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Statement

Write $s(n)=\sigma(n)-n$; an $s$-preimage of $m$ is an $n$ with $s(n)=m$.

Theorem 1.4 (manuscript p. 2). There is a constant $c>0$ with the following property. For all positive real numbers $\alpha$ and $\epsilon$, there are infinitely many $m$ having at least

$$\exp\!\left(c\,\frac{\log m}{\log\log m}\right)$$

$s$-preimages in the interval $(\alpha(1-\epsilon)m,\;\alpha(1+\epsilon)m)$.

The authors state after the theorem (p. 2) that their proof shows $c=1/7$ is admissible. The proof of Theorem 3.2 is written with $c=1/10$, and the remark after it (p. 5) allows any $c<(5/24)\log2$, in particular $c=1/7$.

Hypothesis 1.3 disproved. Hypothesis 1.3 (p. 2), from Erdős, Granville, Pomerance and Spiro, posits for each $\theta>0$ a constant $C_\theta$ such that, for every positive integer $m$, at most $C_\theta$ numbers $n\leq\theta m$ satisfy $s(n)=m$. Taking $\alpha=\theta/2$ and $\epsilon=1/2$, say, the theorem gives infinitely many $m$ with an unbounded number of preimages below $\theta m$, so the hypothesis fails for every $\theta>0$; the paper records the disproof as the purpose of Section 3 (p. 2).

A remark on small fibers. The remark after the proof (p. 6) notes that the same construction with $n_0=2$ gives infinitely many even $m$ with more than $\exp(c\log m/\log\log m)$ preimages of the form $2pq$. This settles what the second author had called difficult in the paper's reference [18]: showing that infinitely many even $m$ have at least three $s$-preimages.

Source. Paul Pollack, Carl Pomerance, and Lola Thompson, Divisor-Sum Fibers, Mathematika 64(2) (2018), 330--342, DOI 10.1112/S0025579317000535. Theorem 1.4 is on p. 2 of the 11-page author manuscript that the source card identifies.

Read depth. Claims checked: the statement, the admissible constant and Hypothesis 1.3 were read clause by clause against the manuscript, and the proof in Section 3 (pp. 4--6) was read for its structure only, not verified.

Proof pointer

Section 3, pp. 4--6. Theorem 3.2 (p. 4) gives, for integers $a\neq0$ and $b>0$, infinitely many $k$ with more than $\exp(c\log k/\log\log k)$ representations as $(bp+a)(bq+a)$ with $p,q$ prime; its proof counts pairs of primes in prescribed residue classes modulo a product $M$ of small primes and applies the pigeonhole principle. For Theorem 1.4 (pp. 5--6), one fixes $n_0>1$ with $s(n_0)/n_0$ within a factor $1\pm\epsilon/2$ of $\alpha^{-1}$, which the density of the values $s(n)/n$ allows. For distinct primes $p,q\nmid n_0$, identity (3.1) on p. 5 writes $s(n_0)s(n_0pq)$ as $(s(n_0)p+\sigma(n_0))(s(n_0)q+\sigma(n_0))$ plus a constant depending on $n_0$, so Theorem 3.2 with $a=\sigma(n_0)$, $b=s(n_0)$ turns many representations of $k$ into many $n=n_0pq$ with the same value $s(n)=m$. Lower bounds on $p$ and $q$ place each such $n$ in $((1-\epsilon)\alpha m,(1+\epsilon)\alpha m)$. This is a map of the proof, not a reconstruction of it.

Dependencies

Theorem 3.1 (p. 4), on primes in arithmetic progressions to moduli free of finitely many exceptional factors, which the authors deduce from Alford, Granville and Pomerance, Ann. of Math. 139 (1994), Theorem 2.1 (the paper's [1]); the density of $\{s(n)/n\}$ in $(0,\infty)$. Theorem 3.2 generalizes ideas of Prachar and Erdős (p. 4).

Bears on

• Problem 955: the problem asks whether $s^{-1}(A)$ has density zero for every density-zero $A$. The authors cite EGPS (p. 2) for Hypothesis 1.3 implying that conjecture; the theorem refutes the hypothesis and so removes that route, but it neither proves nor disproves the problem's assertion.