One paper I'm looking at says,
$n(M, z) \, dM \, dz$, the number of sources with mass $M$ at a redshift $z$, in the mass interval $dM$ occurring in the redshift interval $dz$.
While another says that,
$\frac{d^2n}{dM dz}$, the number of sources with mass and redshift in the range $[M, M+dM]$ and $[z, z+dz]$ respectively.
What is the difference (conceptually and technically, please) between these formalisms -- and the (different?) meaning(s) of the distribution $n$?
Perhaps unrelated: sometimes when integrating numerically (e.g. to find the total number of sources over all parameters) we have to include the differential itself, i.e. multiply by $\sum n_i \Delta z_i = n_i (z_{i+1} - z_i)$, whereas other times we just sum the distribution $\sum n_i$ --- is this because the latter version implicitly includes the differential-elements within it?