We investigate the surface width W of solid-on-solid surfaces in the v
icinity of the roughening temperature T(r). Above T(r), W2 is expected
to diverge with the system size L like ln L. However, close to T(r) a
clean ln L behavior can only be seen on extremely large lattices. Sta
rting from the Kosterlitz-Thouless renormalization group, we derive an
improved formula that describes the small L behavior on both sides of
T(r). For the discrete Gaussian model, we used the valleys-to-mountai
ns-reflections cluster algorithm in order to simulate the fluctuating
solid-on-solid surface. The base plane above which the surface is defi
ned is an L x L square lattice. In the simulation we took 8 less-than-
or-equal-to L less-than-or-equal-to 256. The improved formula fits the
numerical results very well. From the analysis, we estimate the rough
ening temperature to be T(r) 0.755(3).