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This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions). There is also a bit of discussion of non-abelian equidistribution results, like the Chebotarev density theorem, and the Sato-Tate conjecture for elliptic curves (and recent progress on same by Clozel, Harris, Shepherd-Barron, Taylor). For a detailed description of the content and structure of the course, please see the first set of lecture notes. (PDF)

Textbooks

There is no required text for this class. Detailed lecture notes are provided in lieu of following a specific text.

Recommended Texts

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.

Prerequisites

Complex analysis (18.112), some background in number theory (at the level of 18.781). There are one or two points where a little algebraic number theory (as in 18.786) may be helpful, but it is in no way required. On the other hand, the 18.112 prerequisite is quite serious; if you do not formally meet it, you must let me know in writing what your equivalent preparation is. Also recommended: some abstract algebra (18.701 and 18.702), though this will probably only make a real difference at the end of the semester.

Homework

There will be roughly weekly assignments throughout the semester.

Collaboration policy

You may (and should) work together on problems, but you must write up solutions individually, and you should indicate on your homework who you were working with. In case of ambiguity, I reserve the right to ask you to defend your solutions individually.

Exams

There are no exams in this class.

Grading

100% homework. This is a graduate course, after all, albeit one which is probably suitable for a sufficiently prepared and motivated undergraduate.

Calendar

Course calendar.LEC # | TOPICS | KEY DATES |
---|

1 | Introduction to the course; the Riemann zeta function, approach to the prime number theorem | |

2 | Proof of the prime number theorem | |

3 | Dirichlet series, arithmetic functions | |

4 | Dirichlet characters, Dirichlet L-series | Problem set 1 due |

5 | Nonvanishing of L-series on the line Re(s)=1 | |

6 | Dirichlet and natural density, Fourier analysis; Dirichlet's theorem | |

7 | Prime number theorem in arithmetic progressions; functional equation for zeta | Problem set 2 due |

8 | Functional equation for zeta (cont.) | |

9 | Functional equations for Dirichlet L-functions | |

10 | Error bounds in the prime number theorem; the Riemann hypothesis | Problem set 3 due |

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

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

13 | von Mangoldt's formula (cont.) | Problem set 4 due |

14 | von Mangoldt's formula; error bounds in arithmetic progressions | |

15 | Error bounds in arithmetic progressions (cont.) | |

16 | Introduction to sieve methods: the sieve of Eratosthenes | |

17 | Guest lecture by Professor Ben Green | Problem set 5 due |

18 | The sieve of Eratosthenes (cont.); Brun's combinatorial sieve | |

19 | Brun's combinatorial sieve (cont.) | |

20 | The Selberg sieve | Problem set 6 due |

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

22 | Introduction to large sieve inequalities | |

23 | A multiplicative large sieve inequality; an application of the large sieve | Problem set 7 due |

24 | The Bombieri-Vinogradov theorem (statement) | |

25 | The Bombieri-Vinogradov theorem (proof) | Problem set 8 due |

26 | The Bombieri-Vinogradov theorem (proof, cont.) | |

27 | The Bombieri-Vinogradov theorem (proof, cont.); prime k-tuples | Problem set 9 due |

28 | Short gaps between primes | |

29 | Short gaps between primes (cont.) | |

30 | Short gaps between primes (proofs) | Problem set 10 due |

31 | Short gaps between primes (proofs, cont.) | |

32 | Short gaps between primes (proofs, cont.) | |

33 | Artin L-functions and the Chebotarev density theorem | Problem set 11 due |

34 | Artin L-functions | |

35 | Equidistribution in compact groups | |

36 | Elliptic curves; the Sato-Tate distribution | |