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## Main Question or Discussion Point

Given a Hamiltonian ##H##, with a spectrum of eigenvalues ##\lambda##, you can define

its zeta function as ##\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}##.

Subsequently, the log determinant of ##H## with a spectral parameter ##m^2## acts as a generating function for the zeta functions:

##ln(\frac{det(H+m^2)}{det(H)}) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}m^{2n} \zeta_{H}(n)##.

I understand that the zeta function for the Hamiltonian is defined in analogy to the Riemann zeta function. However, I do not understand how the log determinant can be used as a generating function for the zeta functions.

What exactly is a generating function? Can somebody prove the second relation, please?

its zeta function as ##\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}##.

Subsequently, the log determinant of ##H## with a spectral parameter ##m^2## acts as a generating function for the zeta functions:

##ln(\frac{det(H+m^2)}{det(H)}) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}m^{2n} \zeta_{H}(n)##.

I understand that the zeta function for the Hamiltonian is defined in analogy to the Riemann zeta function. However, I do not understand how the log determinant can be used as a generating function for the zeta functions.

What exactly is a generating function? Can somebody prove the second relation, please?