Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Gafni's restricted partition asymptotics connect density of allowed parts to exponential growth rates, with Dedekind eta function as the modular backbone. | MATH: For $\mathcal{A}\subset\mathbb{N}$ with counting function $A(x)=\alpha x + O(x^\beta)$, $\beta<1$, Gafni's result (arXiv:1407.4443) gives $\log p_{\mathcal{A}}(n) \sim \frac{\pi}{\sqrt{3}}\sqrt{\alpha n} + \frac{1}{4}\log n + O(1)$ — the leading term scales as $\sqrt{\alpha n}$, generalizing Hardy–Ramanujan ($\alpha=1$). The generating function $\sum p_{\mathcal{A}}(n)q^n = \prod_{a\in\mathcal{A}}(1-q^a)^{-1}$ and modularity enters via $\eta(\tau) = q^{1/24}\prod_{n=1}^\infty(1-q^n)$, where $q=e^{2\pi i\tau}$. The constant $\pi/\sqrt{3} \approx 1.8138$ is the saddle-point residue from the Dedekind eta transformation $\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau)$. | CONNECTION: The exponent $\alpha$ in the density $A(x)\sim\alpha x$ directly multiplies $n$ inside the square root — this is a *linear scaling* of the modular Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222796175
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