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 | Is independently able to build a problem in the area of study, solve the problem by developing solution techniques and assess the solutions. | X | ||||
| 2 | Is capable of creating a groundwork in the fundamental branches of mathematics as well as in his/her research area | X | ||||
| 3 | follows the latest national and international literature in Mathematics and in his/her area of research; and uses them in his/her related studies | X | ||||
| 4 | observes and adopts the scientific ethical values in his/her professional and social life | X | ||||
| 5 | presents in Turkish and English in academic/scientific events the results of his/her research or the latest studies and findings on a special topic and participates in discussions | X | ||||
| 6 | Develops skills to work independently or as a member of a team | X | ||||
| 7 | Develops competences in the areas of creative and critical thinking, problem solving and producing original studies. Follows recent scientific studies, is capable of making an analysis, synthesis and assessment of the knowledge acquired | X | ||||
| 8 | Is open to lifelong improvement of his/her acquired knowledge, skills and competences. | X | ||||
| 9 | Is able to apply the acquired knowledge and problem-solving skills to interdisciplinary studies, proposes different solution methods to problems in terms of mathematical models and from a mathematical point of view | X | ||||
| 10 | Uses the mathematical based softwares, informatics and communication technologies for scientific purposes | 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 | ||