Resurgence and Partial Theta Series

Abstract

We consider partial theta series associated with periodic sequences of coefficients, of the form $Θ(τ) := \sum_{n>0} n^νf(n) e^{iπn^2τ/M}$, with $ν$ non-negative integer and an $M$-periodic function $f : \mathbb{Z} \rightarrow \mathbb{C}$. Such a function is analytic in the half-plane $\{Im(τ)>0\}$ and as $τ$ tends non-tangentially to any $α\in\mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $Θ(τ)$, depending on the parity of $ν$ 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 $Θ$, 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 motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.