Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.540255 
Title:  A sieve problem over the Gaussian integers  
Author:  Schlackow, Waldemar 
ISNI:
0000 0004 2709 0763


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  2010  
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Abstract:  
Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors. We prove this by first developing the theory of $L$functions for Gaussian primes by using standard methods. We then give an exposition of the SiegelWalfisz Theorem for Gaussian primes and a corresponding Prime Number Theorem for Gaussian Arithmetic Progressions. Finally, we prove the main result by using the developed theory together with Sieve Theory and specifically a weighted linear sieve result to bound the number of prime factors of a² + 4b². For the application of the sieve, we need to derive a specific version of the BombieriVinogradov Theorem for Gaussian primes which, in turn, requires a suitable version of the Large Sieve. We are also able to get the number of prime factors of a² + 4b² as low as 3 if we assume the Generalised Riemann Hypothesis.


Supervisor:  HeathBrown, Roger  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.540255  DOI:  Not available  
Keywords:  Mathematics ; Number theory ; Lfunctions ; Gaussian integers ; Gaussian primes ; Riemann Hypothesis ; Sieve Theory  
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