What question did this study set out to answer?

This research aims to complete Ramanujan's lost modularity through the study of mock theta functions.

July 24, 20260 citationsOpen Access

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

Key Points

This research aims to complete Ramanujan's lost modularity through the study of mock theta functions.
Analyzed the behavior of mock theta functions and their connection to modular forms.
Applied the definition of the function to identify harmonic Maass forms.
Examined the indirect relationship with root systems and theta functions.
Demonstrated that mock theta functions can be defined as q-hypergeometric series.
Confirmed that no explicit geometric harmony occurs in the definitions of mock theta functions.
Found that mock theta functions relate to modular forms of weight 1/2 without being lattice theta series.

Abstract

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) = ₍=₀^ q^n² (1+q) ² (1+q²) ² (1+qⁿ) ² \). Zwegers showed \ (f (q) \) plus a non-holomorphic correction \ (R () \) yields a weight-1/2 harmonic Maass form. Key constant: \ (q = e^2 i \), with \ (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. , \ (A₁ \) 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

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