Fast check: **the AI’s answer is not correct**. The structur...

Prompt

Fast check: **the AI’s answer is not correct**. The structure looks plausible, but the coefficient/ζ terms are wrong. For example, at \(s=4\), numerical evaluation of the integral does **not** match its formula. A good way to challenge it is to send: $$ \boxed{ \text{Derive the integral from first principles. Expand both Bose-Einstein factors as geometric series,} } $$ $$ \boxed{ \frac1{e^x-1}=\sum_{m=1}^{\infty}e^{-mx}, \qquad \frac1{e^{2x}-1}=\sum_{n=1}^{\infty}e^{-2nx}, } $$ and then evaluate the resulting double sum exactly. **Do not assume the proposed formula is correct.** Then additionally require: $$ \boxed{\text{Evaluate the exact result at }s=5.} $$ That should expose whether the AI can actually derive the result rather than pattern-match a zeta-function formula.