MATH 406: Analysis and Number Theory:  MWF 9 - 9:50am Bronfman 104

(click on the picture for a larger version)

Professor Steven Miller (Steven.J.Miller AT, 202 Bronfman Science Center (413-597-3293)

Office hours: M-F 10:05-10:50 and M-F 4-4:30 and whenever I'm in my office (click here for my schedule)

take aways for the course (if you are interested in adding to this, please let me know)

course description    syllabus/general    objectives    course notes / handouts    supplemental reading    other links    additional comments

COURSE DESCRIPTION: Gauss said "Mathematics is the queen of the sciences and number theory the queen of mathematics"; in this class we shall meet some of her subjects. We will discuss many of the most important questions in analytic and additive number theory, with an emphasis on techniques and open problems. Topics will range from Goldbach's Problem and the Circle Method to the Riemann Zeta Function and Random Matrix Theory. Other topics will be chosen by student interest, coming from sum and difference sets, Poissonian behavior, Benford's law, the dynamics of the 3x+1 map as well as suggestions from the class. We will occasionally assume some advanced results for our investigations, though we will always try to supply heuristics and motivate the material. No number theory background is assumed, and we will discuss whatever material we need from probability, statistics or Fourier analysis.
Format: lecture/discussion and almost surely presentations. Evaluation will be based on scholarship, discussions, homework and examinations (and if there is student interest, papers and presentations in place of some of the exams).
Prerequisites: Multivariable calculus, linear algebra, Math 301 or 305, Math 312 or 315. No enrollment limit (expected: 21). 

SYLLABUS / GENERAL: The textbook will be Miller and Takloo-Bighash’s `An Invitation to Modern Number Theory’ (errata for the book is here; additional comments from each day's lecture is here). On the first day of class we will describe many of the topics (including Goldbach's Problem and the Circle Method, the Riemann Zeta Function and Random Matrix Theory, Benford's Law, Poissonian behavior, the 3x+1 map and sum and difference sets), and then determine which topics to explore in detail. Please feel free to swing by my office or mention before, in or after class any questions or concerns you have about the course. If you have any suggestions for improvements, ranging from method of presentation to choice of examples, just let me know. If you would prefer to make these suggestions anonymously, you can send email from (the password is the first seven Fibonacci numbers, 11235813). Grading will be: 25% homework, 25% projects, 25% midterm, 25% final.

OBJECTIVES: There are two main goals to this course: to explore modern number theory and see the connections between various problems, and to learn problem solving skills. We will constantly emphasize the techniques we use to solve problems, as these techniques are applicable to a wide range of problems in the sciences.

COURSE NOTES AND HANDOUTS: Below are scanned copies of my lecture notes for the course (click here for additional comments from each day's lecture). Skimming these notes is a good way to prepare for lecture and to review the material; however, it is not the case that everything said in lecture will be in these notes for two reasons: (1) I hope to have a lot of class discussion, and these comments will undoubtably influence which direction we pursue; (2) detailed explanations of many arguments are given in the book, so often I have just jotted down notes to remind myself of what I wanted to mention.

SUPPLEMENTAL READING: For more on each topic, click below to go to a subpage with handouts and reading recommendations:




This is my first year at Williams; click here for some personal information about me, my family and my research interests.

For extra credit, find the flaw (or flaws) in any of the following papers (or, to be fair, convince me that they're correct or correct them and receive a Fields medal):


course description    syllabus/general    objectives    course notes / handouts    supplemental reading    other links (summer research)