  We develop high temperature series expansions for $\ln{Z}$ and the uniform
structure factor of the spin-half Heisenberg model on the hyperkagome lattice
to order $\beta^{16}$. These expansions are used to calculate the uniform
susceptibility ($\chi$), the entropy ($S$), and the heat capacity ($C$) of the
model as a function of temperature. Series extrapolations of the expansions
converge well down to a temperature of approximately $J/4$. A comparison with
the experimental data for Na$_4$Ir$_3$O$_8$ shows that its magnetic
susceptibility is reasonably well described by the model with an exchange
constant $J\approx 300 K$, but there are also additional smaller terms present
in the system. The specific heat of the model has two peaks. The lower
temperature peak, which is just below our range of convergence contains about
40 percent of the total entropy. Despite being a 3-dimensional lattice, this
model shares many features with the kagome lattice Heisenberg model and the
material must be considered a strong candidate for a quantum spin-liquid.
