numbers

- The fundamental theorem of arithmetic

- The Euclidean algorithm

2. Arithmetic Functions

- The Mobius function

- The Euler function

- The Dirichlet convolution

- The Mangoldt function

- Multiplicative functions

3. Averages of Arithmetic Functions

- The big oh notation

- Euler's summation formula

- Some elementary asymptotic formulas

- Average order of some other arithmetic functions

4. Elementary Theorems of the Distribution of Prime Numbers

- Chebyshev's functions

- Equivalent forms of the prime number theorme

- Shapiro's Tauberian theorem

- Sums over primes

5. Dirichlet Characters, Dirichlet Series and Euler Products

- Definitions

- The character group

- Dirichlet characters

- Dirichlet series

- The non-vanishing of L-functions

6. Dirichlet’s Theorem on Primes in Arithmetic Progression

- Dirichlet's theorem for primes of the form 4n-1

- The plan of the proof of Dirichlet's theorem

- Distribution of primes in arithmetic progressions

7. Riemann zeta function

- Definitions

- Analytic continuation

- Functional equation

- Non-vanishing on Re(s) = 1

8. Proof of the Prime Number Theorem

9. The Riemann Hypothesis and its significance

*Some of the topics above are subject to change

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
---|

Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures |

Guided independent study | 117 | Assessment preparation; private study |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Not applicable. | |||

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours | All | Exam mark. Results released online. Individual feedback upon request |

Coursework - problem sheets | 20 | Variable | All | Written and verbal |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-assessment |
---|---|---|---|

All Above | Written Examination (100%) | All | August Ref/Def Period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

**Basic reading:**

**ELE: http://vle.exeter.ac.uk/**

**Web based and Electronic Resources:**

**Other Resources:**

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | T. Apostol | Introduction to Analytic Number Theory | Undergraduate Texts in Mathematics, Springer-Verlag | 1976 | [Library] | ||

Set | H. Davenport | Multiplicative Number Theory | Springer-Verlag, Graduate Texts in Mathematics | 2000 | [Library] | ||

Set | G.H. Hardy and E.M. Wright | An Introduction to the Theory of Numbers | Oxford University Press | 2008 | [Library] | ||

Set | G.J.O. Jameson | The Prime Number Theorem | LMS Student texts vol. 53 | Cambridge | 2003 | [Library] | |

Set | H.L. Montgomery and R.C. Vaughan | Multiplicative number theory, I. Classical theory | Cambridge advanced maths vol 97 | Cambridge University Press | 2007 | [Library] | |

Set | J. Stopple | A Primer of Analytic Number Theory | Cambridge | 2003 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM3703 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 7 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Thursday 28 February 2019 |

KEY WORDS SEARCH | Analytic Number Theory; Prime Number Theorem; the Riemann zeta-function |
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