Kullback-Leibler divergence and primitive non-deficient numbers

TL;DR

This paper explores inequalities involving the product of prime factors of primitive non-deficient numbers and connects these to the Kullback-Leibler divergence applied to divisor distributions.

Contribution

It introduces new inequalities for primitive non-deficient numbers using Kullback-Leibler divergence and analyzes their implications.

Findings

Established inequalities of the form H(n) > 2 + f(n) for primitive non-deficient numbers.

Connected these inequalities to information-theoretic measures on divisor distributions.

Provided insights into the structure of primitive non-deficient numbers through divergence measures.

Abstract

Let $H(n) = \prod_{p|n}\frac{p}{p-1}$ where $p$ ranges over the primes which divide $n$. It is well known that if $n$ is a primitive non-deficient number, then $H(n) > 2$. We examine inequalities of the form $H(n)> 2 + f(n)$ for various functions $f(n)$ where $n$ is assumed to be primitive non-deficient and connect these inequalities to applying the Kullback-Leibler divergence to different probability distributions on the set of divisors of $n$.