The Weyl Series and the Trace formula: can we add them?
  Periodic orbit expressions for the density of states lead to spurious results when directly used to calculate quantities of thermodynamic interest. This is because the trace formula is usually valid only for large energies while the calculations make use of it at all energies. We present a prescription for circumventing this problem by isolating contributions which arise from the inaccurate low-energy behavior, and are spurious, from other subdominant contributions which are physical and are not contained in the Weyl series. The method is tested by analyzing the fermionic grand partition function for a disk billiard and may readily be extended to other dynamical systems. 
R. K. Bhaduri, N. D. Whelan, M. Brack, H. G. Miller and M. V. N.   Murthy