Erdős–Granville–Pomerance–Spiro: does density 0 pull back to density 0 under s(n)? (Erdős #955)
Statement
Let $$s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d$$ be the sum of the proper divisors of $n$. Prove or disprove the conjecture of Erdős, Granville, Pomerance and Spiro: if $A\subseteq\mathbb{N}$ has natural density $0$, then its preimage $s^{-1}(A)=\{n\in\mathbb{N}: s(n)\in A\}$ also has density $0$.
Acceptance. FULLY RESOLVES: a complete proof of the EGPS conjecture (every density-$0$ set $A$ has $s^{-1}(A)$ of density $0$), OR a disproof (an explicit density-$0$ set $A$ for which $s^{-1}(A)$ has positive lower density, with proof), written or machine-checkable. ADVANCES (each strictly beyond the background): (a) enlarge the class of admissible $A$ by pushing the Pollack–Pomerance–Thompson threshold from $\lvert A\cap[1,x]\rvert\leq x^{1/2+o(1)}$ up to $\leq x^{1/2+c}$ for a fixed constant $c>0$, with proof; (b) prove the conjecture for a new natural family of density-$0$ sets beyond the primes, sums of two squares, integers with abnormal $\omega(n)$, and integers with missing digits, with proof; (c) a conditional proof of the full conjecture under a clearly stated hypothesis. Deliver the proof, specifying the new admissible size threshold or family and why it strictly extends the $x^{1/2+o(1)}$ bound above.
Background
A conjecture of Erdős, Granville, Pomerance and Spiro [EGPS90]; listed as open on erdosproblems.com/955 (fetched 2026-07-21, status 'open'). The zero-density hypothesis on $A$ is essential: $s$ can send a density-$0$ set to a positive-density set (e.g. $A=\{pq: p<q\text{ prime}\}$ has density $0$ yet $s(A)$ can be dense), and Erdős [Er73b] proved there exist positive-density sets $A$ with $s^{-1}(A)$ empty. Known partial cases where the conjecture holds: $A=$ the primes (Pollack [Po14b]); $A=$ integers with abnormally many prime factors (Troupe [Tr15]); $A=$ integers that are sums of two squares (Troupe [Tr20]); and $A=$ integers with missing digits (Benli, Cesana, Dartyge, Dombrowsky and Thompson, 2023). The strongest general result is Pollack, Pomerance and Thompson [PPT18]: if $\epsilon(x)=o(1)$ and $\lvert A\cap[1,x]\rvert\leq x^{1/2+\epsilon(x)}$, then $\#\{n\leq x: s(n)\in A\}=o(x)$; using $s(n)\ll n\log\log n$, any $A$ with $\lvert A\cap[1,x]\rvert\leq x^{1/2+o(1)}$ therefore has $s^{-1}(A)$ of density $0$. The gap is that a density-$0$ set may be far larger than $x^{1/2+o(1)}$ (e.g. of size $x/\log x$). Related: integers $k$ with $s(n)=k$ unsolvable are the 'untouchable' numbers (Guy's UPINT, problem B10); SciNet's iterated-$\sigma$ problems (Erdős #410 at erdosproblems.com/410, #412 at erdosproblems.com/412) treat related but distinct dynamics of the same divisor function. Attacker's tool: analytic number theory on the anatomy of $s$-preimages and the distribution of $\sigma(n)/n$, aimed at pushing the PPT size threshold past exponent $1/2$ or at new structured sparse families $A$; computational anatomy of preimage sizes to locate the true barrier.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #955 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.