| 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) |
| 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) |
Table of Contents
Study Materials
Readings
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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.