Syllabus

Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Course Description

This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions). There is also a bit of discussion of non-abelian equidistribution results, like the Chebotarev density theorem, and the Sato-Tate conjecture for elliptic curves (and recent progress on same by Clozel, Harris, Shepherd-Barron, Taylor). For a detailed description of the content and structure of the course, please see the first set of lecture notes. (PDF)

Textbooks

There is no required text for this class. Detailed lecture notes are provided in lieu of following a specific text.

Recommended Texts

Buy at Amazon Davenport, H., and H. L. Montgomery. Multiplicative Number Theory. New York, NY: Springer-Verlag, 2000. ISBN: 9780387950976.

Buy at Amazon Iwaniec, Henryk, and Emmanuel Kowalski. Analytic Number Theory. Providence, RI: American Mathematical Society, 2004. ISBN: 9780821836330.

Prerequisites

Complex analysis (18.112), some background in number theory (at the level of 18.781). There are one or two points where a little algebraic number theory (as in 18.786) may be helpful, but it is in no way required. On the other hand, the 18.112 prerequisite is quite serious; if you do not formally meet it, you must let me know in writing what your equivalent preparation is. Also recommended: some abstract algebra (18.701 and 18.702), though this will probably only make a real difference at the end of the semester.

Homework

There will be roughly weekly assignments throughout the semester.

Collaboration policy

You may (and should) work together on problems, but you must write up solutions individually, and you should indicate on your homework who you were working with. In case of ambiguity, I reserve the right to ask you to defend your solutions individually.

Exams

There are no exams in this class.

Grading

100% homework. This is a graduate course, after all, albeit one which is probably suitable for a sufficiently prepared and motivated undergraduate.

Calendar

LEC # TOPICS KEY DATES
1 Introduction to the course; the Riemann zeta function, approach to the prime number theorem  
2 Proof of the prime number theorem  
3 Dirichlet series, arithmetic functions  
4 Dirichlet characters, Dirichlet L-series Problem set 1 due
5 Nonvanishing of L-series on the line Re(s)=1  
6 Dirichlet and natural density, Fourier analysis; Dirichlet's theorem  
7 Prime number theorem in arithmetic progressions; functional equation for zeta Problem set 2 due
8 Functional equation for zeta (cont.)  
9 Functional equations for Dirichlet L-functions  
10 Error bounds in the prime number theorem; the Riemann hypothesis Problem set 3 due
11

Zeroes of zeta in the critical strip; a zero-free region

 
12 A zero-free region; von Mangoldt's formula  
13 von Mangoldt's formula (cont.) Problem set 4 due
14 von Mangoldt's formula; error bounds in arithmetic progressions  
15 Error bounds in arithmetic progressions (cont.)  
16 Introduction to sieve methods: the sieve of Eratosthenes  
17 Guest lecture by Professor Ben Green Problem set 5 due
18 The sieve of Eratosthenes (cont.); Brun's combinatorial sieve  
19 Brun's combinatorial sieve (cont.)  
20 The Selberg sieve Problem set 6 due
21 The Selberg sieve (cont.); applying the Selberg sieve  
22 Introduction to large sieve inequalities  
23 A multiplicative large sieve inequality; an application of the large sieve Problem set 7 due
24 The Bombieri-Vinogradov theorem (statement)  
25 The Bombieri-Vinogradov theorem (proof) Problem set 8 due
26 The Bombieri-Vinogradov theorem (proof, cont.)  
27 The Bombieri-Vinogradov theorem (proof, cont.); prime k-tuples Problem set 9 due
28 Short gaps between primes  
29 Short gaps between primes (cont.)  
30 Short gaps between primes (proofs) Problem set 10 due
31 Short gaps between primes (proofs, cont.)  
32 Short gaps between primes (proofs, cont.)  
33 Artin L-functions and the Chebotarev density theorem Problem set 11 due
34 Artin L-functions  
35 Equidistribution in compact groups  
36 Elliptic curves; the Sato-Tate distribution