On the Digits of the Sum of Proper Divisors

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theorem_1_4: Gives a quantitative density-zero bound for inputs whose sum of proper divisors has all digits in a fixed proper subset, including base two.

theorem_1_5: Gives a stretched-exponential upper bound for composite inputs whose sum of proper divisors omits a fixed nonzero base-g digit.

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Kübra Benli, Cécile Dartyge, Charlotte Dombrowsky, Paul Pollack, and Lola Thompson, On the Digits of the Sum of Proper Divisors, arXiv:2607.18981v1 (21 July 2026).

Local artifact. The selected 18-page arXiv v1 PDF was retrieved for the source review from 2026-09-07T11:21:20.751914000Z through 2026-09-07T11:21:21.097356000Z; this acquisition interval is distinct from the later reading recorded in the payload receipt. The arXiv record (https://arxiv.org/abs/2607.18981, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Theorem 1.1 says that for every fixed base $g\geq2$ and every $k(x)\to\infty$, almost all $n\leq x$ have every base-$g$ digit among both the first and the last $k(x)$ digits of $s(n)$. Theorem 1.2 establishes Benford's law for $s(n)$ with respect to logarithmic density. These typical digit results motivate the paper's treatment of unusually restrictive digit patterns.

Conjecture 1.3 restates the EGPS conjecture and explicitly says it remains open. Theorem 1.4 records the missing-digit preimage bound for every base $g\geq2$: its $g\geq3$ part is the earlier Benli--Cesana--Dartyge--Dombrowsky--Thompson theorem, while the binary case is supplied by the separate argument in Appendix A.

Theorem 1.5 obtains a stronger bound when inputs are required to be composite and the omitted digit is a fixed nonzero digit $a_0\in\{1,\ldots,g-1\}$:

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

where $c=c(g)>0$. Prime inputs must be restored separately before drawing an E955 consequence, since $s(p)=1$.

Overview

Page numbers below are the PDF pages of the selected arXiv v1 PDF. Theorem 1.1 (p. 1) is proved separately as Theorems 2.2 (p. 3) and 2.5 (p. 5) in §2. For trailing digits, Lemma 2.1 (p. 3, cited from [11]) makes $g^k\mid\sigma(n)$ typical; then $s(n)\equiv-n\pmod{g^k}$ reduces the count to sparse residue classes. For leading digits, Lemmas 2.3–2.4 (pp. 4–5) control $s(n)/n$ through the small prime factors of $n$ and compare the digit lengths of $s(n)$ and $n$; Theorem 2.5 (p. 5) then counts inputs in short intervals attached to forbidden leading blocks.

Theorem 1.2 (p. 2), established through Theorem 3.5 (p. 7) in §3, gives strong Benford frequencies in logarithmic density: a valid leading base-$g$ block $D$ occurs with frequency $\log_g(1+1/D)$. Theorem 3.5 (p. 7) proves the stronger cancellation $\frac1{\log N}\sum_{1<n\leq N}s(n)^{i\alpha}/n\to0$ for each fixed real $\alpha\ne0$. Its proof expands $s(n)^{i\alpha}=\sigma(n)^{i\alpha}(1-n/\sigma(n))^{i\alpha}$, groups integers by the special prime divisor of Definition 3.4 (p. 7), and applies the cited weighted Halász estimate, Proposition 3.3 (p. 7), to multiplicative terms; the finite expansion and factorization appear in (3.1)–(3.2) (p. 9). Proposition 3.6 (p. 11) shows that Benford's law fails for natural density, using the concentration estimate (3.3) (p. 12).

Section 4 (pp. 12–16) proves Theorem 1.5 (p. 2), the composite-input bound displayed above, using smooth-number and repeated-largest-prime bounds (Lemmas 4.1–4.2, p. 13), the congruence-counting Lemma 4.3 (p. 13), and the identity $s(mP)=P s(m)+\sigma(m)$ when $P\nmid m$, equation (4.1) (p. 14). Theorem 1.4 (p. 2) is explicitly a cited result of [1], extended to base $2$ in Appendix A (pp. 16–17); it bounds the preimage of any fixed proper digit alphabet without restricting inputs to composites. Conjecture 1.3 (p. 2) is the still-open general density-zero preimage claim.

The printed proof of Theorem 1.5 (§4.2, pp. 14–16 of the arXiv v1 PDF, read clause by clause on the page images) has gaps in the compilation's reading. As defined on p. 14, $E_1$ counts the $mP\le x$ with $m\le x^{\alpha_g}$ and so literally includes $m=1$, where the injectivity in $P$ asserted on p. 15 (for fixed $m$ and $a$, at most one $P\le x/m$ has $s(mP)=a$) fails, since $s(P)=1$ for every prime $P$; the theorem counts only composite $n=mP$, which have $m\geq2$, so this is a slip in the definition rather than in the argument. The material gap is the small-gcd calculation (pp. 15–16): the displayed summation over $m$ and $d$ ends with an unexplained factor $1/y$ (top of p. 16), and its final bound $x/y^{1/2-\log(g-1)/\log g}$ (p. 16) gives no saving for $g\geq3$, where $\log(g-1)/\log g>1/2$. The quantitative composite bound is therefore a theorem stated by the paper; this is the compilation's reading of the printed argument, not a published erratum, and the displayed argument needs repair before that rate can be used as proved.

Relation to E955

This source bears on Problem 955.

E955 is exactly Conjecture 1.3 with $A=\mathcal A$: it asks whether $d(A)=0$ always implies $d(s^{-1}(A))=0$. For a fixed proper digit alphabet $\mathcal D\subsetneq\{0,\ldots,g-1\}$, the paper's $\mathcal W_{\mathcal D}$ (§1.1, p. 3) is a density-zero target. The cited Theorem 1.4 (p. 2) supplies $\#\{n\leq x:s(n)\in\mathcal W_{\mathcal D}\}\ll x\exp(- (\log\log x)^\gamma)$ for each fixed $0<\gamma<1$, including prime inputs. Theorem 1.1 (p. 1) also gives a qualitative zero-density preimage conclusion for each fixed missing digit. These are special target sets and provide no estimate for an arbitrary density-zero $A$.

Theorem 1.5 (p. 2) would sharpen the count for composite inputs when $A$ consists of numbers omitting a fixed nonzero digit, subject to the §4.2 proof gaps noted above. If that digit is $1$, prime inputs contribute nothing since $s(p)=1$; if it differs from $1$, every prime belongs to the preimage, giving a lower bound of $\pi(x)$. The decomposition $n=mP$ and (4.1) (p. 14), followed by congruences modulo $g^k$ and Lemma 4.3 (p. 13), suggest a route for other targets with controlled residue counts and divisibility conditions. Density zero alone supplies neither property. The Benford result concerns logarithmic density, and Proposition 3.6 (p. 11) rules out its natural-density analog; neither establishes E955.

Bears on. #955.

Results to transcribe.

• Theorem 1.4: the missing-digit preimage bound for every base $g\geq2$, with the binary case proved in Appendix A.
• Theorem 1.5: the sharper composite-input bound for omission of a fixed nonzero digit.

Living verification. Needs review. The selected version, Conjecture 1.3, Theorems 1.4 and 1.5, the nonzero-digit restriction, and the relevant proof sections were checked against the selected arXiv v1 PDF. No complete proof is supplied, reconstructed, or independently certified here.