On the closure of the image of the generalized divisor function

Abstract

For any real number $s$, let ${\sigma }_s$ be the generalized divisor function, i.e., the arithmetic function defined by ${\sigma }_s(n):={\sum }_{d\mid n}{d}^s$, for all positive integers $n$. We prove that for any $r>1$ the topological closure of ${\sigma }_{-r}({\mathbf{N}}^{+})$ is the union of a finite number of pairwise disjoint closed intervals $I_1,\dots ,I_{\ell }$. Moreover, for $k=1,\dots ,\ell$, we show that the set of positive integers $n$ such that ${\sigma }_{-r}(n)\in I_k$ has a positive rational asymptotic density $d_k$. In fact, we provide a method to give exact closed form expressions for $I_1,\dots ,I_{\ell }$ and $d_1,\dots ,d_{\ell }$, assuming to know $r$ with sufficient precision. As an example, we show that for $r=2$ it results $\ell =3$, $I_1=[1,{\pi }^2/9]$, $I_2=[10/9,{\pi }^2/8]$, $I_3=[5/4,{\pi }^2/6]$, $d_1=1/3$, $d_2=1/6$, and $d_3=1/2$.

arXiv comment10 pages, 1 figure