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

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 $$\alpha $$ α -fold divisor function, for any complex number $$\alpha \not \in \{1\}\cup -\mathbb {N}$$ α ∉ { 1 } ∪ - N , even when considering a sequence of parameters $$\alpha $$ α 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 $$d_k(n)$$ d k ( n ) , with $$k\in \mathbb {N}_{\ge 2}$$ k ∈ N ≥ 2 .