Further Number Theory
| Code | School | Level | Credits | Semesters |
| MATH4088 | Mathematical Sciences | 4 | 20 | Autumn UK |
- Code
- MATH4088
- School
- Mathematical Sciences
- Level
- 4
- Credits
- 20
- Semesters
- Autumn UK
Summary
Number theory concerns the solution of polynomial equations in whole numbers, or fractions. For example, the cubic equation
x3 + y3 = z3 with x, y, z non-zero
has infinitely many real solutions yet not a single solution in whole numbers. Equations of this sort are called Diophantine equations, and were first studied by the Greeks. What makes the study of these equations so fascinating is the seemingly chaotic distribution of prime numbers within the integers. We shall establish the basic properties of the Riemann zeta-function to find out how evenly these primes are distributed in nature. This course will present several methods to solve Diophantine equations including analytical methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.
• Möbius inversion
• The Riemann zeta function
• The distribution of primes
• Dirichlet series
• Theta functions
• Application to arithmetic problems
• Introduction to modular forms
Target Students
Single and Joint Honours students from the School of Mathematical Sciences. Available to Postgraduate Taught students and Postgraduate Research students.
Classes
- One lecture each week for 11 weeks
- Two lectures each week for 11 weeks
Assessment
- 20% Coursework 1
- 80% Exam 1 (3-hour): Written in-person exam
Assessed by end of autumn semester
Educational Aims
It follows on from Algebra and Number Theory, and presents a modern analytic approach to the solution of classical Diophantine equations. It provides a valuable insight into the systematic study of integers, primes and polynomials using techniques from both analysis and algebra. For students intending to go on to study Algebraic Number Theory, it builds a bridge between the elementary concepts in the prerequisite modules and the more advanced ideas.Learning Outcomes
L1 - state the basic theorems about Dirichlet series and theta functions
L2 - apply analytic techniques to obtain information about the distribution of primes
L3 - solve some types of Diophantine equations
L4 - define and state basic properties of modular forms