  We consider Mn-doped bulk zinc-blende semiconductors described by the 4 $
\times$ 4 Luttinger Hamiltonian. In these semiconductors, Mn atom acts as an
acceptor providing the system a mobile hole, and also acts like a magnetic
impurity of spin $S=5/2$. We obtain exact analytical expressions of the hole
mediated Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interaction between two
Mn${}^{2+}$ ions. The RKKY interaction of the Luttinger system consists of
collinear Heisenberg-like and Ising-like interactions. The characteristic
beating patterns appear in the range functions of the RKKY interaction owing to
the presence of multiple Fermi wave-vectors of the underlying $j=3/2$ states.
As an application of the analytical form of the range function, from the finite
temperature evaluation of the correlation functions, we calculate the
contribution of RKKY interaction to the Curie-Weiss temperatures of a
particular dilute magnetic semiconductor ZnMnTe where $4\times4$ Luttinger
Hamiltonian is valid.
