Measuring Abundance with Abundancy Index

Abstract

A positive integer $n$ is called perfect if $ \sigma(n)=2n$, where $\sigma(n)$ denote the sum of divisors of $n$. In this paper we study the ratio $\frac{\sigma(n)}{n}$. We define the function Abundancy Index $I:\mathbb{N} \to \mathbb{Q}$ with $I(n)=\frac{\sigma(n)}{n}$. Then we study different properties of the Abundancy Index and discuss the set of Abundancy Index. Using this function we define a new class of numbers known as superabundant numbers. Finally, we study superabundant numbers and their connection with Riemann Hypothesis.