Problem 955

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claims/: The 8 claim pages of Problem 955, one per claimant's result; the problem's standing derives from them.

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Statement. Let

$$s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d$$

be the sum of proper divisors function.

If $A\subset \mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$.

Status. Open. The site labels the problem OPEN (page last edited 30 September 2025). Its commentary credits proofs for $A$ the primes, the integers with unusually many prime factors, the sums of two squares and the sets of size $x^{1/2+o(1)}$, each an accepted partial claim; the general conjecture is open.

Source. erdosproblems.com/955, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #955, https://www.erdosproblems.com/955.

References.

• [EGPS90] Erdős, P. and Granville, A. and Pomerance, C. and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989) (1990), 165-204.
• [Er73b] Erdős, P., [[../library/arithmetic_functions/erdos_1973_uber_die_zahlen_der_form_und/_index|Über die Zahlen der Form $\sigma (n)-n$ und $n-\phi(n)$]]. Elem. Math. (1973), 83-86.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Part B has no passage stating the density-zero preimage assertion; the nearest is section B10 "Untouchable numbers", p. 100, which records that the untouchable numbers have positive lower density. Library home: guy_2004_unsolved_problems_number_theory.
• [PPT18] Pollack, Paul and Pomerance, Carl and Thompson, Lola, Divisor-sum fibers. Mathematika (2018), 330-342.
• [Po14b] Pollack, Paul, Some arithmetic properties of the sum of proper divisors and the sum of prime divisors. Illinois J. Math. (2014), 125-147.
• [Tr15] Troupe, Lee, On the number of prime factors of values of the sum-of-proper-divisors function. J. Number Theory (2015), 120-135.
• [Tr20] Troupe, Lee, Divisor sums representable as the sum of two squares. Proc. Amer. Math. Soc. (2020), 4189-4202.

Formalization. Statement in formal-conjectures.

Current assessment

Search scope. A bounded literature check found no full solution among its sampled sources. It covered the 2023 and 2026 arXiv records, the authors' publication pages, exact-title and EGPS searches, and indexed recent announcements including X. The July 2026 paper retains the general assertion as its Conjecture 1.3 and says it remains open; its arXiv record lists v1, and Thompson's publication page lists the work as submitted. No status-changing item was found on these routes. This is a bounded assessment, not an exhaustive literature or announcement search.

Proof coverage. The cited statements and source versions below are taken from the complete cited pages. The truncation, digit-target interpretation and prime-input split are elementary derivations recorded here. Full proofs of the cited theorems are not reconstructed here; each claim page records the evidence its result rests on, and the general conjecture remains the mathematical gap.

Progress

The arbitrary density-zero assertion remains open. It is the preimage form of Erdős--Granville--Pomerance--Spiro's Conjecture 4: a set of positive upper density has an image under $s$ of positive upper density. The 1990 paper states that image form on printed p. 169 and explicitly uses the preimage form on printed p. 200. The equivalence follows from $s(s^{-1}(A))\subseteq A$ and $B\subseteq s^{-1}(s(B))$; it does not prove either formulation.

The strongest structure-free result compiled here requires substantially more than density zero: a target counting function at most $x^{1/2+o(1)}$. Missing-digit sets admit separate results, including the 2026 extension to base two and a sharper bound, stated after prime inputs are excluded, whose printed proof has a gap for $g\geq3$. None treats an arbitrary density-zero target.

Known Results

Single targets. Pollack's Theorem 1.11 [Po14b] shows that $s(n)$ is prime for only $O(x/\log x)$ of the $n\le x$, so the preimage of the primes has density zero (claim page). Troupe's Theorem 1.3 [Tr15] shows that $\omega(s(n))$ and $\Omega(s(n))$ lie within $\epsilon\log\log s(n)$ of $\log\log s(n)$ for all but $o(x)$ of the $n\le x$, which settles the targets of integers with abnormally many or few prime factors (claim page). Pollack and Troupe's Erdős--Kac law for $\omega(s(n))$ (Proc. Amer. Math. Soc. 151 (2023), 977--988) refines this to the targets $\{m:|\omega(m)-\log\log m|>h(m)(\log\log m)^{1/2}\}$ for every $h(m)\to\infty$ (claim page). Troupe's Theorem 1.2 [Tr20] counts the $n\le x$ with $s(n)$ a sum of two squares as of order $x/(\log x)^{1/2}$ (claim page). Pollack's Theorem 1 (Integers 15A (2015), A13) shows that $s(n)$ is a base-$g$ palindrome only for a density-zero set of $n$ (claim page).

