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I am not satisfied with the explanations of $$\sum_n \log \lambda_n = - \frac{d}{ds} \sum_n \lambda_n^{-s}\bigg|_{s=0}$$ "trick" used in zeta function regularization, discussed here and here, or the references therein. If someone could give a step-by-step explanation or a reference if the explananation is more complicated than I'm guessing, that would be greatly appreciated.

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    $\begingroup$ For fun, an other basic trick, used for entropy, is $S = - Tr(\rho \log \rho) = -\frac{d}{ds} Tr(\rho^s)_{|s=1}$ $\endgroup$
    – Trimok
    Commented Oct 17, 2013 at 16:02

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Consider a function of $s$ as $$ f(s) = \frac{1}{a^s} $$ Now take the derivative with respect to $s$: $$ f'(s)=\frac{d}{ds}a^{-s}=-\frac{\log(a)}{a^s} $$ What happens at $s=0$? $$ f'(s=0)=-\frac{\log(a)}{a^0}=-\log(a) $$

What happens if $a=a_0+a_1+a_2+\cdots$ (i.e., $s$-independent constants)? We simply stick in a sum: $$ f(s) =\sum_n \frac{1}{a_n^s} $$ What does this do to the final answer? Nothing, we still can stick in the sum: $$ f'(s=0)=-\sum_n\frac{\log(a_n)}{a_n^0}=-\sum_n\log(a_n) $$

Thus, we get our relation, $$ \left.\frac{d}{ds}\sum_n\lambda_n^{-s}\right|_{s=0}=-\sum_n\log\left(\lambda_n\right) $$

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