In the canonical ensemble for an ideal gas of $N$ bosons, the partition function for $T\to 0$ scales like $$Z\sim e^{-\beta\epsilon_0N},$$ when $\epsilon_0$ is the lowest (non-degenerate) single particle energy niveau, $\beta = 1/k_B T$ and the Boltzmann constant $k_B$. The energy in this limit is $E=\epsilon_0 N$ and the entropy can be obtained by
$$S=k_B(\log Z+\beta E),$$
which is independent of $T$ and identical zero in this case.
What is the next order approximation of the partition function with respect to $T$, such that there remains a temperature dependency of the entropy?
EDIT:
The general partition function of the $N$-particle system for arbitrary $T>0$ is the restricted sum
$$Z(\beta,N)=\sum_{\{N=n_0+n_1+...\}}\, x_0^{n_0}\,x_1^{n_1}\,...$$
with $x_i=e^{-\beta \epsilon_i}$, for $i=0,1,2,...$ Note, that $i$ is the quantum number and $\epsilon_i<\epsilon_{i+1}$, for all $i$.