Kullback-Leibler divergence and primitive non-deficient numbers

TL;DR

The paper links information-theoretic quantities to divisor structures of integers by using the Kullback–Leibler divergence between two probability distributions on the divisors of $n$. By defining $P(d)=\frac{\phi(d)}{n}$ and a split-weighted $Q(d)$ based on $H(d)=\frac{n}{\phi(n)}$, it forms $\mathrm{KL}(n)=\sum_{d|n} d\log\frac{d}{2\phi(d)}$ and investigates the surplus $S(n)=H(n)-2$. For odd primitive non-deficient $n$, the positivity of $\mathrm{KL}(n)$ yields explicit lower bounds on $S(n)$, notably $S(n) > \frac{2\log(24\sqrt{2}/25)}{\sqrt{n}}$ (with refinements for the perfect vs abundant cases) and a corollary $S(n) > \frac{3}{5\sqrt{n}}$ for odd non-deficient $n$. The work further introduces concepts such as $KL$-primitive numbers and record-setter sets, discusses potential improvements via Pinsker-type inequalities, and explores alternative divisor-weighted divergences, connecting entropy-like ideas to classical divisor sums and odd-perfect-number theory. The results provide a quantitative, information-theoretic lens on divisor sums and primitive non-deficient numbers, with open questions about infinite families and sharper bounds.

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 $n$.

Figures (3)

• Figure 1: $h(n)$ and ${\rm KL}(n)$
• Figure 2: $h(n)$ and ${\rm KL}(n)$ zoomed in near $h(n)=2$
• Figure 3: $h(n)$ and ${\rm KL}(n)$ zoomed in further near $h(n)=2$

Theorems & Definitions (37)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof : Proof of Theorem \ref{['Main new surplus theorem with square root bound']}