This is an archived course. A more recent version may be available at ocw.mit.edu.

Readings

The recommended texts for this class are:

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.

In the table below, they are referred to as "Davenport" and "Iwaniec", respectively, followed by the section number.

LEC # TOPICS READINGS
1 Introduction to the course; the Riemann zeta function, approach to the prime number theorem

The Prime Number Theorem (PDF)

Davenport: 8 and 18.

Iwaniec: 5.4 and 5.6.
2 Proof of the prime number theorem See Lec #1
3 Dirichlet series, arithmetic functions

Dirichlet series and arithmetic functions (PDF)

Iwaniec: 1.
4 Dirichlet characters, Dirichlet L-series

Dirichlet characters and L-functions (PDF)

Davenport: 4.

Iwaniec: 2.3.
5 Nonvanishing of L-series on the line Re(s)=1 See Lec #4
6 Dirichlet and natural density, Fourier analysis; Dirichlet's theorem

Primes in arithmetic progressions (PDF)

Davenport: 4.

Iwaniec: 2.3 and 3.2.
7 Prime number theorem in arithmetic progressions; functional equation for zeta

See Lec #6

The functional equation for the Riemann zeta function (PDF)

Davenport: 20, 22, and 8.

Iwaniec: 4.6 and 5.6.
8 Functional equation for zeta (cont.) See Lec #7
9 Functional equations for Dirichlet L-functions

Functional equations for Dirichlet L-functions (PDF)

Davenport: 9.

Iwaniec: 4.6.
10 Error bounds in the prime number theorem; the Riemann hypothesis

Error bounds in the prime number theorem (PDF)

Davenport: 17.
11 Zeroes of zeta in the critical strip; a zero-free region

More on the zeroes of zeta (PDF)

Davenport: 11 and 13.
12 A zero-free region; von Mangoldt's formula

See Lec #11

von Mangoldt's formula (PDF)

Davenport: 17.
13 von Mangoldt's formula (cont.) See Lec #12
14 von Mangoldt's formula; error bounds in arithmetic progressions

See Lec #12

Error bounds in the prime number theorem in arithmetic progressions (PDF)

Davenport: 14 and 19.

Iwaniec: 5.4 and 5.6.
15 Error bounds in arithmetic progressions (cont.) See Lec #14
16 Introduction to sieve methods: the sieve of Eratosthenes

Revisiting the sieve of Eratosthenes (PDF)

Iwaniec: 6.1 and 6.2.
17 Guest lecture by Professor Ben Green No readings
18 The sieve of Eratosthenes (cont.); Brun's combinatorial sieve

See Lec #16

Brun's combinatorial sieve (PDF)

Iwaniec: 6.2 and 6.3.
19 Brun's combinatorial sieve (cont.) See Lec #18
20 The Selberg sieve

The Selberg sieve (PDF)

Iwaniec: 6.5.
21 The Selberg sieve (cont.); applying the Selberg sieve

See Lec #20

Applying the Selberg sieve (PDF)

Iwaniec: 6.6-6.8.
22 Introduction to large sieve inequalities

Introduction to large sieve inequalities (PDF)

Davenport: 27.

Iwaniec: 7.3 and 7.4.
23 A multiplicative large sieve inequality; an application of the large sieve

A multiplicative large sieve inequality (PDF)

Davenport: 27.

Iwaniec: 7.5.
24 The Bombieri-Vinogradov theorem (statement)

The Bombieri-Vinogradov theorem (statement) (PDF)

Davenport: 28.

Iwaniec: 17.1-17.4.
25 The Bombieri-Vinogradov theorem (proof)

The Bombieri-Vinogradov theorem (proof) (PDF)

Davenport: 28.

Iwaniec: 17.1-17.4.
26 The Bombieri-Vinogradov theorem (proof, cont.) See Lec #25
27 The Bombieri-Vinogradov theorem (proof, cont.); prime k-tuples

See Lec #25

Prime k-tuples (PDF)
28 Short gaps between primes

Small gaps between primes (after Goldston-Pintz-Yildirim) (PDF)

Soundararajan Article

Goldston, et al. Article
29 Short gaps between primes (cont.) See Lec #28
30 Short gaps between primes (proofs)

Small gaps between primes (proofs) (PDF)

See Lec #28
31 Short gaps between primes (proofs, cont.) See Lec #30
32 Short gaps between primes (proofs, cont.) See Lec #30
33 Artin L-functions and the Chebotarev density theorem Artin L-functions and the Chebotarev density theorem (PDF)
34 Artin L-functions See Lec #33
35 Equidistribution in compact groups The Sato-Tate distribution (PDF)
36 Elliptic curves; the Sato-Tate distribution

See Lec #35

Elliptic curves and their L-functions (PDF)