Sums of Proper Divisors with Missing Digits

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theorem_1_8: Gives a quantitative density-zero bound for inputs whose sum of proper divisors uses only digits from a fixed proper subset in base g at least three.

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Kübra Benli, Giulia Cesana, Cécile Dartyge, Charlotte Dombrowsky, and Lola Thompson, Sums of Proper Divisors with Missing Digits, arXiv:2307.12859v1 (24 July 2023). The work was subsequently published in Research Directions in Number Theory, Association for Women in Mathematics Series 32 (Springer, 2024), 93--110; those publication details do not identify the bytes selected here as the published edition.

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The paper studies the preimage under the sum-of-proper-divisors function $s(n)=\sigma(n)-n$ of sets whose members use only a prescribed proper subset of the base-$g$ digits. Theorem 1.8 fixes $g\geq3$, $\gamma\in(0,1)$, and a nonempty proper digit set $D\subsetneq\{0,1,\ldots,g-1\}$, and proves

$$\#\{n\leq x:\text{ every base-}g\text{ digit of }s(n)\text{ lies in }D\} =O\!\left(x\exp\bigl(-(\log\log x)^\gamma\bigr)\right).$$

Because the target digit set has asymptotic density zero, this verifies the EGPS preimage conjecture for this structured class. It does not settle the conjecture for an arbitrary density-zero target set. The authors note that when $1\in D$, prime inputs give a lower bound $\pi(x)$ because $s(p)=1$. Thus digit sets containing $1$ show that the logarithmic exponent cannot be uniformly improved over the full class of fixed $g$ and $D$; this is not a matching lower bound for every fixed digit set $D$.

Section 4 proves Theorem 1.8. The main split is according to whether a chosen power $g^k$ divides $\sigma(n)$: Lemma 1.9 controls the nondivisible case, and in the divisible case the digit restriction confines $n$ to at most $|D|^k$ residue classes modulo $g^k$. This is a proof pointer, not a complete proof.

Bears on. #955.

Results to transcribe.

• Theorem 1.8: the quantitative missing-digit preimage bound for $g\geq3$.

Living verification. Needs review. The identity, selected version, Theorem 1.8 statement, special-case transfer, and proof route were checked against the selected arXiv v1 PDF. No complete proof is supplied, reconstructed, or independently certified here.