General information
Meeting Time 
Mon., Wed., Fri., 10:00  10:50 (may be negotiated otherwise) 
Location  Building 380 (Math), room 381T

Professor 
Ben Brubaker (brubaker@math.stanford.edu)
Office: 382F (2nd floor, Math building 380)
Office Phone: 34507
Office Hours: Monday 23:30, Wednesday 23:30, or by appointment.

Course Assistant 
Ryan Vinroot (vinroot@math.stanford.edu)
Office: 380G (Basement, Math building)
Office hours: By appointment only.

Textbooks 
Multiplicative Number Theory, by Harold Davenport.
The Theory of the Riemann Zeta Function, by E.C. Titchmarsh. 
Course Content  We will review the basics of class field theory in the first week through the lens of reciprocity. Then we'll head to Davenport, covering the first 8 chapters, essential and classical analytic number theory from which all else follows. We'll break and do several weeks on the zeta function. Classical stuff again, but often shoved under the rug. We'll end with a potpourri of topics, including Siegel's theorem and proofs of algebraic facts using analytic information. We conclude by discussing automorphic forms in general and their role in solving fundamental questions in algebraic number theory. It will be sweet. 
Prerequisites 
There are essentially no prerequisites for this course. If you took Rubin's 248A in the fall, or know the equivalent information, you'll have a better view from which to understand the arc of the course, but its not at all necessary in understanding the day to day material. 
Syllabus
Date 
Material covered 
Relevant Reading

Wed., 1/7 
Reciprocity Laws and Class Field Theory I 
Cassels/Frohlich, Lang, "Alg. Num. Thy." 
Fri., 1/9 
Reciprocity Laws and Class Field Theory II 
Cassels/Frohlich, Lang, "Alg. Num. Thy." 
Mon., 1/12 
Primes in Arithmetic Progression (prime modulus) 
Davenport, Ch. 1, Lang, "Alg. Num. Thy." 
Wed., 1/14 
Gauss Sums and Cyclotomy 
Davenport Ch. 2,3 
Fri., 1/16 
Primes in Arithmetic Progression (general case) 
Davenport, Ch. 4 
Mon., 1/19  No Class (MLK) 
Wed., 1/21 
Primitive Characters and Quadratic Forms 
Cox, "Primes of the Form x^2 + ny^2," Davenport, Ch. 5 
Fri., 1/23 
Dirichlet's Class Number Formula 
Davenport, Ch. 6 
Mon., 1/26 
"Two Weeks of Zeta" Festival Begins! Introduction, Proofs of Functional Equation, Facts about Gamma 
Titchmarsh, "Theory of the Riemann Zeta Function," Chs. 1 & 2 
Wed., 1/28 
Infinite Products, Zerofree regions 
Titchmarsh, Ch. 3 
Fri., 1/30 
The Prime Number Theorem 
Titchmarsh, Ch. 3 
Mon., 2/9 
Week of Individual Projects Begin (see "Announcements"). 
Titchmarsh, Chs. 415 
Wed., 2/11 
WrapUp of Zeta Topics, Even Moments > Lindelof 
Titchmarsh, Patterson 
Fri., 2/13 
Siegel's Theorem 
Titchmarsh, Course Notes (available soon in handouts section) 
Mon., 2/16 
NO CLASS (Presidents' Day) 
Wed., 2/18 
Finish Proof of Siegel's Theorem 
Titchmarsh, Notes 
Fri., 2/20 
Begin Discussing Zeta Functions on Number Fields 
Class Notes 
Mon., 2/23 
Begin Hecke's proof of Dedekind zeta functional equation 
Class Notes 
Wed., 2/25 
Theta Functions 
Class Notes 
Fri., 2/27 
Finish Hecke's proof of Dedekind zeta functional equation 
Class Notes 
Final 23 Weeks 
Hecke Lfunctions, Basics of Automorphic Forms 
Class Notes, Bump Ch. 1 
