More on the density zero ideal

The main result of this paper is an improvement of the upper bound on the cardinal invariant ${}^{\ast }({\mathcal{Z}}_0)$ 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 ${}^{\ast }({\mathcal{Z}}_0)$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.