# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2023/2024

### Timetable information in the Course Catalogue may be subject to change.

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# Undergraduate Course: Analytic Number Theory (MATH11226)

 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 L-functions they define. 4. Dirichlet's theorem on infinitely many primes in an arithmetic progression. 5. Further topics
 Pre-requisites Students MUST have passed: Introduction to Number Theory (MATH10071) AND Honours Complex Variables (MATH10067) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites None High Demand Course? Yes
 Not being delivered
 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.
 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, Springer-Verlag, 1976, Chapters 2,3,11,12 and 13). 3. Multiplicative number theory by Harold Davenport (third ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, 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).
 Graduate Attributes and Skills Not entered Keywords ANT
 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
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