Finite sparse targets. Pollack, Pomerance, and Thompson's Theorem 1.2 is a proved theorem. In the 11-page 2017 author manuscript, its p. 2, it fixes a function $\varepsilon(x)\to0$ and assumes that a finite set $A$ of positive integers has total cardinality

$$|A|\leq x^{1/2+\varepsilon(x)}.$$

The number of $n\leq x$ with $s(n)\in A$ is then $o_\varepsilon(x)$, uniformly over such $A$. Equivalently, there is a function $\delta_\varepsilon(x)\to0$, independent of $A$, such that this count is at most $\delta_\varepsilon(x)x$. Density zero alone does not imply the finite total-cardinality hypothesis.

The infinite-set consequence is a separate truncation argument. If a fixed set $B$ satisfies $|B\cap[1,y]|\leq y^{1/2+o(1)}$, then for sufficiently large $x$ every relevant value $s(n)$, $n\leq x$, is below $2x\log\log x$. Apply the finite theorem to $A_x=B\cap[1,2x\log\log x]$, whose total cardinality is $x^{1/2+o(1)}$ or smaller. It follows that $s^{-1}(B)$ has density zero (claim page). The manuscript's page numbers are distinct from the published Mathematika 64 (2018), 330--342 pagination.

Large individual fibers. The same paper's Theorem 1.4, also on its p. 2, proves that there is an absolute $c>0$ such that, for every $\alpha,\eta>0$, infinitely many $m$ have at least $\exp(c\log m/\log\log m)$ distinct $s$-preimages in $(\alpha(1-\eta)m,\alpha(1+\eta)m)$. The authors give $c=1/7$. This disproves the proposed uniform bound on the number of solutions $n\leq\theta m$ to $s(n)=m$ for fixed $\theta>0$. It does not disprove the density-zero preimage conjecture: large individual fibers do not supply a density-zero target whose preimage has positive upper density.

Missing digits. Fix a base $g$, a nonempty proper digit set $D\subsetneq\{0,\ldots,g-1\}$, and $\gamma\in(0,1)$. Benli, Cesana, Dartyge, Dombrowsky, and Thompson's Theorem 1.8 (arXiv:2307.12859v1, p. 2; claim page) gives, for $g\geq3$,

$$\#\{n\leq x:\text{all base-}g\text{ digits of }s(n)\text{ lie in }D\} \ll_{g,D,\gamma}x\exp\bigl(- (\log\log x)^\gamma\bigr).$$

These fixed digit targets have density zero. Benli, Dartyge, Dombrowsky, Pollack, and Thompson's 2026 preprint records the bound for every $g\geq2$ in Theorem 1.4 (arXiv:2607.18981v1, p. 2). Its Appendix A, pp. 16--17, supplies the binary case omitted from the earlier theorem (claim page).

The 2026 paper's Theorem 1.5 (same version, p. 2) fixes a nonzero digit $a_0\in\{1,\ldots,g-1\}$ and states, for some $c=c(g)>0$,

$$\#\{n\leq x:n\text{ composite and }s(n)\text{ omits }a_0\} \ll x\exp(-c\sqrt{\log x}).$$

This sharper estimate is restricted to composite inputs and omission of a nonzero digit. Since $s(p)=1$, restoring prime inputs adds no primes when $a_0=1$, and at most $\pi(x)$ otherwise; $n=1$ contributes at most one exception. The derived all-input bound is therefore still $o(x)$, but the displayed sharper estimate is not asserted for all inputs. The printed proof of this rate for $g\geq3$ has a gap in its small-gcd step (pp. 15--16), as the library card records; the qualitative density-zero conclusion for these targets already follows from Theorem 1.4.

