  In this paper, we study the stochastic calculus for the theta process. The
theta process is a stochastic process of number theoretical origin arising as a
scaling limit of quadratic Weyl sums. It has several properties in common with
Brownian motion such as its H\"older regularity, uncorrelated increments and
quadratic variation. However, crucially, we show that the theta process is not
a semimartingale, making It\^o calculus techniques inapplicable. Instead, we
use the celebrated rough paths theory to develop the stochastic calculus for
the theta process. We do so by constructing the iterated integrals - the
``rough path" - above the theta process. Rough paths theory takes a signal and
its iterated integrals and produces a vast and robust theory of stochastic
differential equations. In addition, the rough path we construct can be
described in terms of higher rank theta sums.
