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.
Date of Degree
Master of Arts (MA)
Mathematics and Statistics
Science, Engineering and Technology
Jahn, R. L. (2016). A Measure Theoretic Approach to Problems of Number Theory with Applications to the Proof of the Prime Number Theorem [Master’s thesis, Minnesota State University, Mankato]. Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato. https://cornerstone.lib.mnsu.edu/etds/589/
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