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).

Schedule

We will meet Tuesdays from 5:45-6:45 pm. The room location is TBD.

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 Kevin Chang Selberg Sieve: TBD [7 (Ch 7.1-12)]
3/12 TBD GPY Sieve: TBD [4, 7 (Ch 7.13-15)]
3/19 N/A Spring Break: No meeting. N/A
3/26 TBD Bounded gaps between primes: TBD [1]
4/2 Aditya Ghosh Small gaps between primes: TBD [2]
4/9 Austin Lei Large gaps between primes: TBD [3]
4/16 Austin Lei A large sieve for a class of non-Abelian functions: TBD [8]
4/23 TBD The large sieve and L-functions over finite fields: TBD [9]