  Compact formulas are obtained for the Casimir energy of a relativistic
perfect fluid confined to a $D$-dimensional hypercube with von Neumann or
Dirichlet boundary conditions. The formulas are conveniently expressed as a
finite sum of the well-known gamma and Riemann zeta functions. Emphasis is
placed on the mathematical technique used to extract the Casimir energy from a
$D$-dimensional infinite sum regularized with an exponential cut-off. Numerical
calculations show that initially the Dirichlet energy decreases rapidly in
magnitude and oscillates in sign, being positive for even $D$ and negative for
odd $D$. This oscillating pattern stops abruptly at the critical dimension of
D=36 after which the energy remains negative and the magnitude increases. We
show that numerical calculations performed with 16-digit precision are
inaccurate at higher values of $D$.
