half-plane ${\mathrm{Re}}(z) > m+v+2h$, and extends to a meromorphic function of the whole complex plane, with at most simple poles in the set $$\{ m+v+2h, m+v+2h - 1, \, \ldots \}$$

The meromorphic extension of the zeta function given by this theorem allows the construction of a Wodzicki-Guillemin trace on ${\mathcal{S}}_{H,c}(M) = \Psi_{H,c}(M)/\Psi_c^{-\infty}(M)$.

*(Connes - Moscovici, [@CM1995])* \[CM residue\] The Wodzicki residue functional $${\mathop{\ooalign{$\displaystyle{\int}$\cr$-$}}}: {\mathcal{S}}_{H,c}(M) \longrightarrow {\mathbb{C}}, \quad P \longmapsto {\mathrm{Res}}_{z=0} {\mathrm{Tr}}(P \Delta^{-z/4})$$ is a trace. Moreover we have the following formula, only depending on the formal symbol $\sigma$ of $P$ up to finite order: $$\label{wodzicki residue symbol 0}
{\mathop{\ooalign{$\displaystyle{\int}$\cr$-$}}}P = \frac{1}{(2\pi)^n} \int_{S_H^*M} \iota_L \left(\sigma_{-(v+2h)}(x,p) \, \frac{ \omega^n}{n!}\right)$$

Here, $S_H^*M$ is the Heisenberg cosphere bundle, that is, the sub-bundle $$S_H^*M = \{(x,p) \in T^*M \, ; \, \vert p \vert' = 1 \}$$ of the cotangent bundle $T^*M$, $\omega$ denotes the standard symplectic form on $T^*M$, $\iota$ stands for the interior product, and $L$ is the generator of the Heisenberg dilations given by the formula $$L = \sum_{i=1}^v p_i \partial_{p_i} + 2\sum_{i=v+1}^n p_i \partial_{p_i}$$ All these results still hold for Heisenberg pseudodifferential operators acting on sections of a vector bundle $E$ over $M$. In this case, the symbol $\sigma_{-(v+2n)}(x,p)$ above 