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Université de Montréal, Fall 2018

Instructor: Dimitris Koukoulopoulos |

- Monday 10:00 - 11:00 (4186 Pav. André Aisenstadt)
- Wednesday 10:00 - 12:00 (4186 Pav. André Aisenstadt)

The course starts on Wednesday September 5 and ends on Wednesday December 5. There is no course on Monday October 1 (Québec election day), on Monday October 8 (Thanksgiving), and during the week of October 22 (Fall break). In order to replace some of the missed classes, the Monday course will be extended till noon on September 24, October 15, and October 29.

I will follow my own course notes (to be distributed by e-mail). My old course notes can be found here. Other useful resources are given by:

- Friedlander, John; Iwaniec, Henryk. Opera de cribro. American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 2010.
- Halberstam, H.; Richert, H.-E. Sieve methods. London Mathematical Society Monographs, No. 4. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974.
- Kevin Ford's online notes

Sieve methods is a technical subject and it is important to develop good intuition about them. For this reason, every Wednesday, starting from Wednesday September 12, there will be two 30' student presentations. The first one will be from 10 to 10:30, followed by discussion, and the second one from 11 to 11:30, again followed by discussion. In order for the discussion to work as well as possible, it is important that you read the assigned material beforehand and you are familiar with the technical details of the proofs, as we will be focusing mainly on the intuition behind the proofs.

There will also be a homework assignment consisting of five problems. You must type your solutions in LaTeX and send them to me by email. The due dates (always on Monday, by 10am) are:

- Problem 1: September 24
- Problems 2-3: October 29
- Problems 4-5: December 3

In the end of the semester, your grades will be determined as follows:

Type of evaluation | Percentage |
---|---|

Presentations | 40% |

Class participation | 10% |

Homework | 50% |

Grade | Points |
---|---|

A+ | 90 |

A | 70 |

A- | 60 |

B+ | 50 |

B | 40 |

B- | 30 |

C+ | 20 |

C | 10 |

F | 0 |

(The timeline is indicative; the content might vary in practice.)

- Monday:
**no course**(labor day) - Wednesday: sieve of Eratosthenes-Legendre (chapter 2), the idea behind Brun's combinatorial sieve (chapter 16)

- Monday: application to twin primes (ch. 16); the Kubilius model of the integers and the Hardy-Ramanujan inequality (ch. 12)
- Wednesday: the method of moments and the Erdos-Kac theorem (Youcef); the saddle-point method and Theorem 12.2 (Peter)

- Monday: averages of multiplicative functions via the LSD method and via differential-delay equations (ch. 13 and 14)
- Wednesday: Presentations by Stelios (ch 13 - LSD) and Frank (ch 14 - DDEs)

- Monday (we meet from 10 to 12):
**HW problem #1 is due**; smooth numbers via differential-delay equations and via the saddle-point method (ch. 14 and 15) - Wednesday: Presentations by Tony (ch 14 - Tony) and Martin (ch 15 - Martin)

- Monday:
**no course**(Québec election day) - Wendesday: Axioms of sieve methods (ch. 17)

- Monday:
**no course**(Thanksgiving) - Wednesday: Constructing general sieve weights (ch. 17); the fundamental lemma and its applications (ch. 18); Presentation of Cor. 18.2 and 18.3 by Simon

- Monday (we meet from 10 to 12): Selberg's sieve(ch 19)
- Wednesday: The parity problem (presentation by Antoine); Bilinear sums

No classes this week (Fall break)

- Monday (we meet from 10 to 12):
**HW problems 2 and 3 are due**; bilinear sums and Linnik's theorem (ch. 21, 22) - Wednesday: Presentations

- Monday: the large sieve
- Wednesday: Presentations

- Monday: the Bombieri-Vinogradov theorem
- Wednesday: Presentations

- Monday: Small gaps between primes
- Wednesday: Presentations

- Monday: Large gaps between primes
- Wednesday: Large gaps between primes

- Monday:
**HW problems 4 and 5 are due**; Large gaps between primes (ctd.); irregularities among primes - Wednesday: The circle method and Bombieri's sieve