Pollack–Roy: Powerfree sums of proper divisors

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proposition_3_1: For each fixed k at least 4, almost always no prime power p^k exceeding (log log x)^0.9 divides the sum of proper divisors s(n), which with the divisibility of sigma(n) by all small integers gives the paper's Theorem 1.2.

theorem_1_2: Pollack and Roy prove that for each fixed k at least 4 there is a set of integers of asymptotic density 1 on which n is k-free if and only if the sum of proper divisors s(n) is k-free, the cases k = 2 and k = 3 staying open.

theorem_3_3: Outside a set of o(x) integers, the number of n at most x with d dividing s(n) is at most a constant times x/(d^{1/4} log x), uniformly for d above x^{1/(2 log_3 x)} whose least prime factor exceeds log log x.

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The arXiv record names arXiv's non-exclusive distribution license (arXiv:2106.14953), every other right reserved.

Paul Pollack, Akash Singha Roy, "Powerfree sums of proper divisors," arXiv:2106.14953 (2021); published in Colloquium Mathematicum, 168(2), 287-295, 2022, https://doi.org/10.4064/cm8616-10-2021. The copy read for this card is arXiv:2106.14953v1 (7 pages), not the published edition; page locators below are its pages.

Overview

For $s(n)=\sigma(n)-n$, the paper asks whether, on a set of density $1$, $n$ is $k$-free exactly when $s(n)$ is $k$-free (Conjecture 1.1, §1, p. 1). Theorem 1.2 (p. 2) proves this for every fixed $k\ge4$; the cases $k=2,3$ remain conjectural here. Consequently, for $k\ge4$, the integers with $k$-free $s(n)$ have density $1/\zeta(k)$, using Gegenbauer's theorem on the density of $k$-free integers cited in §1.

The proof separates small and large prime powers. The result pages are Theorem 1.2 (p. 2, with Conjecture 1.1 of p. 1), Proposition 3.1 (p. 4) and Theorem 3.3 (p. 4). Lemma 2.2 shows that almost always every $d\le(\log_2 x)^{1-\epsilon}$ divides $\sigma(n)$, so $n$ and $s(n)$ have the same divisibility by such $d$. Proposition 3.1 supplies the other step: for $y=(\log_2 x)^{0.9}$ and $k\ge4$, almost always no $p^k>y$ divides $s(n)$. For $y<p^k\le x^{1/(2\log_3 x)}$, this follows by summing Lemma 3.2’s uniform $O(x/d^{0.9})$ bound outside an $o(x)$ exceptional set (§3.1).

For larger powers, Theorem 3.3 gives, outside an $o(x)$ exceptional set, $\#\{n\le x:d\mid s(n)\}\ll x/(d^{1/4}\log x)$ uniformly for $d>x^{1/(2\log_3 x)}$ with $P^-(d)>\log_2 x$ (§3.2). Its proof uses Lemmas 2.3–2.4 to obtain a typical largest-prime-factor factorization $n=mP$. When $P$ is large, $d\mid s(mP)$ places $P$ in one coprime residue class modulo $d$. In the remaining case, a unitary squarefree divisor $A$ is separated from $B=n/A$; the bound on $B$ is displayed as (1). The congruence modulo $d$ and $\sigma(A)<d^{1/2}$ force all admissible $A$ for a fixed $B$ to have the same value of $\sigma(A)/A$. The quoted Wirsing bound (Lemma 2.5) limits the number of such $A$. Summing Theorem 3.3 over $d=p^k$ completes Proposition 3.1 and Theorem 1.2. Lemma 2.1 is a cited result used to prove Lemma 2.2; Lemma 2.5 is likewise quoted, not proved in this paper.

Relation to E955

Bears on. Problem 955: by p. 2 and Remark 3.4 (p. 6), the problem's assertion would give the conclusion of Proposition 3.1 and Conjecture 1.1 for every $k\ge2$; Proposition 3.1 and Theorem 1.2 prove them unconditionally for $k\ge4$ only, through Theorem 3.3. No result of the paper is a preimage statement for a density-zero set, and the paper does not decide the problem.

E955 asks whether $s^{-1}(A)$ has density zero for every fixed density-zero set $A\subseteq\mathbb N$. This is the Erdős–Granville–Pomerance–Spiro conjecture identified in Remark 3.4. That remark gives the specific density-zero target $A_k=\{m:\ p^k\mid m\text{ for some }p^k>\log_3(100m)\}$. An affirmative answer to E955 would imply Proposition 3.1 for each $k\ge2$, and hence Conjecture 1.1 for those $k$ (§1; Remark 3.4).

In the other direction, Theorem 1.2 supplies a special distributional result for the power-free property, and Theorem 3.3 supplies a uniform bound for divisibility by a large integer with sufficiently large least prime factor. These can enter an E955 argument when the target set is controlled through such divisibility conditions. They give no bound for preimages of an arbitrary density-zero set: such a set need not be describable by a summable family of the divisibility conditions covered by Lemma 3.2 or Theorem 3.3. Remark 3.4 cites the separate $x^{1/2+o(1)}$ target-counting result as background, not as a result proved here. The paper does not resolve E955.

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