### 2017　Advanced topics in Algebra C1

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Instructor(s)
Suzuki Masatoshi
Course component(s)
Lecture
Day/Period(Room No.)
Thr5-6(H137)
Group
-
Course number
MTH.A407
Credits
1
2017
Offered quarter
3Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
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### 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.A408 ： Advanced topics in Algebra D1

None required.

### Other

None in particular.