Asymptotics of the d'Arcais Numbers at Small $k$

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 $C(k){\sigma }_{2k-1}(n)/n^k$ where the divisor sum function is ${\sigma }_p(n)={\sum }_{d|n}{d}^p$ and $C(k)=(\zeta (2){)}^k/(\Gamma (k)\zeta (2k))$. 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.

15 pages, 1 figure