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
|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.
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.
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
Information for Visiting Students
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Course Delivery Information
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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
|Course organiser||Prof Jim Wright
Tel: (0131 6)50 8570
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427