  A poly-log time method to compute the truncated theta function, its
derivatives, and integrals is presented. The method is elementary, rigorous,
explicit, and suited for computer implementation. We repeatedly apply the
Poisson summation formula to the truncated theta function while suitably
normalizing the linear and quadratic arguments after each repetition. The
method relies on the periodicity of the complex exponential, which enables the
suitable normalization of the arguments, and on the self-similarity of the
Gaussian, which ensures that we still obtain a truncated theta function after
each application of the Poisson summation. In other words, our method relies on
modular properties of the theta function. Applications to the numerical
computation of the Riemann zeta function and to finding the number of solutions
of Waring type Diophantine equations are discussed.
