Resurgence and partial theta series

We consider partial theta series associated with periodic sequences of coefficients, namely, $\Theta(\tau):= \sum_{n>0} n^\nu f(n) e^{i\pi n^2\tau/M}$, where $\nu\in\mathbb{Z}_{\ge0}$ and $f\colon\mathbb{Z} \to \mathbb{C}$ is an $M$-periodic function. Such a function $\Theta$ is analytic in the half-plane $\{\operatorname{Im}\tau>0\}$ and in the asymptotics of $\Theta(\tau)$ as $\tau$ tends nontangentially to any $\alpha\in\mathbb{Q}$ a formal power series appears, which depends on the parity of $\nu$ and $f$. We discuss the summability and resurgence properties of these series; namely, we present explicit formulas for their formal Borel transforms and their consequences for the modularity properties of $\Theta$, or its “quantum modularity” properties in the sense of Zagier's recent theory. The discrete Fourier transform of $f$ plays an unexpected role and leads to a number-theoretic analogue of Écalle's “bridge equations.” The main thesis is: (quantum) modularity $=$ Stokes phenomenon $+$ discrete Fourier transform.