# Question on Dixmier trace

1. Mar 25, 2010

### zetafunction

if the dixmier trace of an operator A is defined as

$$\frac{\sum_{i=0}^{N}\mu_{i}}{logN}$$ in the limit N-->oo (infinity)

where the eigenvalues are ordered in decreasing order

would not it better to be defined as

$$\frac{\sum_{i=0}^{N}\mu_{i}^{s}}{logN}$$

here 's' is a parameter to be FIXED mathematically so

$$\sum_{i=0}^{N}\mu_{i}^{s}=T.log$$ then 'T' is just the Dixmier trace

the idea is to define the number 's' to be real of complex so the zeta function of operator A

$$Z=Tr(A^{s})$$ has a pole or a logarithmic divergence , then the normal definition of dixmier trace is recovered when s=1