MA4011: Combinatorial and Analytic Number Theory
School | Cardiff School of Mathematics |
Department Code | MATHS |
Module Code | MA4011 |
External Subject Code | 100405 |
Number of Credits | 20 |
Level | L7 |
Language of Delivery | English |
Module Leader | Dr Matthew Lettington |
Semester | Spring Semester |
Academic Year | 2015/6 |
Outline Description of Module
Number theory is the branch of mathematics concerned with the structure of the positive integers. This lecture based module begins with the study of elementary and combinatorial number theory, before introducing the fascinating world of analytic number theory.
The first part of this module studies how properties of divisibility can be used to characterise the integers and some special numbers, such as Bernoulli numbers, using elementary and combinatorial techniques.
The second part of the module introduces the standard arithmetic functions and studies how these counting functions vary and what their typical and extreme values are. The most basic questions concern prime numbers and how they are distributed amongst the integers. For example, a forty-digit number usually only has five or six prime factors, one of them much bigger than all of the others. The course culminates with the derivation of the prime number theorem which states that the n-th prime is roughly n log n.
Recommended Modules: MA0216 Elementary Number Theory II, MA3004 Combinatorics
On completion of the module a student should be able to
- Understand and use congruences, solve power congruences using primitive roots.
- Understand and describe some standard counting functions in number theory.
- Derive some standard number theoretic relations using combinatorial functions.
- Manipulate `multiplicative’ arithmetic functions.
- Understand and estimate the average sizes and the extreme values of arithmetic functions using order of magnitude notation.
- Understand the distribution of prime numbers.
- Construct and use Dirichlet characters and Dirichlet L functions.
How the module will be delivered
30 - 50 minute lectures
Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.
Students are also expected to undertake at least 120 hours private study including preparation of worked solutions for problem classes.
Skills that will be practised and developed
Skills:
Working with congruences. Using generating functions. Deriving number theoretic properties of combinatorial relations. Manipulating multiplicative functions. Calculating sums and averages as main term plus an error term whose order of magnitude is estimated. Finding the table of Dirichlet characters for a given modulus.
Transferable Skills:
Recognising patterns. Reasoning logically. Understanding orders of magnitude. Analysing complex expressions as a main term plus smaller order terms.
How the module will be assessed
Formative assessment is carried out in the problem classes. Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.
The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.
The examination paper has a choice of four from five equally weighted questions.
Assessment Breakdown
Type | % | Title | Duration(hrs) |
---|---|---|---|
Exam - Spring Semester | 100 | Combinatorial And Analytic Number Theory | 3 |
Syllabus content
- Euler’s phi function
- Primitive roots modulo prime powers
- Generating Functions
- Bernoulli and Euler Numbers/Polynomials
- Von Staudt Clausen's Theorem
- The Riemann zeta function at integer values – Euler and Ramanujan’s formula
- Multiplicative functions
- Mobius inversion
- Orders of magnitude
- Sizes and averages of arithmetic functions
- Chebychev’s bounds
- The Riemann zeta function revisited
- Dirichlet characters
- The prime number theorem
- Dirichlets theorem on primes in arithmetic progression
Essential Reading and Resource List
Introduction to the Theory of Numbers, Hardy G H and Wright E M, OUP
The Higher Arithmetic: An Introduction to the Theory of Numbers, Davenport H, CUP
The Distribution of Prime Numbers, Ingham A E ed. Vaughan R C, CUP
The Prime Number Theorem, Jameson G J O, CUP