Measuring Abundance with Abundancy Index

TL;DR

This paper explores the properties of the Abundancy Index, introduces superabundant numbers, and investigates their potential link to the Riemann Hypothesis, advancing understanding of number classification and divisor functions.

Contribution

It defines the Abundancy Index, studies its properties, introduces superabundant numbers, and examines their connection to the Riemann Hypothesis, offering new insights into divisor ratios.

Findings

Properties of the Abundancy Index analyzed

Superabundant numbers characterized and defined

Connections between superabundant numbers and Riemann Hypothesis explored

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.