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

Key Points

• 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.