Some Observations about the "Generalized Abundancy Index"

TL;DR

The paper investigates the generalized abundancy index arising from commuting $\ell$-tuples in $S_n$ by introducing $A(\ell,n,k)$ and $B(\ell,n)$ and studying the generalized abundancy index $B(\ell,n)/n^{\ell-1}$. Building on the Abdesselam probability model, it proves the Cesàro limit $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N} B(\ell,n)n^{-\ell+1}=\zeta(2)\cdots\zeta(\ell)$ and develops a generating-function framework $\mathcal{G}_{\ell}$ and $\mathcal{H}_{\ell,n}$ to analyze large-$n$ behavior and deviations. The authors provide rigorous results on Cesàro averages, large deviations, and a central limit theorem (to appear in a forthcoming preprint), supplemented by extensive numerical evidence for the case $\ell=2$, and propose a refined conjecture for the asymptotics of $-\zeta(2) + (1/N)\sum_{n=1}^{N} (B(2,n)/n)$. They conclude with heuristic saddle-point calculations that motivate a broader conjectural picture for $A(\ell,n,k)/n!$, including logarithmic corrections and a constant $\mu$ in the $\ell=2$ case, linking analytic, probabilistic, and combinatorial perspectives.

Abstract

Let $\mathcal{A}(\ell,n) \subset S_n^{\ell}$ denote the set of all $\ell$-tuples $(π_1,\dots,π_{\ell})$, for $π_1,\dots,π_{\ell} \in S_n$ satisfying: $\forall i<j$ we have $π_iπ_j=π_jπ_i$. Considering the action of $S_n$ on $[n]=\{1,\dots,n\}$, let $κ(π_1,\dots,π_{\ell})$ be equal to the number of orbits of the action of the subgroup $\langle π_1,\dots,π_{\ell} \rangle \subset S_n$. There has been interest in the study of the combinatorial numbers $A(\ell,n,k)$ equal to the cardinalities $|\{(π_1,\dots,π_{\ell}) \in \mathcal{A}(\ell,n)\, :\, κ(π_1,\dotsπ_{\ell})=k\}|$. 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 $B(2,n) = \sum_{d|n} d = σ_1(n)$ the sum-of-divisors function. Then $A(2,n,1)/n!=B(2,n)/n$ is called the abundancy index: $σ_1(n)/n$. 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 $ζ(2)\cdots ζ(\ell)$. Motivated by this we state a more precise conjecture for the asymptotics of $-ζ(2) + N^{-1}\sum_{n=1}^{N} (B(2,n)/n)$.

Figures (3)

• Figure 1: A histogram of $\sum_{N=1}^{\mathcal{N}} \left(\sum_{n=1}^{N} \frac{B(2,n)}{n}-\zeta(2) N+\frac{1}{2}\, \ln(N)+\mu\right)$ for $\mathcal{N}$ up to $10^6$. There are 250 bins.
• Figure 2: Four discrete $\ell=2$ dimensional tori with a twist. We have $n=24$, $f_1=4$, $f_2=6$. Reading left-to-right the choices of $\phi^{(2)}_1 \in [f_1]$ are $1,2,3,4$. The white circles represent the vertices on the opposite face, repeated so as to draw the graphs without extraneous crossings.
• Figure 3: An indication of a further twist in case $\ell=3$. Suppose we choose $f_1=4$, $f_2=6$ and $f_3=n/(f_1f_2)$ for some number $n \in f_1 f_2 \mathbb{N}$. We take $\phi^{(2)}_1 = 4 \in [f_1]$. Now let us take $(\phi^{(3)}_1,\phi^{(3)}_2) = (2,3) \in [f_1] \times [f_2]$. Then we give an indication of the bijection $\pi$ on $[f_1]\times[f_2]$ such that to go from $(x,y,f_3)$ to a point on the opposite face we use $(\pi(x,y),1)$. The permutation is shown in cycle decomposition. There are two cycles for this particular permutation: one shown in blue, one in red.

Theorems & Definitions (7)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• Corollary 2.2
• Conjecture 2.3
• Conjecture 2.4: Main conjecture
• Remark 4.1