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Information
| Code | MT0013 |
| Name | Advanced Topics in number theory |
| Term | 2023-2024 Academic Year |
| Term | Fall |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Doktora Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | |
| Course Instructor |
1 |
Course Goal / Objective
To mention the prime numbers in arithmetic progressions. To show the distribution of these numbers. To give the properties related to Riemann zeta function and L-functions.
Course Content
Prime numbers, arithmetic progressions, Dirichlet's Theorem and Dirichlet characters, Riemann Zeta Function, L-functions, Dirichlet series, The Prime Number Theorem and its analogue for arithmetic progressions, basic sieve theory and its consequences on some sets of prime numbers
Course Precondition
None.
Resources
Multiplicative Number Theory, H. Davenport, 1980, New York, Springer-Verlag.
Notes
None.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | (S)he understands the primes in arithmetic progressions. |
| LO02 | (S)he learns Gauss' sums. |
| LO03 | (S)he gains the knowledge about the primitive characters. |
| LO04 | (S)he learns the distribution of the prime numbers. |
| LO05 | (S)he learns the Riemann zeta function and the Dirichlet L-functions. |
| LO06 | (S)he learns the prime number theorem for arithmetic progressions. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows the results of previous research in a special field of mathematics | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 4 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Establishes new mathematical models with the help of the knowledge gained in the field of specialization. | 5 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Has basic knowledge in all areas of mathematics | 4 |
| PLO05 | Bilgi - Kuramsal, Olgusal | It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way. | |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics | 5 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Sets up original problems in her field and offers different solution techniques | |
| PLO08 | Bilgi - Kuramsal, Olgusal | It carries out original and qualified scientific studies on the subject related to its field. | 4 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | 3 |
| PLO10 | Beceriler - Bilişsel, Uygulamalı | Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. | 2 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | To have foreign language knowledge at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. | 4 |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | It presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. | |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title | 4 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Primes in arithmetic progressions | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Primes in arithmetic progressions 2 | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Gauss' Sum | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | Properties of the roots of unity | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Primes in arithmetic progressions in a general modulus | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | Primitive characters and the Dirichlet Class Number Formula | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | Asal sayıların dağılımı | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 8 | Mid-Term Exam | Reviewed of the topics discussed in the lecture notes and source again | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Riemann's memoir | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | The functional equation of the L functions | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | The properties of the gamma function | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | A zero-free region for the Riemann zeta function | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Zero-free regions for the L functions | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | The Prime Number Theorem | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | The prime number theorem for arithmetic progressions | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 16 | Term Exams | Reviewed of the topics discussed in the lecture notes and source again | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Reviewed of the topics discussed in the lecture notes and source again | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 5 | 70 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 15 | 15 |
| Final Exam | 1 | 30 | 30 |
| Total Workload (Hour) | 157 | ||
| Total Workload / 25 (h) | 6,28 | ||
| ECTS | 6 ECTS | ||