This is the webpage for the section of Undergraduate Seminars II focused on additive number theory. The syllabus can be found here.

**Instructor:** Avi Zeff

**Time/place:** Wednesday 11:40 AM - 1:30 PM, in 528 Mathematics

Homework 2, solutions

Homework 3 (

Final project guidelines

Note that the quality of the video is unfortunately not very good, so it can be difficult to read what's on the board; I recommend watching the video with the notes up in order to follow along. I'll try and fix this for future meetings.

Date |
Speaker |
Topic |
Sources |
Materials |

January 24 | Avi | Introduction and logistics | [N, §1.2] | Notes |

January 31 | Jinoo | Quadratic forms and sums of three squares | [N, §1.3-1.5] | Slides, notes |

January 31 | Keila | Waring's problem for cubes | [N, §2.1-2.3] | Notes, video |

February 7 | Connor | Quantitative estimates and sums of two cubes | [N, §1.6-1.7, 2.4] | Notes |

February 7 | Robert | Waring's problem in general | [N, §3] | Notes, video |

February 14 | Katie | Tools and "easier Waring's problem" | [N, §4.1-4.3] | Notes |

February 14 | Melody | Weyl's inequality and Hua's lemma | [N, §4.4-4.5] | Notes, video |

February 21 | Lily | Generating series | [LR] | Notes |

February 21 | Akash | The circle method | [A, §1] | Notes, video |

February 28 | Johnny | The Hardy-Littlewood asymptotic formula | [N, §5] | Notes |

February 28 | Avi | Dirichlet series and arithmetic functions | Notes, video | |

March 6 | Johnny | Introduction to sieves | [C, §1] | Notes |

March 6 | Julie | Further applications of the circle method | [A, §2] | Notes, video |

March 20 | Connor | Brun's combinatorial sieve and twin primes | [N, §6.4], [C, §2.1] | Notes, video |

March 20 | Akash | More Brun's sieve | [C, §2.2-2.6] | Notes, video |

March 27 | Jinoo | Selberg's sieve and Goldbach's conjecture | [N, §7.1-7.3] | Notes |

March 27 | Keila | The Shnirelman-Goldbach theorem | [N, §7.4-7.5] | Notes, video |

April 3 | Lily | The large sieve and the Bombieri-Vinogradov theorem | [MV], [P] | Notes, video |

April 3 | Julie | Vinogradov's theorem | [N, §8] | Notes, video |

April 10 | Katie | The linear sieve and the Jurkat-Richert theorem | [N, §9] | Notes, video |

April 10 | Robert | Chen's theorem | [N, §10] | Notes, video |

April 17 | Melody | Bounded gaps between primes | [Z] | Video |

April 24 | Matthew | Curves on hypersurfaces via the circle method | Video |

[A] Lambert A'Campo. The Circle Method, Applications to the Partition Function, and Beyond, 2006.

[C] Denis Xavier Charles. *Sieve methods*. Department of Computer Science, State University of New York at Buffalo, 2000.

[LR] Tom Leighton and Ronitt Rubinfield. 6.042/18.062J Mathematics for Computer Science, Fall 2006.

[MV] Hugh L. Montgomery and Robert C. Vaughan. The Large Sieve. *Mathematika* 20(2): p. 119-134, 1973.

[N] Melvyn B. Nathanson. *Additive number theory*. New York: Springer, 1996.

[P] Peter S. Park. The Bombieri-Vinogradov theorem. Expository, preprint available http://web.math.princeton.edu/pspark/papers/bv.pdf, 2016.

[Z] Yitang Zhang. Bounded gaps between primes. *Annals of Mathematics*: p. 1121-1174, 2014.