Solution
= Solution
For $k_BT\ll\hbar\omega_D$, extend the <integral> to infinity:
$$
\log Z=-\int_0^\infty g(\omega)\log(1-e^{-\beta\hbar\omega})d\omega
=\frac{A k_B^2T^2}{\pi c_s^2\hbar^2}\sum_{n\ge1}\frac1{n^3}.
$$
\b[Thus $X=\zeta(3)/\pi$.]
= Solution
For $k_BT\ll\hbar\omega_D$, extend the <integral> to infinity:
$$
\log Z=-\int_0^\infty g(\omega)\log(1-e^{-\beta\hbar\omega})d\omega
=\frac{A k_B^2T^2}{\pi c_s^2\hbar^2}\sum_{n\ge1}\frac1{n^3}.
$$
\b[Thus $X=\zeta(3)/\pi$.]