Large Deviations for the d'Arcais Numbers

TL;DR

This work proves a Bahadur-Rao type local large deviation formula for the d'Arcais numbers $A(2,n,k)$ in the regime $k_n/n\to\kappa\in[0,1)$ with $k_n\to\infty$, expressing the asymptotics of $\frac{k_n!\,A(2,n,k_n)}{n!}$ via the generating function $F(y)=-\ln((e^{-y};e^{-y})_{\infty})$ and its Legendre transform. The rate function arises as the Legendre–Fenchel transform of $f(y)=\ln F(y)$, with $y_n$ solving $-F'(y_n)/F(y_n)=1/\kappa_n$, yielding a precise BR-type asymptotic including a nontrivial prefactor. The paper also establishes asymmetry of the rate function, relates the result to the classical Bahadur–Rao framework, and derives a corollary on log-concavity for large $n$; in addition, it discusses the abundancy-index asymptotics and re-derives $\,\ell=2$ asymptotics via modular symmetry and a Hardy–Ramanujan style circle method. The results illuminate the probabilistic structure behind the d'Arcais numbers and connect to the underlying number-theoretic objects via the abundancy index and modular forms.

Abstract

The d'Arcais polynomials $P_n(z)$ for $n\in\{0,1,\dots\}$ are defined as $\sum_{n=0}^{\infty} P_n(z) q^n = \exp(-z\ln((q;q)_{\infty}))$ where the $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k)$ for $|q|<1$. Denoting the coefficients for $n \in \mathbb{N}$ by the formula $P_n(z) = \sum_{k=1}^{n} A(2,n,k) z^k/n!$, we prove that $k_n! A(2,n,k_n)/n!$ satisfies a Bahadur-Rao type large deviation formula in the limit $n \to \infty$ with $k_n/n \to κ\in [0,1)$ as long as $k_n \to \infty$. The large deviation rate function is the Legendre-Fenchel transform $g^*(-κ)$ where $g(κ) = f^{-1}(κ)$ for the function $f : (0,\infty) \to \mathbb{R}$ given by $f(y)= \ln(-\ln((e^{-y};e^{-y})_{\infty}))$. We relate this fact to information about the abundancy index.

Figures (4)

• Figure 1: In the first plot, the blue dots are a list-plot of $\ln(a(n,k)) = \ln(k! A(2,n,k)/n!)$ versus $k$, for $n=150$ and $k \in \{1,\dots,150\}$. The red curve is the logarithm of the approximation from Theorem \ref{['thm:BR1']}. In the second plot, in blue, we have plotted $(a(n,k)-a(n,n+1-k))/(a(n,k)+a(n,n+1-k))$ for $n=150$ and for $k$ from $3$ to $147$.
• Figure 2: In the plot, the blue dots are $\ln((a(n,k))^2/(a(n,k-1)a(n,k+1)))$ versus $k/n$, for $n=150$ and $k \in \{2,\dots,149\}$. The red curve is a plot of $1/(n (\mathcal{K}(y))^3 \mathcal{V}(y))$ versus $\mathcal{K}(y)$ for a range of $y$ between $0.01$ and $10$.
• Figure 3: Here we plot $F(y) F(y+\ln(F(y)))$ versus $y$.
• Figure 4: Here we plot $\ln(|F(y+i\theta)|^2/(F(y))^2)$ for $y=10^{-3}$. We truncated the sum defining $F$ at $n=10^5$. In red is the logarithm of the bound $1-\beta(y;\theta)$. The inset is a zoom-in near the maximum peak, at $\theta=0$.

Theorems & Definitions (8)

• Theorem 3.1
• Corollary 3.2
• Lemma 4.1
• Proposition 4.2
• Lemma 5.1
• Lemma 5.2
• Proposition 6.1
• Lemma 1.1