A lower bound for the variance in arithmetic progressions of some multiplicative functions close to $1$

Abstract: 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 $\Omega(n)$.