  We have studied the pseudogap formation in the single-particle spectra of the
half-filling two-dimensional Hubbard model. Using a Green's function with the
one-loop self-energy correction of the spin and charge fluctuations, we have
numerically calculated the self-energy, the spectral function, and the density
of states in the weak-coupling regime at finite temperature. Pseudogap
formations have been observed in both the density of states and the spectral
function at the Fermi level. The pseudogap in the spectral function is
explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous
behavior in the self-energy is caused by both the strong antiferromagnetic spin
fluctuation and the nesting condition on the non-interacting Fermi surface. In
the present approximation, we find a logarithmic singularity in the integrand
of the imaginary part of the self-energy. Concerning the energy dependence of
the spectral function and the self-energy, two theorems are proved. They give a
necessary condition in the self-energy to produce the pseudogap at the Fermi
level. The pseudogap in the spectral function is highly momentum dependent on
the Fermi surface. It opens initially in the $(\pm \pi,0)$, $(0,\pm \pi)$
regions as the normal state pseudogap observed in the high-$T_c$
superconductors and if the interaction is increased, it spreads to other Fermi
surface sectors. The anisotropy of the pseudogap is produced by the low-energy
enhancement of the spin excitation around ${\bf Q}=(\pi,\pi)$ and the flatness
of the band dispersion around the saddle point.
