### Special Topics in MATH: Analytic Number Theory (MATH 68005)

2020 Spring
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
10.00
Kağan Kurşungöz kursungoz@sabanciuniv.edu,
English
Doctoral, Master
--
Formal lecture

### CONTENT

1 - Ability to utilize the properties of arithmetic functions, compute their average orders. 2 - Ability to apply elementary theorems on distribution of primes, knowledge of the prime number theorem. 3 - Thorough understanding of Dirichlet's theorem on primes in arithmetic progressions. 4 - Basic ability to manipulate Dirichlet series, Euler products, zeta-and L-functions. 5 - Basic understanding of the components of the proof of the prime number theorem. * Arithmetic functions. * Big-oh notation and average orders of arithmetic functions. * Elementary theorems on the distribution of prime numbers, introduction to the prime number theorem. * Characters of finite abelian groups. * Dirichlet's theorem on primes in arithmetic progressions. * Dirichlet series and Euler products. * Zeta- and L-functions. * Analytic proof of the prime number theorem.

### OBJECTIVE

Upon completion of the course, student should be able to

1 ? Use arithmetical functions and the relations between them.
2 ? Calculate averages of various arithmetical functions.
3 ? Understand some elementary theorems on the distribution of primes.
4 ? Understand characters of finite abelian groups and the related L-functions.
5 ? Know basics of Dirichlet series and Euler products.

### PROGRAMME OUTCOMES

1. Develop and deepen the current and advanced knowledge in the field with original thought and/or research and come up with innovative definitions based on Master's degree qualifications 4

2. Conceive the interdisciplinary interaction which the field is related with ; come up with original solutions by using knowledge requiring proficiency on analysis, synthesis and assessment of new and complex ideas. 1

3. Evaluate and use new information within the field in a systematic approach. 5

4. Develop an innovative knowledge, method, design and/or practice or adapt an already known knowledge, method, design and/or practice to another field; research, conceive, design, adapt and implement an original subject. 1

5. Critical analysis, synthesis and evaluation of new and complex ideas. 1

6. Gain advanced level skills in the use of research methods in the field of study. 5

7. Contribute the progression in the field by producing an innovative idea, skill, design and/or practice or by adapting an already known idea, skill, design, and/or practice to a different field independently. 2

8. Broaden the borders of the knowledge in the field by producing or interpreting an original work or publishing at least one scientific paper in the field in national and/or international refereed journals. 1

9. Demonstrate leadership in contexts requiring innovative and interdisciplinary problem solving. 1

10. Develop new ideas and methods in the field by using high level mental processes such as creative and critical thinking, problem solving and decision making. 2

11. Investigate and improve social connections and their conducting norms and manage the actions to change them when necessary. 1

12. Defend original views when exchanging ideas in the field with professionals and communicate effectively by showing competence in the field. 4

13. Ability to communicate and discuss orally, in written and visually with peers by using a foreign language at least at a level of European Language Portfolio C1 General Level. 4

14. Contribute to the transition of the community to an information society and its sustainability process by introducing scientific, technological, social or cultural improvements. 1

15. Demonstrate functional interaction by using strategic decision making processes in solving problems encountered in the field. 3

16. Contribute to the solution finding process regarding social, scientific, cultural and ethical problems in the field and support the development of these values. 1

1. Develop the ability to use critical, analytical, and reflective thinking and reasoning 5

2. Reflect on social and ethical responsibilities in his/her professional life. 3

3. Gain experience and confidence in the dissemination of project/research outputs 4

4. Work responsibly and creatively as an individual or as a member or leader of a team and in multidisciplinary environments. 5

5. Communicate effectively by oral, written, graphical and technological means and have competency in English. 5

6. Independently reach and acquire information, and develop appreciation of the need for continuously learning and updating. 4

1. Design and model engineering systems and processes and solve engineering problems with an innovative approach. 1

2. Establish experimental setups, conduct experiments and/or simulations. 1

3. Analytically acquire and interpret data. 3

1. Employ mathematical methods to solve physical problems and understand relevant numerical techniques. 1

2. Conduct basic experiments or simulations. 1

3. Analytically acquire and interpret data. 3

4. Establish thorough understanding of the fundamental principles of physics. 1

1. Develop abstract mathematical thinking and mathematical intuition. 5

2. Demonstrate a broad understanding of several areas of advanced mathematics and of their interrelations. 4

3. Have knowledge of the fundamental and advanced concepts, principles and techniques from a range of topics. 5

4. The ability to tackle complex problems, reveal structures and clarify problems, discover suitable analytical and/or numerical methods and interpret solutions. 4

5. Analyze problems of the area of specialization, plan strategies for their solution, and apply notions and methods of abstract and/or applied mathematics to solve them. 5

1. Apply knowledge of mathematics, science, and engineering in computer science and engineering related problems. 1

2. Display knowledge of contemporary issues in computer science and engineering and apply to a particular problem. 1

3. Demonstrate the use of results from interpreted data to improve the quality of research or a product in computer science and engineering. 1

1. Apply software, modeling, instrumentation, and experimental techniques and their combinations in the design and integration of electrical, electronic, control and mechanical systems. 1

2. Interact with researchers from different disciplines to exchange ideas and identify areas of research collaboration to advance the frontiers of present knowledge and technology; determine relevant solution approaches and apply them by preparing a research strategy. 1

3. Take part in ambitious and highly challenging research to generate value for both the industry and society. 1

1. Use advanced Math (including probability and/or statistics), advanced sciences, advanced computer and programming, and advanced Electronics engineering knowledge to design and analyze complex electronics circuits, instruments, software and electronic systems with hardware/software. 1

2. Analyze and design advanced communication networks and systems, advanced signal processing algorithms or software using advanced knowledge on diff. equations, linear algebra, complex variables and discrete math. 1

1. Apply knowledge of key concepts in biology, with an emphasis on molecular genetics, biochemistry and molecular and cell biology. 1

2. Display an awareness of the contemporary biological issues in relation with other scientific areas. 1

3. Demonstrate hands-on experience in a wide range of biological experimental techniques. 1

1. Establish a strong theoretical background in several of a broad range of subjects related to the discipline, such as manufacturing processes, service systems design and operation, production planning and control, modeling and optimization, stochastics, statistics. 1

2. Develop novel modeling and / or analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy, also using an interdisciplinary approach whenever appropriate. 1

3. Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 1

4. Acquire skills to independently explore and tackle problems related to the discipline that were not encountered previously. Develop appropriate modeling, solution, implementation strategies, and assess the quality of the outcome. 1

1. Assess and identify developments, strategies, opportunities and problems in energy security and energy technologies. 1

2. Define and solve technical, economic and administrative problems in energy businesses. 1

3. Establish knowledge and understanding of energy security, energy technologies, energy markets and strategic planning in energy enterprises. 1

4. Demonstrate an awareness of environmental concerns and their importance in developing engineering solutions and new technologies. 1

5. Acquire a series of social and technical proficiencies for project management and leadership skills. 1

1. Apply a broad knowledge of structure & microstructure of all classes of materials, and the ability to use this knowledge to determine the material properties. 1

2. Apply a broad understanding of the relationships between material properties, performance and processing. 1

3. Apply a broad understanding of thermodynamics, kinetics, transport phenomena, phase transformations and materials aspects of advanced technology. 1

4. Demonstrate hands-on experience using a wide range of materials characterization techniques. 1

5. Demonstrate the use of results from interpreted data to improve the quality of research, a product, or a product in materials science and engineering. 1

### ASSESSMENT METHODS and CRITERIA

 Percentage (%) Assignment 100