Connecting arithmetic functions and continuous distribution functions

Abstract

In this thesis, we discuss some classical results in probabilistic number theory, focusing on when the outputs of an arithmetic function, usually multiplicative, attain a continuous distribution function. We study the inception of these theories around the early 20th century in the work of Schoenberg, who inspired Davenport to show that abundant numbers have a continuous distribution. It was not until 2013 that Jennings, Pollack and Thompson looked at this problem from a different perspective an