Coefficient Transfer for the Prime-Power Model of Strongly Irreducible Partitions

This paper isolates the coefficient-transfer problem for the prime-power lower model of strongly irreducible partitions. The model has the exact Euler product (q) =ₚ (1+₀₁q^pᵃ) (|q|<1), and the preceding real-axis analysis gives, under the prime number theorem, (e^-t) ²121t (1/t) (t0^+). The purpose of the present paper is deliberately not to convert this radial statement into an unconditional Hardy–Ramanujan formula by fiat. A radial first-order singular estimate does not determine local Gaussian behavior, minor-arc decay, or the bounded residual exponent needed for a numerical prefactor. We therefore prove exactly what follows without further complex input, formulate coefficient-free admissibility hypotheses for the prime-power Euler product, and compute the constants forced by those hypotheses. In particular, the real-axis estimate alone gives (n) (23+o (1) ) n{ n}, while first-order coefficient admissibility gives the matching logarithmic asymptotic. A full equivalent has the conditional form (n) 12 (²24) ^1/4 n^-3/4 (n) ^-1/4 \! (23n{ n}+ₙ), provided a second-order saddle exponent datum εnεn and the first-order arc estimates are supplied. The paper also records why the frequently guessed constants 2A2A, 3A3A, or π2π2/9π2π2/9% are not the constants produced by the saddle equation for the scale A/ (tlog⁡ (1/t) ) A/ (tlog (1/t) ).