On gaps in the closures of images of divisor functions

Abstract

Given a complex number c, define the divisor function σc : N → C by σc(n) = d|n d c . In this paper, we look at σ−r(N), the topological closures of the image of σ−r, when r > 1. We exhibit new lower bounds on the number of connected components of σ−r(N), bringing this bound from linear in r to exponential. Finally, we discuss the general structure of gaps of σ−r(N) in order to work towards a possible monotonicity result. t .Furthermore, for understanding the behavior of C r , it has proven useful to study the gaps of σ −r (N), where by a gap we mean a bounded connected component of R \σ −r (N). In this paper, we extend Zubrilina's work by showing that 6 is also a Zubrilina number and by improving her lower bound for C r in both the asymptotic and small-r cases. At the end, we will look at the general structure of gaps of σ −r (N) before finishing with some open problems.