Quantum Modularity of Mock Theta Functions from the Alpha Hierarchy
Year: 2026
Type: article
Abstract: This is Part 2 of [1]. We study the radial limits of the mock theta functions M_k(τ) constructed from the alpha hierarchy introduced in [1]. For k=1 we compute these limits numerically at roots of unity and show they satisfy quantum modular transformation properties in the sense of Zagier. The difference L_1(-1)-L_1(1) is observed to equal π²/30 - i/144 plus higher terms involving π⁴, π⁶ and odd zeta values ζ(3), ζ(5), ζ(7), ... with rational coefficients matching those of the Lambda functions Λ_k from [1]. This provides numerical evidence that the functions M_k(τ) give rise to quantum modular forms whose special values involve π² and odd zeta values in a structured way. Conjectures are stated for general k. [1] Hussain, M.R. (2026). Alpha Functions: A New Hierarchy with Connections to Elliptic Integrals and Mock Theta Functions. Zenodo. DOI:10.5281/zenodo.19025728
Author: Hussain, Muhammad Rayyan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7136047360
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