Sieving Techniques in Number Theory


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Organizers: Alan Zhao, Austin Lei

As the set of primes eludes a global pattern, two approaches to studying their behavior stand out in the literature: sieves manufactured to target certain primes and the probabilistic approach that captures data on the distribution of all primes. In this seminar, we will follow the recent revival of the targeted sieving approach, initiated first by Zhang and quickly after by Maynard.

If you are not a graduate student and interested in attending, or interested in being added to a mailing list, please email Austin (email can be found on website).

References

  1. Bounded gaps between primes (Zhang)
  2. Small gaps between primes (Maynard)
  3. Large gaps between primes (Maynard)
  4. Small gaps between Primes or Almost Primes (Goldston-Graham-Pintz-Yildrim)
  5. Sieves in Number Theory (Greaves)
  6. Sieve Methods (Richert and Halberstam)
  7. Opera de Cribro (Friedlander and Iwaniec)
  8. A Large Sieve for a Class of non-Abelian L-functions (Goldfeld)
  9. The large sieve and L-functions over finite fields (Kowalski)
  10. An Introduction to Sieve Methods and their Applications (Cojocaru and Murty)
  11. "An exposition of Selberg's Sieve" (Dalton)
  12. "Applications of the Selberg Sieve" (Oh)
  13. "Sieving Using Dirichlet Series" (Murty)
  14. "A Large Sieve Zero Density Estimate for Maass Cusp Forms" (Lewis)

Schedule

We will meet Tuesdays from 5:45-6:45 pm in Math 507.

Date Speaker Abstract References Notes
1/19 Austin Lei Organization and Introduction: We will talk about and decide on a schedule for the following weeks. N/A
1/30 Alan Zhao Basic Setup: We will review and discuss the basic techniques of sieves, such as switching divisors, prime sprigs, and sifting weights, in preparation for the later sieves discussed in the seminar. [7 (Ch. 1, 2, 5.1-4)] notes
2/6 Alan Zhao Basic Setup (continued): We continue discussion of basic techniques of sieves, such as sieve dimension, compositions of sieves, and reduced compositions of sieve-twisted sums. [7 (Ch. 5.5-10)] notes
2/13 N/A No Meeting N/A
2/20 Austin Lei Bombieri Sieve: We discuss how the techniques we have been building up can be used, as well as with improvements, to create the Bombieri Sieve. [7 (Ch. 3)]
2/27 Wenqi Li Brun Sieve: We will discuss Brun's pure sieves and Brun's beta sieves. We will motivate the construction of these sieves from the Buchstab formula, establish general estimates of the main terms, and see an application. In particular, we will give a proof of a twin prime conjecture type of result but for almost primes. [7 (Ch. 6)] notes
3/5 N/A No meeting. N/A
3/12 N/A Spring Break: No meeting. N/A
3/19 Vidhu Adhihetty Selberg Sieve: We will discuss the Selberg sieve, its benefits and limitations, as well as some applications. Some of the applications we aim to discuss include Chebycheff's theorem, the Brun-Titchmarsh theorem, twin primes and more (twin primes and more are time-permitting). [10, 11, 12] notes
3/26 Kevin Chang Sieving methods in arithmetic statistics: I will give an overview of the Davenport-Heilbronn method for counting cubic fields by counting binary cubic forms and sieving.
4/2 Kevin Chang Parametrizations of higher rank rings: I will give an overview of Bhargava's parametrizations of quartic and quintic rings with resolvent by orbits in prehomogeneous vector spaces.
4/9 Aditya Ghosh Small gaps between primes: We'll look at results on short gaps between primes - the GPY sieve and improvements by Zhang and Maynard. [2] notes
4/16 Alan Zhao A Large Sieve Zero Density Estimate for Maass Cusp Forms: We discuss the large sieve and large sieve inequalities and its applications to Maass forms. [14]
4/23 Austin Lei A large sieve for a class of non-Abelian functions: We show another application of the large sieve, this time to compute global density of zeroes of Artin L-functions of specific Kummer fields. [8]