# Courses:

## Analytic Number Theory >> Content Detail

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The recommended texts for this class are: Davenport, H., and H. L. Montgomery. Multiplicative Number Theory. New York, NY: Springer-Verlag, 2000. ISBN: 9780387950976. 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.

1Introduction 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.

2Proof of the prime number theoremSee Lec #1
3Dirichlet series, arithmetic functions

Dirichlet series and arithmetic functions (PDF)

Iwaniec: 1.

4Dirichlet characters, Dirichlet L-series

Dirichlet characters and L-functions (PDF)

Davenport: 4.

Iwaniec: 2.3.

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

Primes in arithmetic progressions (PDF)

Davenport: 4.

Iwaniec: 2.3 and 3.2.

7Prime 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.

8Functional equation for zeta (cont.)See Lec #7
9Functional equations for Dirichlet L-functions

Functional equations for Dirichlet L-functions (PDF)

Davenport: 9.

Iwaniec: 4.6.

10Error bounds in the prime number theorem; the Riemann hypothesis

Error bounds in the prime number theorem (PDF)

Davenport: 17.

11Zeroes of zeta in the critical strip; a zero-free region

More on the zeroes of zeta (PDF)

Davenport: 11 and 13.

12A zero-free region; von Mangoldt's formula

See Lec #11

von Mangoldt's formula (PDF)

Davenport: 17.

13von Mangoldt's formula (cont.)See Lec #12
14von 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.

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

Revisiting the sieve of Eratosthenes (PDF)

Iwaniec: 6.1 and 6.2.

17Guest lecture by Professor Ben GreenNo readings
18The sieve of Eratosthenes (cont.); Brun's combinatorial sieve

See Lec #16

Brun's combinatorial sieve (PDF)

Iwaniec: 6.2 and 6.3.

19Brun's combinatorial sieve (cont.)See Lec #18
20The Selberg sieve

The Selberg sieve (PDF)

Iwaniec: 6.5.

21The Selberg sieve (cont.); applying the Selberg sieve

See Lec #20

Applying the Selberg sieve (PDF)

Iwaniec: 6.6-6.8.

22Introduction to large sieve inequalities

Introduction to large sieve inequalities (PDF)

Davenport: 27.

Iwaniec: 7.3 and 7.4.

23A multiplicative large sieve inequality; an application of the large sieve

A multiplicative large sieve inequality (PDF)

Davenport: 27.

Iwaniec: 7.5.

Davenport: 28.

Iwaniec: 17.1-17.4.

Davenport: 28.

Iwaniec: 17.1-17.4.

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

See Lec #25

Prime k-tuples (PDF)

28Short gaps between primes

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

Soundararajan Article

Goldston, et al. Article

29Short gaps between primes (cont.)See Lec #28
30Short gaps between primes (proofs)

Small gaps between primes (proofs) (PDF)

See Lec #28

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

See Lec #35

Elliptic curves and their L-functions (PDF)