Exact Partition Asymptotics via Circle Method and Modular L-Value Links — E8 Intelligence Research

Abstract

FINDING: Hardy-Ramanujan circle method yields exact asymptotic for partition function p(n), with modular-form connections to critical L-values. MATH: p(n) ~ (1/(4n√3))·exp(π√(2n/3)); exact series p(n) = (1/(π√2)) Σ_{k=1}^∞ A_k(n)·√k·[d/dn](sinh(π√(2/3(n−1/24)))/k) / √(n−1/24); A_k(n) = Σ_{h mod k, gcd(h,k)=1} exp(πi·s(h,k) − 2πinh/k); Dedekind eta η(τ) = q^{1/24}Π(1−q^n) enters via modular transformation; Zagier's interpolation: Apéry-like sequences = critical L-values of weight-4 modular forms. CONNECTION: The 1/24 in the exponent (n−1/24) is the modular anomaly — the same shift appears in the Dedekind eta function, linking to the Leech lattice (24-dimensional) and the Golay code. The √3 in the denominator connects to the hexagonal lattice (root system A₂, crystallographic symmetry 6-fold). The π in the exponential is the circle constant — the "circle method" literally integrates over circles in the complex plane, and the dominant saddle point sits at the unit circle's cusp. The r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com