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

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