Preimages of the sum of proper divisors function

Let $s(n)$ denote the sum of proper divisors of an integer $n$. The function $s(n)$ has been studied for thousands of years, due to its connection with the perfect numbers. In 1992, Erdös, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. This has been confirmed for certain specific sets $\mathcal{A}$, but remains open in general. In this talk, we will give a survey of recent progress towards the EGPS conjecture. This talk is based on joint work with Paul Pollack and Carl Pomerance, and also on joint work with Kübra Benli, Giulia Cesana, Cécile Dartyge, and Charlotte Dombrowsky.