\theta(t) \defi \sum_{k=0}^\infty \e ^{-tE_k} \qquad \Re(t) > 0 \\$$ can be asymptotically expanded in increasing (real) powers $\{t^\rho \}$ for $t\downarrow 0$: $$\label{tas}
\theta(t) \sim \sum_\rho c_\rho t^\rho  \quad (t\downarrow 0), \quad
\mbox{with } \mu \defi -\min {\{\rho\}} >0 
\mbox{ (the ``growth order").}$$ (In summations etc., Latin indices will systematically mean [*integers*]{}, and Greek indices mean more general real indices, namely real-valued functions over the natural integers, strictly $\uparrow +\infty$ or $\downarrow -\infty$.)

### Spectral zeta functions [@VN; @VZ; @Ez]

Eq.(\[tas\]) implies that the “Hurwitz", resp. “plain" spectral zeta functions $$\label{zeta}
Z(s,\lambda) \defi \sum_k (E_k+\lambda)^{-s} \quad
(|\arg(\lambda+E_0)|< \pi-\delta), \quad
\mbox{resp. } Z(s) \defi Z(s,0)$$ converge for $\Re(s) > \mu$, that $Z(s)$ has a meromorphic continuation to all complex $s$, with [@P] $$\label{res}
\mbox{polar set: } \{-\rho\}, \quad \mbox{residue formula: }
\lim_{s \to -\rho} (s+\rho) Z(s) = c_\rho /\G(-\rho),$$ $$\label{TR}
\mbox{``trace identities": for }m \in \Bbb N,\  Z(-m) = (-1)^m m! \, c_m   
\quad (c_m\defi 0 \mbox{ for } m \notin \{\rho\})$$ and similarly for $Z(s,\lambda)$, just by substituting $\e^{-\lambda t} \theta(t)$ for $\theta(t)$; for general $\lambda$, the leading trace identity (specially useful for eq.(\[Scal\]) below) is then $$\label{Tr0}
Z(0,\lambda) = \sum_{0 \le n \le \mu} c_{-n}{(-\lambda)^n \over n!} \equiv
