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July 29, 2026Open Access

Higher-Order Saddle Asymptotics, Gibbs Limit Laws, and Rational-Arc Obstructions for Ones-Free Pairwise-Coprime Prime-Power Partitions

Key Points

The aim is to study the asymptotic behavior of ones-free pairwise-coprime prime-power partitions using advanced mathematical techniques.
Utilized the saddle-point method to derive asymptotic formulae for the generating function.
Applied uniform far-arc decay techniques alongside established number-theoretic theorems such as Siegel–Walfisz and Vinogradov–Vaughan estimates.
Investigated Gibbs measures and established large-deviation principles for total size and number of parts.
Established that the coefficient saddle exists uniquely, where q^{\circ}(n) shows complete saddle expansion.
Proved a bivariate central limit theorem yielding mixed cumulants with explicit functions for Gibbs measures.
Demonstrated boundary behavior of Euler products with simple negative zeros converging at -1.

Abstract

We study the ones-free pairwise-coprime prime-power partition model with generating function\ Q^ (q) =ₚ (1+₀₁q^pᵃ) =₍₀q^ (n) qⁿ. the classical prime number theorem, we prove asymptotic formulae for every fixed derivative of \ (H (t) = Q^ (e^-t) \), establish the existence and uniqueness of the coefficient saddle \ (tₙ\), and construct a zero-free growing principal arc. A prime-phase energy inequality reduces the complementary arcs to exponential sums over primes. Combining rational-scale profiles, the Siegel--Walfisz theorem, and the Vinogradov--Vaughan estimate yields uniform far-arc decay. Consequently, for every fixed order, \ (q^ (n) \) has a complete saddle expansion; in particular, ^ (n) (H (tₙ) +ntₙ) 2 H'' (tₙ), q^ (n) 23n{ n}. correction terms are universal Gaussian polynomials in the standardized derivatives of \ (H\), with an explicit finite combinatorial formula at every order. The same analysis gives a lattice Edgeworth expansion, Gaussian ratio asymptotics, and a moderate-deviation local law. For the associated Gibbs measure, we prove a bivariate central limit theorem for total size and number of parts, determine all fixed mixed cumulants, and establish joint canonical and microcanonical large-deviation principles with explicit dilogarithmic cumulant functions. Fixed prime bases are occupied with probability tending to one, while missing bases on the scale \ (p (1/tₙ) \) converge to a homogeneous Poisson point process. Conditional on occupancy, the logarithmic exponents of finitely many fixed bases converge to independent uniform laws. Finally, we analyze the boundary geometry of the Euler product. Individual odd-prime factors have simple negative zeros accumulating at \ (-1\), with Lambert-\ (W\) asymptotics, boundary exponent of convergence \ (1/2\), and an associated genus-zero canonical product. Fixed rational arcs of the prime-only comparison model have explicit Ramanujan-sum constants and a multiplicative Dirichlet series. These results explain why the present singular theory yields all fixed-order saddle expansions but not a classical Rademacher series. Keywords Integer partitions; pairwise coprime parts; ones-free partitions; prime powers; Euler products; saddle-point method; Edgeworth expansions; prime exponential sums; Gibbs measures; large deviations; Poisson point processes; rational arcs; canonical products. MSC 2020 **11P82; 05A16; 05A17; 11L03; 30D15; 41A60; 60F05; 60F10; 60G55. **

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