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.

Accepted in Mathematics Exchange (Ball State University), Vol 15, 2021