The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-03-12 , DOI: 10.1007/s11139-020-00374-8 Daniele Mastrostefano
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 \(k\in \mathbb {N}_{\ge 2}\).
中文翻译:
算术级数中广义除数函数方差的下界
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\(k \ in \ mathbb {N} _ {\ ge 2} \)。