This course is an introduction to analytic number theory. Particularly, we will study modern tools and concepts in the theory of zeta- and L-functions. This course is followed by Advanced topics in Algebra D1.
Zeta- and L-functions appear in many areas of number theory, and are studied very actively. This course hopes to provide solid background for students intending to learn advanced topics on zeta- and L-functions. We begin with the classical Riemann zeta function and Dirichlet L-functions.
Students are expected to:
-- understand fundamental notions and methods of analytic number theory
-- be familiar with modern tools and concepts in the theory of zeta- and L-functions.
Riemann zeta function, Dirichlet characters, Dirichlet L-functions, Prime Number Theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | Distribution of prime numbers, Chebyshev's inequality | Details will be provided during each class. |
Class 2 | The Riemann zeta function | Details will be provided during each class. |
Class 3 | Dirichlet characters, Gauss sums | Details will be provided during each class. |
Class 4 | Dirichlet L-functions, Dirichlet's class number formula | Details will be provided during each class. |
Class 5 | Properties of the gamma function | Details will be provided during each class. |
Class 6 | Functional equations of the Riemann zeta-function and Dirichlet L-functions | Details will be provided during each class. |
Class 7 | Zero-free region of the Riemann zeta-function and Dirichlet L-functions | Details will be provided during each class. |
Class 8 | Prime Number Theorem (in arithmetic progressions) | Details will be provided during each class. |
None required.
H. Davenport, Multiplicative Number Theory, GTM 74 (3rd revised ed.), New York: Springer-Verlag
H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I : Classical Theory, CSAM 97. Cambridge University Press
Assignments (100%).
None required.
None in particular.