Resurgence and Partial Theta Series

Abstract

We consider partial theta series associated with periodic sequences of coefficients, of the form $\hat{I}˜(\ddot{I}„):={\sum }_{n>0}{n}^{\hat{I}}½f(n)e^{i\ddot{I}€n^2\ddot{I}„/M}$, with $\hat{I}½$ non-negative integer and an $M$-periodic function $f:\mathbb{Z}\to \mathbb{C}$. Such a function is analytic in the half-plane $\{Im(\ddot{I}„)>0\}$ and as $\ddot{I}„$ tends non-tangentially to any $\hat{I}\pm \in \mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $\hat{I}˜(\ddot{I}„)$, depending on the parity of $\hat{I}½$ 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 $\hat{I}˜$, 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.

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