  Let F/F_q be an algebraic function field of genus g defined over a finite
field F_q. We obtain new results on the existence, the number and the density
of dimension zero divisors of degree g-k in F where k is a positive integer. In
particular, for q=2,3 we prove that there always exists a dimension zero
divisor of degree \gamma-1 where \gamma is the q-rank of F. We also give a
necessary and sufficient condition for the existence of a dimension zero
divisor of degree g-k for a hyperelliptic field F in terms of its Zeta
function.
