Send the lecturer (R. Hough) email at:
Office: 4-118 Mathematics Building
Office hours: 6-7pm Monday in the MLC, 9-11 am, Friday in 4-118.
Lectures: MW 10:00-11:20pm Physics P122
This course combines an introduction to analytic number theory with a selection of modern topics in prime and combinatorial number theory. Roughly the first third of the course will cover the Prime Number Theorem in arithmetic progressions, following the treatment of Davenport's Multiplicative number theory. The remaining topics are to include the proof of Linnik's Theorem on the least prime in an arithmetic progression and the Bombieri-Vinogradov Theorem, following the treatment of Bombieri's The large sieve in analytic number theory, Maynard's proof of bounded gaps between primes, and the Green-Tao Theorem on arbitrarily long arithmetic progressions in the primes.
| Mon 8/27 | 1. | Davenport Chap. 1 | Primes in arithmetic progressions |
| Wed 8/29 | 2. | Davenport Chap. 2-3 | Gauss sums, cyclotomy |
| Mon 9/3 | No class | Labor day | |
| Wed 9/5 | 3. | Davenport Chap. 4-5 | Primes in arithmetic progressions, primitive characters |
| Mon 9/10 | 4. | Davenport Chap. 6 | Dirichlet's Class Number Formula |
| Wed 9/12 | 5. | Davenport Chap. 7-9 | The distribution of primes, Riemann's memoir, the functional equation |
| Mon 9/17 | 6. | Davenport Chap. 10-12 | Properties of the Gamma function, integral functions of order 1, the infinite product representations |
| Wed 9/19 | 7. | Davenport Chap. 13-14 | The classical zero free region |
| Mon 9/24 | 8. | Davenport Chap. 15-17 | The zero counting functions, and the explicit formula |
| Wed 9/26 | 9. | Davenport Chap. 18-20 | The prime number theorem, PNT in AP I |
| Mon 10/1 | 10. | Davenport Chap. 21-22 | Siegel's theorem, PNT in AP II |
| Wed 10/3 | 11. | Bombieri Chap. 0-2 | The analytic large sieve |
| Mon 10/8 | No class | Fall break | |
| Wed 10/10 | 12. | Bombieri Chap. 3-4 | Selberg's sieve, the multiplicative large sieve |
| Mon 10/15 | 13. | Montgomery Chap. 5 | Turan's method |
| Wed 10/17 | 14. | Bombieri Chap. 6 | Linnik's theorem |
| Mon 10/22 | 15. | Bombieri Chap. 7 | Bombieri-Vinogradov Theorem |
| Wed 10/24 | 16. | Maynard Sec. 4 | The Goldston-Pintz-Yilidirim method |
| Mon 10/29 | 17. | Maynard Sec. 5-6 | Exposing the main terms |
| Wed 10/31 | 18. | Maynard Sec. 7 | The optimization |
| Mon 11/5 | 19. | Roth's theorem | |
| Wed 11/7 | 20. | Gowers sec. 3 | The uniformity norms |
| Mon 11/12 | 21. | Green-Tao sec. 3 | Pseudo-random measures |
| Wed 11/14 | 22. | Green-Tao sec. 5 | The generalized von Neumann inequality |
| Mon 11/19 | 23. | Green-Tao sec. 6 | The dual norms |
| Wed 11/21 | No class | Thanksgiving | |
| Mon 11/26 | 24. | Green-Tao sec. 7-8 | Koopman-von Neumann Theorem |
| Wed 11/28 | 25. | Green-Tao sec. 8 | Proof of the relative Szemerédi Theorem |
| Mon 12/3 | 26. | Green-Tao sec. 9 | Construction of the pseudo-random measure |
| Wed 12/5 | 27. | The Lovász Local Lemma | |
| Mon 12/10 | 28. | Hough | Covering systems of congruences |
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