Coefficient Transfer from Real-Axis Singular Scales for Strongly Irreducible Partitions
Year: 2026
Type: preprint
Abstract: This paper is the coefficient-transfer sequel to the real-axis singular theory for strongly irreducible partitions. Its purpose is deliberately narrower than a formal announcement of a Hardy–Ramanujan formula. The preceding papers provide, under their stated hypotheses, the real-axis first-order singular estimate $\log \overline P(e^{-t})\sim \frac{\pi^2}{12}\frac{1}{t\log(1/t)}\qquad(t\to0^+)$. Such an estimate alone does not determine a pointwise coefficient asymptotic: minor-arc control and local Gaussian behavior are additional analytic inputs. We therefore prove the exact coefficient consequences that genuinely follow from the real-axis estimate, formulate a noncircular saddle-point transfer criterion, and compute the constants forced by that criterion. In particular, the real-axis estimate implies the sharp logarithmic upper bound $\log \overline p(n)\le \left(2\sqrt{\frac{\pi^2}{6}}+o(1)\right)\sqrt{\frac{n}{\log n}}$, and, under explicit Hayman-type locality and decay hypotheses, it yields the logarithmic coefficient asymptotic $\log \overline p(n)\sim 2\sqrt{\frac{\pi^2}{6}}\sqrt{\frac{n}{\log n}}$. A full equivalent with a numerical prefactor is isolated as a stronger conditional theorem requiring a second-order singular expansion. The paper also explains why the frequently guessed expression C $n^{-3/4}\exp(c\sqrt{n/\log n})$, with a constant independent of $\log n$, is not justified by first-order singular information and has the wrong saddle scale unless further terms are supplied. Keywordsinteger partitions; pairwise coprime partitions; strongly irreducible partitions; saddle-point method; Hayman admissibility; coefficient asymptotics
Author: Wang, Jianming
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161474450
Cites:
Cited by:
Related to: