Asymptotics of the d'Arcais Numbers at Small $k$
Shannon Starr
TL;DR
This work analyzes the small-$k$ asymptotics of the d'Arcais numbers $A(2,n,k)$ by establishing that $\alpha(n,k)=\frac{k!\,A(2,n,k)}{n!}$ satisfies $\alpha(n,k)\sim\beta(n,k)$ with $\beta(n,k)=\frac{n^{k-1}(\zeta(2))^{k}\sigma_{-2k+1}(n)}{\Gamma(k)\zeta(2k)}$. The authors derive this via circle-method heuristics rooted in the modularity of the Dedekind eta function, then provide a rigorous induction confirming the asymptotics and exploring log-concavity: $\liminf_{n\to\infty}\frac{\alpha(n,k)^2}{\alpha(n,k-1)\alpha(n,k+1)}=0$ for $k=2$, and a strictly positive limit for $k\ge3$. They further show that for $k\ge3$, the liminf exceeds 1, implying asymptotic log-concavity for large $n$, while $k=2$ yields counterexamples. The work connects these numbers to divisor sums and Ramanujan sums, and outlines open problems for higher $\ell$ and remainder-term analyses using modern analytic techniques.
Abstract
The d'Arcais numbers are the triangular array $\{A(2,n,k)\, :\, n=0,1,\dots,\, k=0,\dots,n\}$, such that $\sum_{n=0}^{\infty} \sum_{k=0}^{n} A(2,n,k) x^k z^n/n! = ((z;z)_{\infty})^{-x}$. The infinite $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{n=1}^{\infty} (1-q^n)$. Holding $k$ fixed and considering large $n$, we note that the ratio $k! A(2,n,k)/n!$ is asymptotic to $C(k) σ_{2k-1}(n)/n^k$ where the divisor sum function is $σ_p(n) = \sum_{d|n} d^p$ and $C(k) = (ζ(2))^k/(Γ(k) ζ(2k))$. This is a slightly generalized version of one of Ramanujan's formulas from his paper, ``On Certain Arithmetical Functions," and it is an immediate consequence of the more recent article of Oliver, Shreshta and Thorne. Heim and Neuhauser made a conjecture, that $A(2,n,k)/A(2,n,k-1)$ is greater than or equal to $A(2,n,k+1)/A(2,n,k)$, for $k=2,3,\dots$ and all $n$. The conjecture is false for $k=2$, and it is true for $k=3,4,\dots$ when $n$ is sufficiently large. We consider the Hardy-Ramanujan circle method as a heuristic step.