Calculate the exact grand partition function for a system of ultra-relativistic bosons.

Let's break down this problem on ultra-relativistic bosons.

1. Calculate the exact grand partition function $\Xi$ for the system.

For a system of non-interacting bosons, the grand partition function $\Xi$ is given by:

$$\ln \Xi = -\sum_k \ln(1 - e^{-\beta(\epsilon_k - \mu)})$$

where $\beta = \frac{1}{k_B T}$, $\mu$ is the chemical potential, and $\epsilon_k$ are the single-particle energy levels.
For ultra-relativistic particles in 3D, the energy-momentum relation is $E = pc$. The density of states in momentum space is $\frac{V}{(2\pi\hbar