  The mean and the variance of the logarithm of the conductance ($ln g$) in the
localized regime in the one-dimensional Anderson model are calculated
analytically for weak disorder, starting from the recursion relations for the
complex reflection- and transmission amplitudes. The exact recursion relation
for the reflection amplitudes is approximated by improved Born approximation
forms which ensure that averaged reflection coefficients tend asymptotically to
unity in the localized regime, for chain lengths $L=Na\to\infty$. In contrast
the familiar Born approximation of perturbation theory would not be adapted for
the localized regime since it constrains the reflection coefficient to be less
than one. The proper behaviour of the reflection coefficient (and of other
related reflection parameters) is responsible for various anomalies in the
cumulants of $\ln g$, in particular for the well-known band center anomaly of
the localization length. While a simple improved Born approximation is
sufficient for studying cumulants at a generic band energy, we find that a
generalized improved Born approximation is necessary to account satisfactorily
for numerical results for the band center anomaly in the mean of $\ln g$. For
the variance of $\ln g$ at the band center, we reveal the existence of a weak
anomalous quadratic term proportional to $L^2$, besides the previously found
anomaly in the linear term. At a generic band energy the variance of $\ln g$ is
found to be linear in $L$ and is given by twice the mean, up to higher order
corrections which are calculated. We also exhibit the $L=$independent offset
terms in the variance, which strongly depend on reflection anomalies.
