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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.
Item Metadata
Title 
Extending ErdősKac and SelbergSathe to Beurling primes with controlled integer counting functions

Creator  
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  
Type  
Language 
eng

Date Available 
20130419

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0071950

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201305

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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Rights
AttributionNonCommercialNoDerivatives 4.0 International