  A prominent feature of quantum spin liquids is fractionalization of the spin
degree of freedom. Fractionalized excitations have their own dynamics in
different energy scales, and hence, affect finite-temperature ($T$) properties
in a peculiar manner even in the paramagnetic state harboring the quantum spin
liquid state. We here present a comprehensive theoretical study of the spin
dynamics in a wide $T$ range for the Kitaev model on a honeycomb lattice, whose
ground state is such a quantum spin liquid. In this model, the
fractionalization occurs to break up quantum spins into itinerant matter
fermions and localized gauge fluxes, which results in two crossovers at very
different $T$ scales. Extending the previous study for the isotropic coupling
case [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. ${\textbf 117}$,
157203 (2016)], we calculate the dynamical spin structure factor, the NMR
relaxation rate, and the magnetic susceptibility while changing the anisotropy
in the exchange coupling constants, by using the dynamical mean-field theory
and the continuous-time quantum Monte Carlo method based on a Majorana fermion
representation. We find that the system exhibits peculiar behaviors below the
high-$T$ crossover whose temperature is comparable to the average of the
exchange constants, reflecting the spin fractionalization in the paramagnetic
region. Among them, the dichotomy between the static and dynamical spin
correlations is unusual behavior hardly seen in conventional magnets. We
discuss the relation between the dichotomy and the spatial configuration of
gauge fluxes. Our results could stimulate further experimental and theoretical
analyses of candidate materials for the Kitaev quantum spin liquids.
