  Using two sets of high-precision Monte Carlo data for the two-dimensional XY
model in the Villain formulation on square $L \times L$ lattices, the scaling
behavior of the susceptibility $\chi$ and correlation length $\xi$ at the
Kosterlitz-Thouless phase transition is analyzed with emphasis on
multiplicative logarithmic corrections $(ln L)^{-2r}$ in the finite-size
scaling region and $(ln \xi)^{-2r}$ in the high-temperature phase near
criticality, respectively. By analyzing the susceptibility at criticality on
lattices of size up to $512^2$ we obtain $r = -0.0270(10)$, in agreement with
recent work of Kenna and Irving on the the finite-size scaling of Lee-Yang
zeros in the cosine formulation of the XY model. By studying susceptibilities
and correlation lengths up to $\xi \approx 140$ in the high-temperature phase,
however, we arrive at quite a different estimate of $r = 0.0560(17)$, which is
in good agreement with recent analyses of thermodynamic Monte Carlo data and
high-temperature series expansions of the cosine formulation.
