MATH 406(S) Analysis and Number Theory (Q) 
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. For more information, see http://www.math.brown.edu/~sjmiller/williams/406.

Format: lecture/discussion and possible 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: 18). 

Below we provide links for some of the possible topics. The textbook for the course is An Invitation to Modern Number Theory (by S. J. Miller and R. Takloo-Bighash). From there you can click on links to the chapters on Poissonian behavior and on the connections between Number Theory and Random Matrix Theory. The homepage for the book also contains numerous links to papers and survey articles on the subject (see, for example, the paper by Lagarias for more on the 3x+1 map). For sum and difference sets, see the paper by Nathanson (if we choose to discuss this subject, we will almost surely examine the papers by Martin and O'Bryant and Hegarty and Miller); standard references for Benford's law are Hill and Raimi. Common advice to all students is to 'read the masters'; thus, see Hardy and Littlewood for more on the (Hardy-Littlewood) Circle Method. If you have any questions or comments, email me at Steven.Miller.MC.96 AT aya.yale.edu.

I'm currently an assistant professor at Brown University. I'm starting at Williams College in Fall '08; click here for some personal information about me, my family and my research interests.