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Extending ErdősKac and SelbergSathe to Beurling primes with controlled integer counting functions Rupert, Malcolm
Abstract
In this thesis we extend two important theorems in analytic prime number theory to a the setting of Beurling primes, namely The Erdős–Kac theorem and a theorem of Sathe and Selberg. The Erdős–Kac theorem asserts that the number of prime factors that divide an integer n is, in some sense, normally distributed with mean log log n and variance log log n. Sathe proved and Selberg substantially refined a formula for the counting function of products of k primes with some uniformity on k. A set of Beurling primes is any countable multiset of the reals with elements that tend towards infinity. The set of Beurling primes has a corresponding multiset of Beurling integers formed by all finite products of Beurling primes. We assume that the Beurling integer counting function is approximately linear with varying conditions on the error term in order to prove the stated results. An interesting example of a set of Beurling primes is the set of norms of prime ideals of the ring of integers of a number field. Recently, Granville and Soundararajan have developed a particularly simple proof of the Erdős–Kac theorem which we follow in this thesis. For extending the theorem of Selberg and Sathe much more analytic machinery is needed.
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Title  Extending ErdősKac and SelbergSathe to Beurling primes with controlled integer counting functions 
Creator  Rupert, Malcolm 
Publisher  University of British Columbia 
Date Issued  2013 
Description 
In this thesis we extend two important theorems in analytic prime number theory to a the setting of Beurling primes, namely The Erdős–Kac theorem and a theorem of Sathe and Selberg. The Erdős–Kac theorem
asserts that the number of prime factors that divide an integer n is, in some sense, normally distributed with mean log log n and variance
log log n. Sathe proved and Selberg substantially refined a formula for the counting function of products of k primes with some uniformity on k. A set of Beurling primes is any countable multiset of the reals with elements that tend towards infinity. The set of Beurling primes has a corresponding multiset of Beurling integers formed by all finite products of Beurling primes. We assume that the Beurling integer counting function is approximately linear with varying conditions on the error term in order to prove the stated results. An interesting example of a set of Beurling primes is the set of norms of prime ideals of the ring of integers of a number field. Recently, Granville and Soundararajan have developed a particularly simple proof of the Erdős–Kac theorem which we follow in this thesis. For extending the theorem of Selberg and Sathe much more analytic machinery is needed.

Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20130419 
Provider  Vancouver : University of British Columbia Library 
Rights  AttributionNonCommercialNoDerivatives 4.0 International 
DOI  10.14288/1.0071950 
URI  
Degree  Master of Science  MSc 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Graduation Date  201305 
Campus  UBCV 
Scholarly Level  Graduate 
Rights URI  
Aggregated Source Repository  DSpace 
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AttributionNonCommercialNoDerivatives 4.0 International