  In this paper, we give an optimal estimate of an average of Hurwitz class
numbers. As an application, we give an equidistribution result of the family
$\{\frac{t}{2q^{\nu/2}} \ | \ \nu \in \mathbb{N}, t \in \mathbb{Z},
|t|<2q^{\nu/2}\}$ with $q$ prime, weighted by Hurwitz class numbers. This
equidistribution produces many asymptotic relations among Hurwitz class
numbers. Our proof relies on the resolvent trace formula of Hecke operators on
elliptic cusp forms of weight $k\ge 2$.
