This question is completely answered, and the result is that the condition involving the Moebius inversion you mention is both necessary and sufficient! See
- K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series, J. Amer. Math. Soc. 13 (2000) 773–806. https://doi.org/10.1090/S0894-0347-00-00342-8
This is really a remarkable and beautiful theorem.
