2017 Advanced topics in Algebra D1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Suzuki Masatoshi 
Course component(s)
Lecture     
Day/Period(Room No.)
Thr5-6(H137)  
Group
-
Course number
MTH.A408
Credits
1
Academic year
2017
Offered quarter
4Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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.

Student learning outcomes

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.

Keywords

general L-functions, approximate functional equations, convexity bounds, zero-free regions, generalized Riemann hypothesis

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

  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

Textbook(s)

None required.

Reference books, course materials, etc.

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

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.A407 : Advanced topics in Algebra C1

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None required.

Other

None in particular.

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