Analytic Number Theory -- Distribution of the Primes
Frank Thorne - Spring 2025
University of South Carolina
This will be an introduction to analytic number theory, focusing on questions about the distribution
of the prime numbers. The course will aim to be simple and self-contained, while explaining the content and flavor
of contemporary research as quickly as possible.
Want to learn how
James Maynard proved the existence of
small gaps between primes?
How Harald Helfgott
settled the ternary Goldbach conjecture?
Then this is the course for you.
Instructor: Frank Thorne, LeConte 447, thorne [at] math [dot] sc [dot] edu
Office Hours: Tuesdays 9:30-11:00, Wednesdays 3:30-5:00, and by appointment.
Course objectives/learning outcomes:
Successful students will:
-
Master the basic technical tools of the subject, such as partial summation, Perron's formula and contour integration, manipulating Dirichlet series, etc.
-
Gain an appreciation of what's known about the distribution of the primes -- from
both classical and contemporary points of view.
-
Begin to develop an appreciation -- from the analytic number theorist's point of view -- of how research projects are conceived, begun, and carried to completion.
-
The student will not master analytic number theory, one semester is far too short for that.
Text :
The Distribution of the Primes by myself and
Robert Lemke Oliver. The book is in progress, will be continuously edited over the course of the semester, and will be made freely available online to all students.
Lectures : 12:00-12:50, MWF, in LC 348.
Homework : Homeworks will be assigned periodically,
roughly every two weeks. Collaboration is encouraged, but you are responsible for writing up your own solutions. Roughly speaking, this means that you should not be looking at someone else's solution when writing up your own.
Optional discussion sections to talk about the homework problems will be organized at least occasionally, if there is sufficient interest.
Grading : Grading is based exclusively on homeworks. You are guaranteed at least the following grades: A = 75+, B+ = 70+, B = 60+, C+ = 50+, C = 40+, D = 30+. Students who have passed the comprehensive exam may arrange a reduced workload with their advisor's permission.
Contacting me :
Please contact me if you have any questions about the course, about my expectations, about my lectures,
about the homeworks, about the reading, or about anything else. I am also very
happy to talk about analytic number theory in general and to recommend further reading,
and I am available to supervise Ph.D. theses. Graduate work is demanding and it is my
job to help you succeed.
Policies :
Late homework may be accepted, especially if arranged in advance, but no guarantees are made, and repeated late homeworks will be accepted only in case of severe extenuating circumstances.
Academic honesty is expected of all students.
Attendance is encouraged, in class and even more so at the
Algebra, Geometty, and Number Theory Seminar, but I will not be taking roll or enforcing any penalty for absences.
If you have any disabilities that require accommodation, please register with
the Student Disability Resource Center.
It is my goal to create a welcoming classroom environment, free of racism, sexism, homo- or transphobia, discrimination, bullying, insults, or harassment. Please bring any concerns to me; major or repeated violations will be reported to the Office of Student Conduct.