MATH 680 Fall 2020
Basic Information
- Classroom: DuSable 212 (yes, this course is face-to-face, not on line!)
- Instructor: Jeff Thunder
- Office: WH 320 (and sometimes WH 362)
- Phone: 753-6725
- e-mail: jthunder@math.niu.edu
- Office Hours: Monday, Wednesday, Friday, 11:00-11:50 a.m. or by appointment.
Course Description and Objectives
This is a graduate course in analytic number theory. The emphasis will be on statistical properties of some important arithmetic functions such as the prime counting function. We will begin the course by defining "arithmetic function," considering some examples, and investigating a few important structural properties thereof. We'll then turn our attention to the prime counting function and early estimates, Dirichlet series, primes in arithmetic progressions, and ultimately a proof of the prime number theorem. We will also study infinite products (as it applies to Dirichlet series), the Riemann zeta function, and the functional equation. The last part of the course deals with more quantitative statements on zero-free regions and the distribution of the zeros of the zeta function.
Students are expected to hone their mathematical skills for constructing and communicating proofs, as well as gain a deeper understanding of number theory and the role analysis plays in the investigations of the integers.
Student assessment will be done via homework (see below), occaisonal in-class presentation of work, and a final exam. The weights for these are 80%, 5%, and 15%, respectively.
Text
Rather than a formal textbook, I'll be making available written up lecture notes. These will be made available in a timely fashion.
Homework
Homework will be assigned somewhat irregularly. I'll announce assignments (and due dates) in class and also post them to this webpage. You are free to work with other students on the homework; in fact, this is encouraged. Sloppy and/or illegible work will be returned back with no credit! Your homework is something of which you should be proud (notice how I didn't end with a preposition there). Expect to spend lots of time on it.
Homework Assignments
- First homework assignment (Due 9/2)
- Second homework assignment (Due 9/9)
- Third homework assignment (Due 9/18)
- Fourth homework assignment (Due 9/30)
- Fifth homework assignment (Due 11/2)
- Sixth homework assignment (Due 11/9)
- Seventh homework assignment (Due 11/18)
- Eighth homework assignment (Due 11/25)
Final Exam
The final exam will take place in DuSable 212 from 10:00-11:50 on Monday, December 7. Hopefully this date won't live in infamy for us.
You will be asked to present a solution to one of the following exercises: final exam questions.
Handouts
- Arithmetic Functions
- Chebyshev's Estimates
- Bertrand's Postulate
- The Riemann Zeta Function
- Slides for Monday, Sept. 14 (and the generating TeX file
- Slides for Wednesday, Sept. 16
- Slides for Friday, Sept. 18
- generic results on infinite products
- A Particular Infinite Product
- Slides for Monday, Sept. 21
- Slides for Wednesday, Sept. 23
- Slides for Friday, Sept. 25
- Dirichlet Series
- Primes in an Arithmetic Progression
- The Prime Number Theorem
- Slides for Friday, Oct. 23
- The Functional Equation
- A Quantitative Prime Number Theorem I: Zero-Free Regions
- Slides for Wednesday, Nov. 18
- Slides for Friday, Nov. 20
- A Quantitative Prime Number Theorem II: The Inverse Mellin Transform
- Stirling's Formula
- Slides for Monday, Nov. 23
- Slides for Monday, Nov. 30
- Hardy's Theorem (zeros along the critical line)
- Slides for Wednesday, Dec. 2
- Slides for Friday, Dec. 4
- Properties of the xi Function
- The Distribution of the Zeros
DRC Statement
If you need an accommodation for this class, please contact the Disability Resource Center as soon as possible. The DRC coordinates accommodations for students with disabilities. It is located on the 4th floor of the Health Services Building, and can be reached at 815-753-1303 or drc@niu.edu. Also, please contact me privately as soon as possible so we can discuss your accommodations. Please note that you will not be required to disclose your disability, only your accommodations. The sooner you let me know your needs, the sooner I can assist you in achieving your learning goals in this course.
Academic Conduct
Academic honesty and mutual respect (student with student and instructor with student) are expected in this course. Mutual respect means being on time for class and not leaving early, being prepared to give full attention to class work, not reading newspapers or other material in class, not using cell phones or pagers during class time, and not looking at another student's work during exams. Academic misconduct, as defined by the Student Judicial Code, will not be treated lightly.
Coronavirus
Your health and safety are my No. 1 priority. We are all members of the Huskie community, and we owe it to each other to protect ourselves and each other. When I come to class, I’ll be wearing a face covering. I expect you to do the same. I also expect you to monitor your health, and you should stay home if you’ve been exposed to someone who recently tested positive for COVID-19 or if you develop any symptoms that might be related to COVID-19. If you do have symptoms, stay home and contact NIU’s COVID helpline (815-753-0444) to report your symptoms and get advice.
Homer does math!
Yes indeed, there's plenty of math humor to be found in the Simpsons. Just look and see!
Last update: Dec. 4, 2020