  Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a
noncompact hyperbolic surface of finite area, and let the neighborhood shrink
into the cusp at a rate of $T^{-1}$ as $T \rightarrow \infty$. We show that a
closed horocycle whose length $\ell$ goes to infinity or even a segment of that
horocycle becomes equidistributed on the shrinking neighborhood when normalized
by the rate $T^{-1}$ provided that $T/\ell \rightarrow 0$ and, for any
$\delta>0$, the segment remains larger than
$\max\left\{T^{-1/6},\left(T/\ell\right)^{1/2}\right\}\left(T/\ell\right)^{-\delta}$.
We also have an effective result for a smaller range of rates of growth of $T$
and $\ell$. Finally, a number-theoretic identity involving the Euler totient
function follows from our technique.
