Sums of proper divisors with missing digits

TL;DR

This paper proves the Erdős-Granville-Pomerance-Spirou conjecture for sets of integers with missing digits, showing their preimages under the sum of proper divisors function have asymptotic density zero, and provides bounds and survey of related work.

Contribution

The paper confirms the EGPS conjecture for missing digit sets and offers a sharp upper bound for the size of the preimage set, advancing understanding of divisor sum behavior.

Findings

EGPS conjecture holds for missing digit sets

Provides a sharp upper bound for preimage set size

Surveys recent progress on sets with missing digits

Abstract

Let $s(n)$ denote the sum of proper divisors of an integer $n$. In 1992, Erd\H{o}s, 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. In this paper we show that the EGPS conjecture holds when $\mathcal{A}$ is taken to be a set of integers with missing digits. In particular, we give a sharp upper bound for the size of this preimage set. We also provide an overview of progress towards the EGPS conjecture and survey recent work on sets of integers with missing digits.