Resurgence and Partial Theta Series

Abstract

We consider partial theta series associated with periodic sequences of coefficients, of the form $\Theta(\tau) := \sum_{n>0} n^\nu f(n) e^{i\pi n^2\tau/M}$, with $\nu$ non-negative integer and an $M$-periodic function $f : \mathbb{Z} \rightarrow \mathbb{C}$. Such a function is analytic in the half-plane $\{Im(\tau)>0\}$ and as $\tau$ tends non-tangentially to any $\alpha\in\mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $\Theta(\tau)$, depending on the parity of $\nu$ and $f$. We discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the 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 \'Ecalle's ``Bridge Equations''. The motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.