Resurgence and Partial Theta Series

Abstract

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.