Divisor-sum fibers

Abstract

Let $s(\cdot )$ denote the sum-of-proper-divisors function, that is, $s(n)={\sum }_{d\mid n,d<n}d$. Erdős-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $\hat{I}\mu (x)$ is any function tending to $0$ as $x\to \infty$, and $\mathcal{A}$ is a set of integers of cardinality at most $x^{\frac{1}{2}+\hat{I}\mu (x)}$, then the number of integers $n\le x$ with $s(n)\in \mathcal{A}$ is $o(x)$, as $x\to \infty$. In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{\frac{1}{2}+\hat{I}\mu (x)})$. We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers $\hat{I}\pm$ and $\hat{I}\mu$, there are integers $n$ with arbitrarily many $s$-preimages lying between $\hat{I}\pm (1-\hat{I}\mu )n$ and $\hat{I}\pm (1+\hat{I}\mu )n$. Finally, we make some remarks on solutions $n$ to congruences of the form $\ddot{I}ƒ(n)\equiv a(\mathrm{mod}n)$, proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions $n\le x$, making it uniform in $a$.