 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-2/37e/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For

k_{B} T ≪ ℏ ω_{D}

, extend the integral to infinity:

lo g Z = - \int_{0}^{\infty} g (ω) lo g (1 - e^{- β ℏ ω}) d ω = \frac{A k _{B}^{2} T ^{2}}{π c _{s}^{2} ℏ ^{2}} \sum_{n \geq 1} \frac{1}{n ^{3}} .

Thus $X = ζ (3) / π$ .

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