  Hecke studies the distribution of fractional parts of quadratic irrationals
with Fourier expansion of Dirichlet series. This method is generalized by
Behnke and Ash-Friedberg, to study the distribution of the number of totally
positive integers of given trace in a general totally real number field of any
degree. When the field is cubic, we show that the asymptotic behavior of a
weighted Diophantine sum is related to the structure of the unit group. The
main term can be expressed in terms of Gr\"ossencharacter $L$-functions.
