  We investigate lower bounds for the variance in arithmetic progressions of
certain multiplicative functions "close" to $1$. Specifically, we consider
$\alpha_N$-fold divisor functions, when $\alpha_N$ is a sequence of positive
real numbers approaching $1$ in a suitable way or $\alpha_N=1$, and the
indicator of $y$-smooth numbers, for suitably large parameters $y$. As a
corollary, we will strengthen a previous author's result on the first subject
and obtain matching lower bounds to some Barban-Davenport-Halberstam type
theorems for $y$-smooth numbers. Incidentally, we will also find a lower bound
for the variance in arithmetic progressions of the prime factors counting
functions $\omega(n)$ and $\Omega(n)$.
