Ramanujan-Type Partition Congruences Extended to Prime Powers via Modular Forms — E8 Intelligence Research

FINDING: Ramanujan-type congruences for partition functions extend from modulus 5, 7, 11 to prime powers p^k via modular forms and generalized Borcherds products. | MATH: p(n) congruences: p(5n+4)≡0 mod 5, p(7n+5)≡0 mod 7, p(11n+6)≡0 mod 11. Generating function: Σp(n)q^n = q^(1/24)/η(τ) where η(τ)=q^(1/24)Π(1-q^n). Borcherds product: f(z)=q^A Π(1-q^n)^c(n²) with c(n) from Maass form coefficients. | CONNECTION: The Dedekind eta function η(τ) is a modular form of weight 1/2 with multiplier system; its 24th power Δ(τ)=η^24 is the unique cusp form of weight 12. The exponent 24 connects to the Leech lattice (24-dimensional, kissing number 196560) and the Ramanujan tau function τ(n) satisfying |τ(p)|≤2p^(11/2) (Deligne). The 1/24 shift in the partition generating function relates to the modular weight and the 24 in the denominator of the Bernoulli number B₁₂ = -691/2730. | DEPTH: 8 FINDING: Generalized Borcherds products convert congruences of modular form coefficients into congruences for Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com