Higher rank sieves and applications

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Authors

Vatwani, Akshaa

Date

2016-04-25

Type

thesis

Language

eng

Keyword

Sieves , Distribution of primes

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Abstract

This thesis focuses on some of the key sieve theoretic ideas behind recent progress on bounded gaps between the primes. One such idea is the notion of higher rank sieve weights, first proposed by Atle Selberg and applied successfully to the context of prime k-tuples by J. Maynard and T. Tao. We develop an axiomatic formulation for a general higher rank sieve, in the spirit of Selberg's own treatment of his classical sieve. We apply this theory to an assortment of problems such as almost prime k-tuples and prime k-tuples in imaginary quadratic fields with class number 1. Another novel idea that was brought to the forefront by the path-breaking work of Yitang Zhang is that of obtaining new equidistribution results for the primes by making the moduli "smooth" or free of large prime factors. We develop a general method to incorporate the technique of smoothing the moduli into the higher rank sieve and apply this to prime k-tuples. In a different vein, the last chapter of the thesis expands upon the well-known parity principle in sieve theory. We show that sufficient "randomness" in the sign of the M\"obius function, combined with another conjecture about the equidistribution of the primes in arithmetic progressions, can be used to break the parity barrier and yield infinitely many twin primes.

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Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-04-23 13:21:39.438

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This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.

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