# Divisor sums representable as the sum of two squares

## TL;DR

This paper proves that the count of natural numbers up to x whose sum of proper divisors is a sum of two squares grows roughly as x divided by the square root of log x, confirming a special case of a conjecture relating densities.

## Contribution

It establishes the asymptotic behavior of numbers with sum-of-divisors as sum of two squares, confirming a conjecture about density transfer for such sets.

## Key findings

- Number of n ≤ x with s(n) as sum of two squares is ~ x/√log x
- The result aligns with the count of n ≤ x that are sums of two squares
- Supports a conjecture on density zero sets and their preimages under s

## Abstract

Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$ which are a sum of two squares. Our result confirms a special case of a conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero (e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two squares), then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.