This course follows Advanced topics in Algebra C.
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. Based on Advanced topics in Algebra C, we study more general L-functions defined axiomatically.
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.
Axiomatic definition of L-functions, analytic properties of L-functions, Selberg trace formula, Weil's explicit formula
✔ 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 | Axiomatic definition of L-functions | Details will be provided during each class session |
Class 2 | Analytic properties of L-functions | Details will be provided during each class session |
Class 3 | Generalized Riemann hypothesis and other conjecturtes | Details will be provided during each class session |
Class 4 | Selberg trace formula | Details will be provided during each class session |
Class 5 | Selberg zeta functions | Details will be provided during each class session |
Class 6 | Weil's explicit formula | Details will be provided during each class session |
Class 7 | Spectral interpretation of zeros I | Details will be provided during each class session |
Class 8 | Spectral interpretation of zeros II | Details will be provided during each class session |
None required
H. Iwaniec and E. Kowalski, Analytic number theory, Colloquium Publications, 53, AMS
H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of L-functions, Geom. Funct. Anal. 2000, 705-741
Course scores are evaluated by homework assignments (100%). Details will be announced during the course.
Basic undergraduate algebra and complex analysis
None in particular.