  The functions $F_{G}(n)$ measures the asymptotic behavior of residual
finiteness for a finitely generated group $G$. In previous work
\cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when
$N$ is a finitely generated nilpotent group. However, a counterexample to the
above claim was communicated to the author, and subsequently, the statement of
the asymptotic characterization of $F_{N}(n)$ is incorrect. In this article, we
introduce new tools to provide lower asymptotic bounds for $F_{N}(n)$ when $N$
is a finitely generated nilpotent group. Moreover, we introduce a class of
finitely generated nilpotent groups for which the upper bound of
\cite{Pengitore_1} can be improved. Finally, we construct of a class of
finitely generated nilpotent groups $N$ for which the asymptotic behavior of
$F_N(n)$ can be fully characterized.
