Some Observations about the "Generalized Abundancy Index"

TL;DR

This paper studies the generalized abundancy index derived from the combinatorial structure of commuting permutations, proving its average value converges to a product of zeta functions and proposing related conjectures.

Contribution

It introduces the generalized abundancy index, connects it to permutation group structures, and establishes its asymptotic behavior using probabilistic methods.

Findings

The average of the generalized abundancy index converges to a8a8a8 a8a8a8 as N a8a8a8 a8a8a8.

The paper links combinatorial permutation structures to analytic number theory via the zeta function.

It formulates a conjecture on the asymptotics of the abundancy index for n=2.

Abstract

Let $\mathcal{A}(\ell,n) \subset S_n^{\ell}$ denote the set of all $\ell$-tuples $(\pi_1,\dots,\pi_{\ell})$, for $\pi_1,\dots,\pi_{\ell} \in S_n$ satisfying: $\forall i<j$ we have $\pi_i\pi_j=\pi_j\pi_i$. Considering the action of $S_n$ on $[n]=\{1,\dots,n\}$, let $\kappa(\pi_1,\dots,\pi_{\ell})$ be equal to the number of orbits of the action of the subgroup $\langle \pi_1,\dots,\pi_{\ell} \rangle \subset S_n$. There has been interest in the study of the combinatorial numbers $A(\ell,n,k)$ equal to the cardinalities $|\{(\pi_1,\dots,\pi_{\ell}) \in \mathcal{A}(\ell,n)\, :\, \kappa(\pi_1,\dots\pi_{\ell})=k\}|$. If one defines $B(\ell,n)=A(\ell,n,1)/(n-1)!$, then it is known that $B(\ell,n) = \sum_{(f_1,\dots,f_{\ell}) \in \mathbb{N}^{\ell}} \mathbf{1}_{\{n\}}(f_1\cdots f_{\ell}) \prod_{r=1}^{\ell-1} f_r^{\ell-r}$. A special case, $\ell=2$, is $B(2,n) = \sum_{d|n} d = \sigma_1(n)$ the sum-of-divisors function. Then $A(2,n,1)/n!=B(2,n)/n$ is called the abundancy index: $\sigma_1(n)/n$. We call $B(\ell,n) n^{-\ell+1}$ the ``generalized abundancy index.'' Building on work of Abdesselam, using the probability model, we prove that $\lim_{N \to \infty} N^{-1} \sum_{n=1}^{N} B(\ell,n) n^{-\ell+1}$ equals $\zeta(2)\cdots \zeta(\ell)$. Motivated by this we state a more precise conjecture for the asymptotics of $-\zeta(2) + N^{-1}\sum_{n=1}^{N} (B(2,n)/n)$.