A lower bound for the variance of generalized divisor functions in arithmetic progressions

Abstract

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $\alpha$-fold divisor function, for any complex number $\alpha\not\in \{1\}\cup-\mathbb{N}$, even when considering a sequence of parameters $\alpha$ close in a proper way to $1$. Our work builds on that of Harper and Soundararajan, who handled the particular case of $k$-fold divisor functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$.