Sources without a claim page. Erdős's [Er73b] Satz I exhibits a set of positive lower density with empty preimage under $s$ and concerns no density-zero target. Fan's Theorem 6 (arXiv:2508.06005) with $f=1$ treats a target that moves with $x$ and follows from Pollack and Troupe's theorem, and its weighted proportions are not the problem's density. Pollack and Singha Roy (Colloq. Math. 168 (2022), 287--295) prove in Proposition 3.1, for $k\ge4$, that almost always no $p^k$ above a threshold depending on $x$ divides $s(n)$; with their Lemma 2.2 this bears on the density-zero target of their Remark 3.4, but the paper states no preimage theorem for a fixed target. Luca and Pomerance's theorem concerns a target of positive lower density, and the congruence estimates of Lebowitz-Lockard and coauthors supply no density-zero target; neither settles an instance.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• benli_2023_sums_proper_divisors_missing_digits
• benli_2023_sums_proper_divisors_missing_digits / theorem_1_8
• benli_2026_digits_sum_proper_divisors
• benli_2026_digits_sum_proper_divisors / theorem_1_4
• benli_2026_digits_sum_proper_divisors / theorem_1_5
• erdos_1973_uber_die_zahlen_der_form_und
• erdos_1973_uber_die_zahlen_der_form_und / satz_i
• erdos_1973_uber_die_zahlen_der_form_und / satz_ii
• erdos_1990_normal_behavior_iterates_arithmetic_functions
• erdos_1990_normal_behavior_iterates_arithmetic_functions / conjecture_4
• erdos_1990_normal_behavior_iterates_arithmetic_functions / theorem_5_2
• fan_2025_hardy_ramanujan_inequality_sifted_sets_its_applications
• fan_2025_hardy_ramanujan_inequality_sifted_sets_its_applications / theorem_1_1
• fan_2025_hardy_ramanujan_inequality_sifted_sets_its_applications / theorem_1_6
• lebowitz_lockard_et_al_2021_distribution_mod_p_eulers_totient_sum_proper_divisors
• lebowitz_lockard_et_al_2021_distribution_mod_p_eulers_totient_sum_proper_divisors / theorem_1_3
• luca_pomerance_2015_range_sum_of_proper_divisors_function
• luca_pomerance_2015_range_sum_of_proper_divisors_function / theorem_1
• pollack_2014_arithmetic_properties_sum_proper_divisors_sum
• pollack_2014_arithmetic_properties_sum_proper_divisors_sum / theorem_1_11
• pollack_2015_palindromic_sums_proper_divisors
• pollack_2018_divisor_sum_fibers
• pollack_2018_divisor_sum_fibers / theorem_1_2
• pollack_2018_divisor_sum_fibers / theorem_1_4
• pollack_roy_2021_powerfree_sums_proper_divisors
• pollack_roy_2021_powerfree_sums_proper_divisors / proposition_3_1
• pollack_roy_2021_powerfree_sums_proper_divisors / theorem_1_2
• pollack_roy_2021_powerfree_sums_proper_divisors / theorem_3_3
• pollack_troupe_2021_sums_proper_divisors_follow_erdos_kac_law
• pollack_troupe_2021_sums_proper_divisors_follow_erdos_kac_law / theorem_1
• pomerance_2018_first_function_iterates
• pomerance_2018_first_function_iterates / conjecture_2_3
• pomerance_2018_first_function_iterates / theorem_2_4
• troupe_2015_number_prime_factors_values_sum_proper
• troupe_2015_number_prime_factors_values_sum_proper / theorem_1_3
• troupe_2015_number_prime_factors_values_sum_proper / theorem_1_4
• troupe_2020_divisor_sums_representable_as_sum_two
• troupe_2020_divisor_sums_representable_as_sum_two / theorem_1_2
• guy_2004_unsolved_problems_number_theory