Mathematics (English) Preview
Bachelor	TR-NQF-HE: Level 6	QF-EHEA: First Cycle	EQF-LLL: Level 6

Course Introduction and Application Information

Course Code:

MATH004

Course Name:

Analytic Number Theory

Semester:

Fall

Course Credits:

ECTS

Language of instruction:

English

Course Condition:

Does the Course Require Work Experience?:

Type of course:

Departmental Elective

Course Level:

Bachelor

TR-NQF-HE:6. Master`s Degree

QF-EHEA:First Cycle

EQF-LLL:6. Master`s Degree

Mode of Delivery:

Face to face

Course Coordinator:

Araş. Gör. GAMZE AKAR UYSAL

Course Lecturer(s):

Assist. Prof. Dr. Doğa Can Sertbaş

Course Assistants:

Course Objective and Content

Course Objectives:	The course aims to introduce arithmetic functions and the Dirichlet product, to teach Abel’s and Euler’s summation formulas and to gain the ability to use these concepts. Also, it aims to give the idea about the calculation of certain sums related to the prime numbers using Chebyshev functions.
Course Content:	The content of the course consists of arithmetic, additive and multiplicative functions, the 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 and Mertens' theorems.

Learning Outcomes

The students who have succeeded in this course;
1) Gain the general knowledge related to the arithmetical functions and their means.
2) Understand the Dirichlet product and the Möbius inversion formula.
3) Learn Abel's and Euler's summation formulas.
4) Gain the knowledge about the certain sums related to prime numbers and the prime counting function.
5) Comprehend the prime number theorem and the Chebyshev functions.
6) Learn Bertrand's Postulate and Mertens' prime number theorems.

Course Flow Plan

Week	Subject	Related Preparation
1)	Divisibility, prime numbers and the fundamental theorem of arithmetic
2)	Arithmetic functions and their properties
3)	Additive and multiplicative functions
4)	The Dirichlet product, Dirichlet inverses and the Möbius Inversion Formula
5)	Abel's and Euler's Summation Formulas
6)	The average order of the divisor function and the Dirichlet Divisor Problem
7)	The average order of the Euler-ϕ function and its applications
8)	Midterm Exam
9)	Euler's Theorem and the Euler product
10)	Properties of the prime counting function, the Prime Number Theorem and its equivalencies
11)	Chebyshev functions and their properties
12)	Bertrand's Postulate and Chebyshev's proof for Bertrand's Postulate
13)	Mertens' Prime Number Theorems
14)	Sufficient conditions for the Prime Number Theorem and the Chebyshev Theorem

Sources

Course Notes / Textbooks:	Introduction to Analytic Number Theory, Tom M. Apostol, Springer-Verlag, New York, 1976.
References:	Cojocaru AC, Murty MR. An Introduction to Sieve Methods and Their Applications. Cambridge: Cambridge University Press; 2005. doi:10.1017/CBO9780511615993

Course - Program Learning Outcome Relationship

Course Learning Outcomes	1	2	3	4	5	6
Program Outcomes
1) Have the knowledge of the scope, history, applications, problems, methods of mathematics and knowledge that will be beneficial to humanity as both scientific and intellectual discipline.
2) Have the ability to establish a relationship between mathematics and other disciplines and develop mathematical models for interdisciplinary problems.
3) Have the ability to define, formulate and analyze real life problems with statistical and mathematical techniques.
4) Have the ability to think analytically and use the time effectively in the process of deduction.
5) Have the ability to search the literature, understand and interpret scientific articles.
6) Have the knowledge of basic software to be able to work in the related fields of computer science and have the ability to use information technologies at an advanced level of the European Computer Driving License.
7) Have the ability to work efficiently in interdisciplinary teams.
8) Have the ability to communicate effectively in oral and written form, write effective reports and comprehend the written reports, make effective presentations.
9) Have the consciousness of professional and ethical responsibility and acting ethically; have the knowledge about academic standards.
10) Have the ability to use a foreign language at least at B1 level in terms of European Language Portfolio criteria.
11) Are aware of the necessity of lifelong learning; have the ability to access information, to follow developments in science and technology and to constantly renew themselves.

Course - Learning Outcome Relationship

No Effect	1 Lowest	2 Average	3 Highest

	Program Outcomes	Level of Contribution
1)	Have the knowledge of the scope, history, applications, problems, methods of mathematics and knowledge that will be beneficial to humanity as both scientific and intellectual discipline.	2
2)	Have the ability to establish a relationship between mathematics and other disciplines and develop mathematical models for interdisciplinary problems.
3)	Have the ability to define, formulate and analyze real life problems with statistical and mathematical techniques.
4)	Have the ability to think analytically and use the time effectively in the process of deduction.
5)	Have the ability to search the literature, understand and interpret scientific articles.
6)	Have the knowledge of basic software to be able to work in the related fields of computer science and have the ability to use information technologies at an advanced level of the European Computer Driving License.
7)	Have the ability to work efficiently in interdisciplinary teams.
8)	Have the ability to communicate effectively in oral and written form, write effective reports and comprehend the written reports, make effective presentations.
9)	Have the consciousness of professional and ethical responsibility and acting ethically; have the knowledge about academic standards.
10)	Have the ability to use a foreign language at least at B1 level in terms of European Language Portfolio criteria.
11)	Are aware of the necessity of lifelong learning; have the ability to access information, to follow developments in science and technology and to constantly renew themselves.

Assessment & Grading

Semester Requirements	Number of Activities	Level of Contribution
Midterms	1	% 40
Final	1	% 60
total		% 100
PERCENTAGE OF SEMESTER WORK		% 40
PERCENTAGE OF FINAL WORK		% 60
total		% 100