Some Observations about the "Generalized Abundancy Index"

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 $-\zeta(2) + N^{-1}\sum_{n=1}^{N} (B(2,n)/n)$.