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|
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
None in particular.