From Odd-Cycle Non-Vanishing of Mock Theta Functions to Non-Vanishing of Twisted L-Functions

This paper introduces a new class of twisted L-functions derived from Ramanujan’s mock theta functions, using quadratic-phase characters and Mellin transforms. Building on prior results showing uniform non-vanishing of mock theta functions at odd roots of unity, the author proves that the associated twisted L-functions are nonzero in the region Re(s)>1\text{Re}(s) > 1, with analytic continuation to Re(s)≥12\text{Re}(s) \geq \frac{1}{2}. A functional equation and Euler product structure are derived, and extensive numerical evidence supports the Odd-Cycle Non-Vanishing Conjecture across both small and large moduli. The work connects mock modular behavior to analytic number theory and opens new directions for studying non-vanishing phenomena in twisted Dirichlet series.