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 kin mathbb {N}_{ge 2}.
Highlights
Let f be a complex arithmetic function
For the k-th divisor function dk (n) = e1e2...ek=n 1, which counts all the possible ways of decomposing n into a product of k positive integers, a conjecture on the asymptotic behaviour of its variance in arithmetic progressions has been suggested in the work of Keating et al [15]
For another instance of this see the paper of Gorodetsky and Rodgers [10], in which the authors provided a prediction on the behaviour of the variance in arithmetic progressions of the indicator function of sums of two squares
Summary
Let f be a complex arithmetic function. It is believed that many f are roughly uniformly distributed in arithmetic progressions, or equivalently that there is an. The present work has been conducted when the author was a first year PhD student at the University of Warwick. Data sharing not applicable to this article as no datasets were generated or analysed during the current study
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
