We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics
of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite
von Neumann algebra. We find for p > 1 that the asymptotics of the zeta function determines an ideal
strictly larger than Lp,∞ on which the Dixmier trace may be defined. We also establish stronger versions
of other results on Dixmier traces and zeta functions.