Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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 𝒜⊂ℕ with counting function A(x)=α x + O(xβ), β<1, Gafni's result (arXiv:1407.4443) gives log p𝒜(n) ∼ π/√3√(α n) + 1/4log n + O(1) — the leading term scales as √(α n), generalizing Hardy–Ramanujan (α=1). The generating function ∑ p𝒜(n)qn = ∏a∈𝒜(1−qa)−1 and modularity enters via η(τ) = q1/24∏n=1∞(1−qn), where q=e2π iτ. The constant π/√3 ≈ 1.8138 is the saddle-point residue from the Dedekind eta transformation η(−1/τ)=√(−iτ) η(τ). | CONNECTION: The exponent α in the density A(x)∼α 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
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BibTeX
@misc{caldin2026densitydriven,
title = {{Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431075},
url = {https://zenodo.org/records/23431075},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431075
UR - https://zenodo.org/records/23431075
AB - 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 𝒜⊂ℕ with counting function A(x)=α x + O(x^β), β<1, Gafni's result (arXiv:1407.4443) gives log p_𝒜(n) ∼ π/√3√(α n) + 1/4log n + O(1) — the leading term scales as √(α n), generalizing Hardy–Ramanujan (α=1). The generating function ∑ p_𝒜(n)q^n = ∏_a∈𝒜(1−q^a)^−1 and modularity enters via η(τ) = q^1/24∏_n=1^∞(1−q^n), where q=e^2π iτ. The constant π/√3 ≈ 1.8138 is the saddle-point residue from the Dedekind eta transformation η(−1/τ)=√(−iτ) η(τ). | CONNECTION: The exponent α in the density A(x)∼α 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
ER -