= Solution
For discrete states $s$ of energy $E_s$, the <canonical partition function> is
$$
Z(\beta)=\sum_s e^{-\beta E_s},\qquad \beta=(k_BT)^{-1}.
$$
For a classical continuous system the sum is replaced by the appropriately normalized phase-space <integral>, including the Gibbs factor $1/N!$ for identical particles.
It packages the equilibrium probabilities $P_s=e^{-\beta E_s}/Z$ and hence all canonical <thermodynamics>. In particular,
$$
F=-k_BT\log Z,\qquad
\langle E\rangle=-\frac{\partial\log Z}{\partial\beta},\qquad
(\Delta E)^2=\frac{\partial^2\log Z}{\partial\beta^2},
$$
and $S=k_B(\log Z+\beta\langle E\rangle)$.
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