Mock Theta Functions: Completing Ramanujan's Lost Modularity — E8 Intelligence Research

FINDING: Mock theta functions are q-hypergeometric series whose coefficients are half the Fourier coefficients of a non-holomorphic modular form, completing Ramanujan's lost modularity. | MATH: Let \( f(q) = \sum_{n=0}^\infty \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots(1+q^n)^2} \). Zwegers showed \( f(q) \) plus a non-holomorphic correction \( R(\tau) \) yields a weight-1/2 harmonic Maass form. Key constant: \( q = e^{2\pi i \tau} \), with \( \tau \in \mathbb{H} \). No explicit golden ratio or base-60 constants appear in core definitions. | CONNECTION: No direct geometric harmony (0.382, 0.618, 1.618, base-60, or crystallographic symmetry) is present in the mock theta function definitions or Zwegers' modular completion. The functions relate to root systems indirectly via modular forms of weight 1/2, which arise in theta functions of lattices (e.g., \( A_1 \) root lattice), but mock theta functions themselves are not lattice theta series. | DEPTH: 8 — Profound for number theory and modular f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com