Luca–Pomerance: The range of the sum-of-proper-divisors function

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theorem_1: The even integers of the form s(n) = sigma(n) - n, for some integer n, form a set of positive lower density.

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The copy read for this card is the authors' 15-page preprint, pages numbered 1–15; page locators below are its pages. It is the authors' preprint, not the published edition, and prints no copyright or license line on any page; the card records no source URL, so no host page was read; the term is unstated.

Florian Luca and Carl Pomerance, "The range of the sum-of-proper-divisors function," Acta Arithmetica, 168(2), 187-199, 2015. https://doi.org/10.4064/aa168-2-6

Overview

The paper asks whether even integers occur as values of the proper-divisor sum $s(n)=\sigma(n)-n$ with positive frequency. Theorem 1 (§1, p. 2) proves that the even values of $s$ have positive lower density. The authors state that the proof adapts to give a positive proportion of values in every fixed residue class and, similarly, for $s_\varphi(n)=n-\varphi(n)$ (§1, pp. 2–3). These extensions are stated in prose rather than as separately numbered theorems. Conjecture 1 (§1, p. 2), which the paper takes from Erdős, Granville, Pomerance and Spiro (its reference [6], 1990), asserts that $s^{-1}(A)$ has asymptotic density zero whenever $A$ does; the paper does not prove it. Its canonical page is that source's Conjecture 4, whose preimage form it is.

The proof starts with a positive-density set of even deficient integers $n=pm=pqrk$, with primes $p,q,r$ in specified ranges and $k\le x^{1/60}$ (§3, p. 6). Lemma 1 (§2, pp. 3–5) gives typical divisibility properties of $\sigma(n)$ and $s(n)$, including control of their small prime factors. Lemmas 2–4 (§2, p. 5) supply, respectively, a deficiency property, the typical size of $\tau(s(n))$, and a bound for the reciprocal sum of the large prime factors of $\sigma(n)$; the paper derives them from the literature. The authors partition their integers by their largest $y$-smooth divisor $d$, where $y=\log\log x/\log\log\log x$. Lemma 1 makes the corresponding image sets disjoint; equations (1)–(3) show that sufficiently populated classes have substantial total weight (§3, pp. 6–7).

The central estimate is the collision bound $\sum_u r_d(u)^2\ll x/(d\log y)$, equation (4) (§3, p. 7), for the number $r_d(u)$ of representations $u=s(n)$ in a selected class. Equations (5)–(6) turn a collision with $m\ne m'$ into a linear equation in two primes; a sieve gives equation (7) (p. 8). Writing $\gcd(s(m),s(m'))=dh$, for $h>x^{1/3}$ congruence (11) and equation (12) force $\ell=\ell'$ (§3.1, pp. 9–10). For smaller $h$, congruence counting and estimates (13)–(14) control the remaining sieve factor (§3.2, pp. 10–13). Cauchy's inequality then yields $\#s(\mathcal A\cap[1,x])\gg x$, proving Theorem 1. The discussion of an even-range density near $1/3$ reports numerical work, not a theorem (§1, p. 3).

Read status. Claims checked for Theorem 1, Conjecture 1 and the statements of Lemmas 1–4, read clause by clause on the preprint; the proof (§3) was read for structure only.

Results

Labels and pages are those of the authors' preprint (pp. 1–15).

• Theorem 1 (p. 2): the even values of $s(n)=\sigma(n)-n$ form a set of positive lower density.

Bears on

• Problem 955: Conjecture 1 (p. 2) states the problem's assertion for asymptotic density. Theorem 1 proves unconditionally the one consequence of it that the paper draws, that the even values of $s$ do not have density $0$, in the stronger form of positive lower density; that target has positive lower density, and the paper settles no density-zero instance of the problem.

Relation to E955

Conjecture 1 is the problem's assertion: for every $A\subseteq\mathbb N$ of asymptotic density zero, $s^{-1}(A)=\{n:s(n)\in A\}$ has asymptotic density zero. The paper derives one consequence of the conjecture: the set of even integers attained by $s$ would not have density $0$. It notes that this target’s preimage has density $1/2$ and gives it explicitly as $\{n\text{ even}:n,n/2\text{ are not squares}\}\cup\{n^2:n\text{ odd}\}$ (§1, p. 2). Theorem 1 proves more than that consequence for this particular target, namely positive lower density; it does not address arbitrary density-zero targets.

The following is the corpus's reading, not a claim of the paper, which says only that its methods may help in proving Conjecture 1 (p. 3). The collision estimate (4) offers a possible ingredient for E955. Summed over the selected smooth-divisor classes, it gives $\sum_u r(u)^2=O(x)$; hence, for any $A$ with $|A\cap[1,x]|=o(x)$, Cauchy–Schwarz gives $\sum_{u\in A}r(u)=o(x)$ on those classes, for each large $x$. Their inputs are deficient, so their outputs lie below $x$. This controls preimages only within the structured classes selected in §3, which cover a positive proportion rather than a density-one set. Extending that control to essentially all inputs is the gap between this paper and E955.

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