  We investigate the phase of the quark determinant with finite chemical
potential in lattice QCD using both analytic and numerical methods. Applying
the winding number expansion and the hopping parameter expansion to the
logarithm of the determinant, we show that the absolute value of the phase has
an upper bound that grows with the spatial volume but decreases exponentially
with an increase in the temporal extent of the lattice. This analytic but
approximate result is confirmed with a numerical study in four-flavor QCD in
which the phase is calculated exactly. Since the phase is well controlled on
lattices with larger time extents, we try the phase reweighting method in a
region beyond $\mu/T=1$ where the Taylor expansion method cannot be applied.
Working in four-flavor QCD, we find a first-order like behavior on a $6^3\times
4$ lattice at $\mu /T\approx 0.8$ which was previously observed by Kentucky
group with the canonical method. We also show that the winding number expansion
has a nice convergence property beyond $\mu/T=1$. We expect that this expansion
is useful to study the high density region of the QCD phase diagram at low
temperatures.
