(4) was included in [@G4].

The relations can also be formulated in terms of the generalized zeta function $\zeta (A,P,s)=\Tr(AP^{-s})$; here we assume $P$ invertible for simplicity. It is known from [@W], [@Gu] that $\zeta
(A,P,s)$ extends meromorphically from large $\operatorname{Re}s$ across $\operatorname{Re}s=0$ with a simple pole at $0$, $$\zeta (A,P,s)=C_{-1}(A,P)s^{-1}+C_0(A,P)+O(s) \text{ for }s\to 0;\tag1.6$$ this follows also by translating (1.1) to a statement on the pole structure of $\zeta (A,P,s)$ (as e.g. in ). Here$$\aligned
C_{-1}(A,P)&=\tilde c'_0=\tfrac 1m \operatorname{res}A,\\
C_0(A,P)&=\tilde c_{\nu +n}+\tilde c''_0,
\endaligned\tag1.7$$ with $\tilde c_{\nu +n}$ defined as 0 if $\nu <-n$ or $\nu
\notin\Bbb Z$. In the cases (1)–(4), $\tilde c_{\nu +n}$ and $\operatorname{res}A$ vanish (for any choice of local coordinates), so that$$\operatorname{TR}A=
\tilde c''_0 =C_0(A,P)
=\zeta (A,P,0).\tag1.8$$

The coefficient $C_0(A,P)$ has an interest also when none of the conditions (1)–(4) is satisfied. Then it has the properties:$$C_0(A,P)-C_0(A,P') \text{ and }C_0([A,A'],P)\text{ are locally
determined}.\tag1.9$$ Moreover, $C_0(A,P)$ can be written as (1.3) plus local terms. Functionals with properties as in (1.9) for a system of auxiliary elliptic operators $P$ will be called quasi-traces.

By use of more functional calculus, defining $\log P$, one can moreover show that the two expressions in (1.9) are noncommutative residues of suitable 