2017　Advanced courses in Algebra C

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Mathematics
Instructor(s)
Suzuki Masatoshi
Course component(s)
Lecture
Day/Period(Room No.)
Thr5-6(H137)
Group
-
Course number
ZUA.A333
Credits
1
2017
Offered quarter
3Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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.

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

Riemann zeta function, Dirichlet characters, Dirichlet L-functions, Prime Number Theorem

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 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.

Reference books, course materials, etc.

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

Assessment criteria and methods

Assignments (100%).

Related courses

• MTH.A407 ： Advanced topics in Algebra C1
• MTH.A408 ： Advanced topics in Algebra D1
• ZUA.A334 ： Advanced courses in Algebra D

None required.

Other

None in particular.