  We study directed percolation at the upper critical transverse dimension
$d=4$, where critical fluctuations induce logarithmic corrections to the
leading (mean-field) behavior. Viewing directed percolation as a kinetic
process, we address the following properties of directed percolation clusters:
the mass (the number of active sites or particles), the radius of gyration and
the survival probability. Using renormalized dynamical field theory, we
determine the leading and the next to leading logarithmic corrections for these
quantities. In addition, we calculate the logarithmic corrections to the
equation of state that describes the stationary homogeneous particle density in
the presence of a homogeneous particle source.
