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 \(k\in \mathbb {N}_{\ge 2}\).

我们证明，对于一大类乘法函数（称为广义除数函数），可以找到算术级数中相应方差的下限。作为主要推论，我们推导出任何\（\ alpha \）折叠除数函数，任何复数\（\ alpha \ not \ in \ {1 \} \ cup-\ mathbb {N} \）的结果，即使考虑以合理的方式将参数\（\ alpha \）的序列关闭到1的情况。我们的工作也建立在Harper和Soundararajan的工作基础上，他们处理了k倍除数函数\（d_k（n）\ ），并带有\（k \ in \ mathbb {N} _ {\ ge 2} \）。