Undergraduate Course: Analytic Number Theory (MATH11226)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Year 5 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  This course develops the analytic aspects and methods used to study the distribution of prime numbers. In this course you will see how the notions and results from Honours Complex Variables can be used in a decisive and unexpected way to unlock many mysterious and deep properties of the primes. 
Course description 
We will investigate the basic properties of Riemann zeta function, exhibiting its close connection to prime numbers, its analytic continuation to the whole complex plane and its fundamental functional equation which will lead us to a precise statement of the Riemann Hypothesis, arguably the most important unsolved problem in mathematics.
Course Description
1. Properties of the Riemann zeta function, its analytic continuation, the functional equation, Euler's product formula, and the divergence of the
reciprocal sum of primes.
2. The Prime Number Theorem.
3. Dirichlet series and their basic properties. Dirichlet characters and the Lfunctions they define.
4. Dirichlet's theorem on infinitely many primes in an arithmetic progression.
5. Further topics

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Honours Complex Variables (MATH10067)

Corequisites  
Prohibited Combinations  
Other requirements  Note that PGT students on School of Mathematics MSc programmes are note required to have taken prerequisite courses, but they are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling. 
Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2024/25, Available to all students (SV1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
0 %,
Coursework
100 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework: 100% 
Feedback 
Not entered 
No Exam Information 
Learning Outcomes
On completion of this course, the student will be able to:
 Demonstrate an understanding of Analytic Number Theory by proving unseen results using the methods of the course.
 Correctly state the main definitions and theorems in the course.
 Produce examples and counterexamples illustrating the mathematical concepts presented in the course.
 Explain their reasoning about rigorous Analytic Number Theory clearly and precisely,using appropriate technical language.

Reading List
There will be no textbook that the lectures will follow. Lecture notes will be provided. However the following are good references for the course:
1. The Prime Number Theorem by G.J.O. Jameson, (LMS Student Texts 53,Cambridge University Press, 2003).
2. Introduction to Analytic Number Theory by T.M. Apostol, (Undergraduate Texts in Mathematics, SpringerVerlag, 1976, Chapters 2,3,11,12 and 13).
3. Multiplicative number theory by Harold Davenport (third ed., Graduate Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000).
4. Problems in Analytic Number Theory by M. Ram Murty (Springer, 2001.Chapters 1  5).
5. An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright (Sixth edition, Oxford University Press, 2008, Chapters 16 ,17 and 18). 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  ANT 
Contacts
Course organiser  Prof Jim Wright
Tel: (0131 6)50 8570
Email: J.R.Wright@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

