# Dixmier traces, Wodzicki residue and residues of zeta functions

Let $$M$$ be an $$n$$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $$P$$ of order $$-n$$. Here are some facts which I had learned so far:
1. There exists a density defined in terms of $$-n$$-th homogenous part of the symbol of $$P$$ such that integrating this density one obtains a trace on the algebra of all (classical) pseudodifferential operators. This is the so called Wodzicki residue, to be denoted by $$Wres$$. 2. If $$n>1$$ this is unique such trace.
3. If $$P$$ is as above that $$P$$ belongs to the Dixmier trace, is measurable (i.e. the Dixmier trace does not depend from the choice of a state) and thus Dixmier trace $$Tr^+P$$ is defined.
4. We have and equality $$Tr^+P=Res_{s=1}(Trace(P^s))$$ and thus $$P \mapsto Res_{s=1} (Trace(P^s))$$ is also a trace.
5. Also the function $$P \mapsto Res_{s=0}(Trace(P\Delta^{-\frac{s}{2}}))$$ is a trace.
Finally the last fact: 6. There is a universal constant $$c>0$$ (depending only on the dimension of $$M$$) such that $$Res_{s=0}(Trace(P\Delta^{-\frac{s}{2}}))=c Wres(P)$$ (note that this is slightly more which can be deduced from the uniqueness of the trace since this constant does not depend on the choice of $$M$$ provided that the dimension remains the same).

To have the full picture of these issues I would like to know

1. How to determine (in the easiest way, provided we know all the above facts) this universal constant $$c$$ (it should be $$\frac{1}{n(2\pi)^n}$$).
2. Is it clear that $$Res_{s=1}(Trace(P^s))$$ coincides with $$Res_{s=0}(Trace(P\Delta^{-\frac{s}{2}}))$$? Is it clear that the former is a trace?

I would be happy in any solution to these two problems (i.e. answering 2. once we know the answer for 1. etc).