Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and ${\bf I}=(I_1,\ldots,I_d)$ be $\mathfrak{m}-$primary ideals in $R$. We prove that $\lambda_R([H^d_{(x_{ii}t_i:1\leq i \leq d)}(\mathcal R^\prime(\mathcal F)]_{\bf n})$ $< \infty$, for all ${\bf n} \in \mathbb N^d$, where $\mathcal F=\{\mathcal F({\bf n}):{\bf n}\in \mathbb Z^d\}$ is an ${\bf I}-$admissible filtration and $(x_{ij})$ is a strict complete reduction of $\mathcal F$ and $\mathcal R^\prime(\mathcal F)$ is the extended multi-Rees algebra of $\mathcal F.$ As a consequence we prove that the normal joint reduction number of $I,J,K$ is zero in an analytically unramified Cohen-Macaulay local ring of dimension $3$ if and only if $\overline{e}_3(IJK)-[\overline{e}_3(IJ)+\overline{e}_3(IK)+\overline{e}_3(JK)]+\overline{e}_3(I)+\overline{e}_3(J)+\overline{e}_3(K)=0.$ This generalizes a theorem of Rees on joint reduction number zero in dimension $2$. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension $3.$