  We estimate the sums \[ \sum_{c\leq x} \frac{S(m,n,c,\chi)}{c}, \] where the
$S(m,n,c,\chi)$ are Kloosterman sums of half-integral weight on the modular
group. Our estimates are uniform in $m$, $n$, and $x$ in analogy with Sarnak
and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman
sums. Among other things this requires us to develop mean value estimates for
coefficients of Maass cusp forms of weight $1/2$ and uniform estimates for
$K$-Bessel integral transforms. As an application, we obtain an improved
estimate for the classical problem of estimating the size of the error term in
Rademacher's formula for the partition function $p(n)$.
