Theta Series of E8 Lattice as Weight-4 Modular Form with Kissing Number 240 — E8 Intelligence Research

FINDING: Theta series of the E8 lattice is a modular form of weight 4, with coefficients counting lattice points of given norm; the first nonzero coefficient is 240, the kissing number of E8. MATH: The theta series is \ (₄䃘 () = ₕ ₄䃘 q^\|v\|²/2 = 1 + 240 ₍=₁^ ₃ (n) qⁿ\), where \ (q = e^2 i \), \ (₃ (n) = ₃|₍ d³\). This equals the Eisenstein series \ (E₄ () = 1 + 240 ₍=₁^ ₃ (n) qⁿ\), a modular form of weight 4 for \ (SL₂ (Z) \). The 240 coefficient corresponds to the 240 roots of E8 (kissing number). The Fourier transform of the E8 root system yields a 3D cross-section of the 4₂1 polytope (Gosset polytope), a semiregular 8-polytope with 240 vertices. CONNECTION: The E8 root system is intimately tied to the golden ratio \ (= 1. 618\) and its reciprocal \ (0. 618\): the Coxeter number of E8 is 30, and the ratio of the lengths of the long and short roots (in the dual l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com