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?
see here.
When using the definition of the functional determinant you arrive at
$ln(\frac{det(H+m)}{det(H)})=-\frac{d}{da} (\zeta_1 - \zeta_2)|_{a=0}$
with $\zeta_1(s)=tr(H+m)^{-a}$ and $\zeta_2(s)=trH^{-a}$. Now you make a taylor series expansion around $m=0$ and you get your series when you set $\zeta_H(n)=trH^{-n}$.
A generating function of a function $f$ is a function $F$ where the series expansion leads to coefficient that represents another function $f$; here the zeta functions are the coefficients of your series expansion.