  The main result of this paper is an improvement of the upper bound on the
cardinal invariant ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0})$ that
was discovered by Raghavan and Shelah in an earlier paper. Here
${\mathcal{Z}}_{0}$ is the ideal of subsets of the set of natural numbers that
have asymptotic density zero. This improved upper bound is also dualized to get
a better lower bound on the cardinal
${\mathord{\mathrm{non}}}^{\ast}({\mathcal{Z}}_{0})$. En route some variations
on the splitting number are introduced and several relationships between these
variants are proved.
