  We study the geometry associated to the distribution of certain arithmetic
functions, including the von Mangoldt function and the M\"obius function, in
short intervals of polynomials over a finite field $\mathbb{F}_q$. Using the
Grothendieck-Lefschetz trace formula, we reinterpret each moment of these
distributions as a point-counting problem on a highly singular complete
intersection variety. We compute part of the $\ell$-adic cohomology of these
varieties, corresponding to an asymptotic bound on each moment for fixed degree
$n$ in the limit as $q \to \infty$. The results of this paper can be viewed as
a geometric explanation for asymptotic results that can be proved using
analytic number theory over function fields.
