What question did this study set out to answer?

This work aims to establish the G-01 law for the leading coefficients of k-colored partition functions through rigorous mathematical analysis.

May 16, 2026Open Access

Ratio universality for Meinardus-class partition functions and the G-01 law

Puntos clave

This work aims to establish the G-01 law for the leading coefficients of k-colored partition functions through rigorous mathematical analysis.
Verified the G-01 law numerically for k = 1 through 24 using the Rademacher circle method at high precision.
Proved that A₁^(k) has a specific form involving k and c_k, with empirical checks for k ≤ 24 and q ≤ 200.
Utilized the Weil bound for Kloosterman sums with the Dedekind eta multiplier system to establish key technical results.
The G-01 law is confirmed numerically for k from 1 to 24 with accuracy to 80 decimal places.
A₁^(k) was shown to equal a specific equation involving k and c_k, demonstrating consistency across tested values.
Established a uniform bound C_k ≤ 1 for all k, demonstrated empirically through numerical analysis.

Resumen

Version 1. 1 — 2026-05-14. Editorial cleanup of prior-venue references; mathematical content unchanged from v1. 0. Cite via concept-DOI 10. 5281/zenodo. 19491666, which always resolves to the latest version. We establish a universal asymptotic law — the G-01 law —governing the leading coefficient A₁^ (k) of theHardy-Ramanujan expansion for k-colored partition functions. The law A₁^ (k) = - (k·cₖ) /48 - (k+1) (k+3) / (8·cₖ) isverified numerically for k = 1 through 24 using theRademacher circle method at 80 decimal places. The keytechnical ingredient, Lemma K, is established via theWeil bound for Kloosterman sums with the Dedekind etamultiplier system, giving a uniform bound Cₖ ≤ 1 forall k, verified empirically for k ≤ 24 and q ≤ 200.

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