Department of Mathematics

Stony Brook University

office: Math Tower 5-111

phone: (631) 632-8287

e-mail: leon.takhtajan@stonybrook.edu

MAT 401 Seminar in Mathematics: Analytic Number Theory

Fall 2021

Fall 2021

The goal of the seminar is to introduce the students to powerful analytic
methods in number theory. We will cover the following three major topics:

Poisson summation formula, Jacobi inversion formula and complex analysis proof of the Gauss quadratic reciprocity law.

Riemann's memoir and beyond: three different proofs of analytic continuation of Riemann zeta function and the functional equation.

Introduction to modular forms and Dedekind eta function.

I will give a few short lectures on each topic and will assign further material for the students to read and to present in class. Corresponding topics and reading material will be regularly posted on the Blackboard.

We will be using the following classic books:

K. Chandrasekharan *Introduction to Analytic Number Theory*,
Springer; softcover reprint of the original 1968 edition (March 1,
2012)

H.M. Edwards, *Riemann's Zeta Function*, Dover edition,
2001.

S.J. Patterson, *An Introduction to the Theory of the Riemann
Zeta-Function*, Cambridge University Press,
1995.

J.-P. Serre
*A Course in Arithmetics*, Springer, 1996.

Be familiar with the contributions of great mathematicians of the past.

Be conversant with the specialized vocabulary of the topic of analytic number theory.

Be ablie to rigorously prove the main results specific to the topic.

Be able to solve routine problems and quote assumptions and main theorems of the topic.

Appreciate the relationship of the topic to the undergraduate mathematics curriculum.

Tuesday and Thursday 1:15pm-2:35pm in Earth & Space 177.

Students need to have completed MAT342 and MAT312 or 313; completion of MAT311 is also recommended.

The grades will be based on the class participation, the oral presentations and on solution of selected homework problems.

If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the Student Accessibility Support Center. For procedures and information go to the following website: https://ehs.stonybrook.edu//programs/fire-safety/emergency-evacuation/evacuation-guide-disabilities and search Fire Safety and Evacuation and Disabilities.

Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html.

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Student Conduct and Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Until/unless the latest COVID guidance is explicitly amended by SBU, during Fall 2021 "disruptive behavior” will include refusal to wear a mask during classes.

Last updated January 2021