Here are the units covered in the lecture notes. The exercises are attached, but the problem sets were subdivided somewhat differently; this division appears on the course home page.
|
Date |
Section(s) |
Topics |
|
Mon 2/5 |
|
(registration day) |
|
Wed 2/7 (KSK away) |
notes; Davenport: 8, 18; Iwaniec: 5.4, 5.6 |
Introduction to the course; the Riemann zeta function, approach to the prime number theorem |
|
Fri 2/9 (KSK away) |
see 2/7 |
Proof of the prime number theorem |
|
Mon 2/12 |
notes; Iwaniec: 1.all |
Dirichlet series, arithmetic functions |
|
Wed 2/14 |
notes; Davenport: 4; Iwaniec: 2.3 |
Dirichlet characters, Dirichlet $L$-series |
|
Fri 2/16 |
see 2/14 |
Nonvanishing of L-series on the line Re(s)=1 |
|
Mon 2/19 |
|
NO LECTURE (Presidents Day) |
|
Tue 2/20 (MIT Monday) |
notes; Davenport: 4; Iwaniec: 2.3, 3.2; |
Dirichlet and natural density, Fourier analysis; Dirichlet's theorem |
|
Wed 2/21 |
see 2/20; notes; Davenport: 20, 22, 8; Iwaniec: 4.6, 5.6 |
Prime number theorem in arithmetic progressions; functional equation for zeta |
|
Fri 2/23 |
see 2/21 |
Functional equation for zeta (continued) |
|
Mon 2/26 |
notes; Davenport: 9; Iwaniec: 4.6 |
Functional equations for Dirichlet L-functions |
|
Wed 2/28 |
notes; Davenport: 17 |
Error bounds in the prime number theorem; the Riemann hypothesis |
|
Fri 3/2 |
notes; Davenport: 11, 13 |
Zeroes of zeta in the critical strip; a zero-free region |
|
Mon 3/5 |
see 3/2; notes; Davenport 17 |
A zero-free region; von Mangoldt's formula |
|
Wed 3/7 |
see 3/5 |
von Mangoldt's formula |
|
Fri 3/9 |
see 3/5; notes; Davenport, 14, 19; Iwaniec, 5.4, 5.6 |
von Mangoldt's formula; error bounds in arithmetic progressions |
|
Mon 3/12 |
|
NO LECTURE (KSK away) |
|
Wed 3/14 |
|
NO LECTURE (KSK away) |
|
Fri 3/16 |
see 3/9 |
Error bounds in arithmetic progressions |
|
Mon 3/19 |
notes; Iwaniec: 6.1, 6.2 |
Introduction to sieve methods: the sieve of Eratosthenes |
|
Wed 3/21 |
|
Guest lecture by Ben Green |
|
Fri 3/23 |
see 3/19; notes; Iwaniec: 6.2, 6.3 |
The sieve of Eratosthenes; Brun's combinatorial sieve |
|
3/26-30 |
|
NO LECTURES (spring break) |
|
Mon 4/2 |
see 3/23 |
Brun's combinatorial sieve |
|
Wed 4/4 |
notes; Iwaniec: 6.5 |
The Selberg sieve |
|
Fri 4/6 |
see 4/4; notes; Iwaniec: 6.6-6.8 |
The Selberg sieve; applying the Selberg sieve |
|
Mon 4/9 |
notes; Davenport: 27; Iwaniec: 7.3, 7.4 |
Introduction to large sieve inequalities |
|
Wed 4/11 |
notes; Davenport: 27; Iwaniec: 7.5 |
A multiplicative large sieve inequality; an application of the large sieve |
|
Fri 4/13 |
notes; Davenport: 28; Iwaniec: 17.1-17.4 |
The Bombieri-Vinogradov theorem (statement) |
|
Mon 4/16 |
|
NO LECTURE (Patriots Day) |
|
Wed 4/18 |
notes; Davenport: 28; Iwaniec: 17.1-17.4 |
The Bombieri-Vinogradov theorem (proof) |
|
Fri 4/20 |
|
NO LECTURE (KSK away) |
|
Mon 4/23 |
see 4/18 |
The Bombieri-Vinogradov theorem (proof) |
|
Wed 4/25 |
see 4/18; notes |
The Bombieri-Vinogradov theorem (proof); prime k-tuples |
|
Fri 4/27 |
Short gaps between primes |
|
|
Mon 4/30 |
see 4/27 |
Short gaps between primes |
|
Wed 5/2 |
notes; also see 4/27 |
Short gaps between primes (proofs) |
|
Fri 5/4 |
see 5/2 |
Short gaps between primes (proofs) |
|
Mon 5/7 |
see 5/4 |
Short gaps between primes (proofs) |
|
Wed 5/9 |
Artin L-functions and the Chebotarev density theorem |
|
|
Fri 5/11 |
see 5/9 |
Artin L-functions |
|
Mon 5/14 |
Equidistribution in compact groups |
|
|
Wed 5/16 |
see 5/14; notes |
Elliptic curves; the Sato-Tate distribution |