ECTS - Analytic Number Theory
Analytic Number Theory (MATH630) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Analytic Number Theory | MATH630 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | |
| Learning and Teaching Strategies | . |
| Course Lecturer(s) |
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| Course Objectives | The main goal in Analytic number theory is to study number theory using analytic tools. In this course ultimate goal will be studying primes and questions related with prime numbers, such as: what is the size of nth prime?, How fast the sum of reciprocals of prime diverges to infinity?, How many primes are there less than a given magnitude?, Is there always a prime between n and 2n? What is the relation between prime numbers and Riemann zeta function? In doing so, students will see the use of analysis (mostly complex) and algebra in different contexts. |
| Course Learning Outcomes |
The students who succeeded in this course;
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| Course Content | Arithmetical functions, Euler?s and Abel?s summation, Dirichlet convolution, some elementary functions concerning primes, Riemann Zeta function and Dirichlet L-functions, primes in arithmetic progressions. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | Greatest common divisors, The Euclidean Algorithm, Congruences, | T.Apostol - Chapter 1 |
| 2 | The Möbius function, The Euler totient function, | T.Apostol - Chapter 2 |
| 3 | The Möbius inversion formula, Multiplicative functions | T.Apostol - Chapter 2 |
| 4 | Dirichlet convolution. | T.Apostol - Chapter 2 |
| 5 | Big and small O notations. Partial summation formula and Euler’s summation formula. Some asymptotic formulas, | T.Apostol - Chapter 3 |
| 6 | The divisor function and Dirichlet’s hyperbola method | T.Apostol - Chapter 3 |
| 7 | Elementary Results on the Distribution of Primes: The function ψ(x), The functions θ(x) and π(x) | T.Apostol - Chapter 4 |
| 8 | Merten’s estimates, Some applications of prime number theorem. The Bertrand Postulate, | T.Apostol - Chapter 4 |
| 9 | L-functions, Riemann zeta function, Euler product representation of ζ(s) and general L-functions. | T.Apostol - Chapter 11 |
| 10 | Analytic continuation of ζ(s) to σ > 0, Non-vanishing of ζ(s) on s = 1, | T.Apostol - Chapter 12 |
| 11 | The proof of the Prime Number Theorem (PNT), | T.Apostol - Chapter 13 |
| 12 | Properties of Riemann-zeta function, such as functional equation and the number of zeroes in the critical strip. | Davenport Chapter 13-14 -15 |
| 13 | Primes in Arithmetic Progressions: Dirichlet’s characters, The orthogonality relations, | T.Apostol - Chapter 6 |
| 14 | Proof of Dirichlet’s Theorem (Elementary proof) | T.Apostol - Chapter 7 |
| 15 | Non-Vanishing of Dirichet L-functions as s = 1 (Analytic Proof) | T.Apostol - Chapter 12 (revisited) |
| 16 | Review and Final Exam |
Sources
| Course Book | 1. T. Apostol, Introduction to Analytic Number Theory, 1st edition. 1976, 5th edition 2010, Springer, 1441928057 |
|---|---|
| Other Sources | 2. Analytic Number Theory for Undergraduates by Heng Huat Chan Monographs in Number Theory ISSN 1793-8341 April 2009 |
| 3. Multiplicative Number Theory, Classical Theory (Cambridge Studies in Advanced Mathematics) Harold Davenport. 1980 3. ed. | |
| 4. Problems in Analytic Number Theory, M. Ram Murty, 2008, e-ISBN 978-0-387-72350-1 | |
| 5. Multiplicative Number Theory I. Classical Theory, H. L. Montgomery & R. C. Vaughan |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 4 | 40 |
| Presentation | - | - |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 25 |
| Final Exam/Final Jury | 1 | 35 |
| Toplam | 6 | 100 |
| Percentage of Semester Work | |
|---|---|
| Percentage of Final Work | 100 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Alanında, bağımsız olarak, bir problem kurgulayabilir, çözüm yöntemi geliştirerek problemi çözebilir ve sonuçları değerlendirebilir | X | ||||
| 2 | Matematiğin temel alanlarında ve kendi uzmanlığı olarak seçtiği alanda gerekli alt yapıyı oluşturur. | X | ||||
| 3 | Matematik literatürünü ve özel olarak kendi araştırma konusu ile ilgili ulusal ve uluslararası güncel yayınları takip edebilir ve bunlardan kendi araştırma konusu ile ilgili olanları çalışmalarında kullanabilir | X | ||||
| 4 | Bilimsel etik değerleri ve kuralları dikkate alır ve mesleki ve toplumsal yaşamda kullanabilir | X | ||||
| 5 | Kendi çalışmalarının sonuçlarını veya belli bir konudaki güncel çalışmaları ve bulguları, çeşitli bilimsel toplantılarda topluluk önünde Türkçe ve İngilizce olarak sunabilir ve tartışmalara katılabilir. | X | ||||
| 6 | Gerek bireysel, gerek bir çalışma grubunun üyesi olarak çalışabilme becerisini geliştirir | X | ||||
| 7 | Yaratıcı ve eleştirel düşünme, problem çözme, özgün bir çalışma üretme becerisini geliştirir. Bilimsel gelişmeleri takip eder, özümsediği bilgilerin analiz, sentez ve değerlendirmesini yapabilir. | X | ||||
| 8 | Kazandığı bilgi, beceri ve yetkinlikleri yaşam boyu geliştirmeye açık olur. | X | ||||
| 9 | Alanında özümsediği bilgiyi ve problem çözme yeteneğini disiplinler arası çalışmalarda uygulayabilir; karşılaşılan problemleri matematiksel modellerle ifade ederek, matematiksel bakış açısı ile farklı çözüm yöntemleri önerir. | X | ||||
| 10 | Matematik temelli yazılımları, bilişim ve iletişim teknolojilerini bilimsel amaçlı kullanabilir. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| Homework Assignments | 5 | 3 | 15 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 10 | 10 |
| Prepration of Final Exams/Final Jury | 1 | 10 | 10 |
| Total Workload | 125 | ||