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.
Contact information: Frank Thorne, LeConte 317-O, thorne [at] math [dot] sc [dot] edu.
Schedule: The class will meet 10:50-11:40, MWF.
Office hours: MW 9:00-10:00 or Tue 11:30-12:30. Alternatively, ask me questions before or after class or seminar.
Learning outcomes. Successful students will: see classical statements about the distribution of the primes, and understand the methods used to prove them; understand how the "basic" statements can themselves be used in a variety of applications; master foundational subjects such as asymptotic analysis, partial summation, contour integration, etc.; improve their knowledge and proficiency in analytic number theory in general.
Prerequisites. Graduate standing or permission of instructor. Students should either be enrolled in 701 and 703 concurrently, or have equivalent background.
Grading. Homework will be given regularly, and counts for 80% of your grade. Collaboration is strongly encouraged, but please write up solutions on your own.
You are also expected to attend at least ten research talks during the seminar. These could be from any of the following: the department colloquium; the algebra, geometry, and number theory seminar, the number theory seminar, the graduate colloquium, the Palmetto Number Theory Series, or any other seminar or conference talks you find interesting.
For each, you are asked to do Ravi Vakil's Three Things Exercise (20% of your grade). I will do the same. Here are mine.
You are guaranteed an A for a 75+ average, B for 50+, C for 40+.
Students who have passed the comprehensive exam may be a negotiate a reduced workload (please ask).
Textbook: The Distribution of the Primes by myself and Robert Lemke Oliver. The book is in preparation; free PDF copies will be uploaded here and updated frequently. COMMENTS, SUGGESTIONS, AND CORRECTIONS FOR THE BOOK ARE STRONGLY ENCOURAGED.
I also strongly recommend Davenport's Multiplicative Number Theory as a general introduction to analytic number theory.