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Information
| Code | MT0020 |
| Name | Introduction to Analytic Number Theory |
| Term | 2022-2023 Academic Year |
| Semester | . Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Yüksek Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator |
Course Goal / Objective
To introduce arithmetical functions and Dirichlet multiplication. To show the applications of Abel's and Euler's summation formulas. To give the idea of deducing some results related to some certain sums related to prime numbers using Chebyshev functions
Course Content
Arithmetic, additive and multiplicative functions, Dirichlet product, Möbius' Inversion Formula, Euler's Summation Formula, averages of arithmetic functions, Dirichlet's Divisor Problem, Abel's Identity, Chebyshev functions, the distribution of prime numbers, Bertrand's Postulate, Mertens' theorems
Course Precondition
None.
Resources
Introduction to Analytic Number Theory, Tom M. Apostol, Springer-Verlag, New York, 1976.
Notes
None.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | (S)he gains general knowledge about the arithmetical functions |
| LO02 | (S)he understands the Dirichlet product and the Möbius inversion formula. |
| LO03 | (S)he learns Abel's and Euler's summation formulas. |
| LO04 | (S)he learns the mean values of some certain arithmetical functions. |
| LO05 | (S)he gains knowledge about sum certain sums related to prime numbers and the prime counting function. |
| LO06 | (S)he understands the prime number theorem and the Chebyshev functions. |
| LO07 | (S)he learns Bertrand's Postulate and Mertens' prime number theorems. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in his 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 | poses original problems related to field and presents different solution techniques. | |
| PLO08 | Bilgi - Kuramsal, Olgusal | 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 knowledge of a foreign language 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 | 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 | Arithmetic functions and their properties | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Additive and multiplicative functions | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Dirichlet's multiplication and their properties | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | Dirichlet inverses and the Möbius Inversion Formula | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Abel's and Euler's Summation Formulas | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | The average order of the divisor function and the Dirichlet Divisor Problem | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | The average order of the Euler-phi function and its applications | 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 | Euler's Theorem and the Euler product | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | The properties of prime numbers and the prime counting function | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | The Prime Number Theorem and its equivalencies | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | Chebyshev functions and their properties | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Bertrand's Postulate and Chebyshev's proof for Bertrand's Postulate | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | Mertens' Prime Number Theorems | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | The necessary conditions for the Prime Number Theorem and the Chebyshev Theorem | 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 | ||