  Hardy and Littlewood's approximate functional equation for quadratic Weyl
sums (theta sums) provides, by iterative application, a powerful tool for the
asymptotic analysis of such sums. The classical Jacobi theta function, on the
other hand, satisfies an exact functional equation, and extends to an
automorphic function on the Jacobi group. In the present study we construct a
related, almost everywhere non-differentiable automorphic function, which
approximates quadratic Weyl sums up to an error of order one, uniformly in the
summation range. This not only implies the approximate functional equation, but
allows us to replace Hardy and Littlewood's renormalization approach by the
dynamics of a certain homogeneous flow. The great advantage of this
construction is that the approximation is global, i.e., there is no need to
keep track of the error terms accumulating in an iterative procedure. Our main
application is a new functional limit theorem, or invariance principle, for
theta sums. The interesting observation here is that the paths of the limiting
process share a number of key features with Brownian motion (scale invariance,
invariance under time inversion, non-differentiability), although time
increments are not independent and the value distribution at each fixed time is
distinctly different from a normal distribution.
