Abstract

In this paper we demonstrate how the principles of measure theory can be applied effectively to problems of number theory. Initially, necessary concepts from number theory will be presented. Next, we state standard concepts and results from measure theory to which we will need to refer. We then develop our repertoire of measure theoretic machinery by constructing the needed measures and defining a generalized version of the multiplicative convolution of measures. A suitable integration by parts formula, one that is general enough to handle various combinations of measures, will then be derived. At this juncture we will be ready to demonstrate the effectiveness of measure theory tactics on number theory problems. Specifically, due to its highly receptive nature to measure theoretic techniques, the prime number theorem will be proved. First, we prove the theorem by what is termed an "elementary" method. Secondly, the Riemann zeta function is employed to enable us to give a much shorter proof. In both cases, we borrow the ideas from several sources and apply them to our proofs. The approach taken in this paper, however, is distinctive in the sense that the driving force of the proofs is measure theory. Although there is greater overhead in learning the appropriate material used in this approach, we argue that once this material is understood it can be beneficially applied to problems of suitable complexity.

Advisor

Wook Kim

First Committee Member

Brian Martensen

Second Committee Member

Dan Singer

Date of Degree

2016

Language

english

Document Type

Thesis

Degree

Master of Arts (MA)

Department

Mathematics and Statistics

College

Science, Engineering and Technology

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

Included in

Number Theory Commons

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