This course follows Advanced topics in Algebra C1.
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 C1, 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.
general L-functions, approximate functional equations, convexity bounds, zero-free regions, generalized Riemann hypothesis
✔ 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 | Arithmetic functions | Details will be provided during each class. |
Class 2 | Axiomatic definition of L-functions | Details will be provided during each class. |
Class 3 | Analytic conductor, Ramanujan–Petersson conjecture | Details will be provided during each class. |
Class 4 | Approximate functional equations | Details will be provided during each class. |
Class 5 | Convexity bounds of L-functions | Details will be provided during each class. |
Class 6 | Zero-free regions of L-functions | Details will be provided during each class. |
Class 7 | Generalized prime number theorem, Weil's explicit formula | Details will be provided during each class. |
Class 8 | Number theoretic consequences of the generalized Riemann hypothesis | Details will be provided during each class. |
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
Assignments (100%).
None required.
None in particular.