18.785 Lecture Plan

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.

In the Sections column of the following table, "Davenport" means Davenport, Multiplicative Number Theory, and "Iwaniec" means Iwaniec-Kowalski, Analytic Number Theory. "Notes" are links to PDF files in the above list.




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



Wed 3/14



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


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


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

notes; Soundararajan article; Goldston et al article

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