  We study the correlation function and mean linear response function of the
velocity Fourier mode of statistically steady-state, homogeneous and isotropic
turbulence in the Eulerian and Lagrangian coordinates through direct numerical
simulation (DNS). As the Lagrangian velocity, we here adopt Kraichnan's
Lagrangian history framework where Lagrangian particles are labelled with
current positions and their velocity are measured at some time before. This
Lagrangian velocity is numerically calculated with a method known as passive
vector method. Our first goal is to study relation between the correlation
function and the mean linear response function in the Eulerian and Lagrangian
coordinates. Such a relation is known to be important in analysing the closed
set of equations for the two functions, which are obtained by
direct-interaction-approximation type closures. We demonstrate numerically that
the fluctuation-dissipation theorem (proportionality between the two functions)
does not hold. The relation is further investigated with general analytical
expressions of the mean linear response function under stochastic settings,
which are known as the fluctuation-response relations in non-equilibrium
statistical mechanics. Our second goal is to identify characteristic times
associated with the two functions and to compare the times between the Eulerian
and Lagrangian coordinates. Our DNS result supports the common view that the
Eulerian characteristic times have the sweeping-time scaling ($\propto k^{-1}$,
where $k$ is the wavenumber) for both functions and the Lagrangian
characteristic times in the inertial range have the Kolmogorov-time scaling
($\propto k^{-2/3}$) for both functions.
