On the Density of Ranges of Generalized Divisor Functions with Restricted Domains

TL;DR

This paper investigates the ranges of generalized divisor functions with restricted domains, establishing bounds, density conditions, and open problems related to their behavior for different parameters.

Contribution

It introduces the functions ta_k and characterizes when their ranges are dense within specific intervals, advancing understanding of generalized divisor functions.

Findings

Range of ta_{-r,k} is a subset of [1, { extstyle rac{\

Range is dense if and only if r f ta_k

Abstract

We begin by defining functions $\sigma_{t,k}$, which are generalized divisor functions with restricted domains. For each positive integer $k$, we show that, for $r>1$, the range of $\sigma_{-r,k}$ is a subset of the interval $\displaystyle{\left[1,\frac{\zeta(r)}{\zeta((k+1)r)}\right)}$. After some work, we define constants $\eta_k$ which satisfy the following: If $k\in\mathbb{N}$ and $r>1$, then the range of the function $\sigma_{-r,k}$ is dense in $\displaystyle{\left[1,\frac{\zeta(r)}{\zeta((k+1)r)}\right)}$ if and only if $r\leq\eta_k$. We end with an open problem.