| 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.
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| 2 |
Proof of the prime number theorem |
See Lec #1 |
| 3 |
Dirichlet series, arithmetic functions |
Dirichlet series and arithmetic functions (PDF)
Iwaniec: 1.
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| 4 |
Dirichlet characters, Dirichlet L-series |
Dirichlet characters and L-functions (PDF)
Davenport: 4.
Iwaniec: 2.3.
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| 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.
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| 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.
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| 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.
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| 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.
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| 23 |
A multiplicative large sieve inequality; an application of the large sieve |
A multiplicative large sieve inequality (PDF)
Davenport: 27.
Iwaniec: 7.5.
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| 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.
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| 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)
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| 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)
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