Weyl series and the trace formula: A prescription for adding 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 that arise from the inaccurate low-energy behavior, and that are spurious, from other subdominant contributions that 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.