Resurgence and Partial Theta Series

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 n2 tau/M), with nu non-negative integer and an M-periodic function f : Z -> C. Such a function is analytic in the half-plane {Im(tau)>0} and as tau tends non-tangentially to any alpha is an element of 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 & Eacute;calle's ``Bridge Equations''. The motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.