bibkey: wu2019subsets authors: Xiaolong Wu year: 2019 title: Subsets of colossally abundant numbers doi: null url: https://arxiv.org/abs/1903.03490v1 claim: The preprint states quantitative improvements from CA3 sources to later CA prefixes; this starting domain excludes the eligible one-step critical sources whose logarithms lie between their own largest prime and the next prime. strata_touched: [] license: citation-only triage: anchor

Existing CA3 improvements and the source domain

The inspected primary is arXiv:1903.03490v1, 23 pages, with PDF SHA-256 64dd95754591cfd0c114516eccbef1efd577ed42ad7bd3f2c44ba4acff2dc6e2. The definitions on printed p.2, Lemma 2, Theorem 2 on p.13 and its argument, and Corollaries 3–4 on pp.14–15 were inspected. The definitions and quantitative statement were also read in page images. This is a preprint citation and applicability assessment, without a full proof audit, reproduction of its finite computations, or Lean verification.

Put $G(n)=\sigma (n)/(n\log \log n)$, $P=P^{+}(n)$, and let $P^{+}$ be the next prime. The paper’s critical-parameter construction includes all threshold layers according to equations (3)–(6). Its three classes are

$$CA1:\log n<P,CA2:P<\log n<P^{+},CA3:P^{+}<\log n.$$

The label CA2 describes this location of $\log n$. It is distinct from the Caveney–Nicolas–Sondow GA2 property, which compares $G(n)$ with every integer multiple’s value.

Theorem 2 takes an actual CA3 prefix $n_i$, puts $p=P^{+}$, and constructs $n_j$ from the critical parameter

$${ϵ}_j=F(p,1)=\frac{\log (1+1/p)}{\log p}.$$

With $t_0>e$ the minimum point of $t^{{ϵ}_j}/\log \log t$, so that ${ϵ}_j\log t_0\log \log t_0=1$, its stated quantitative comparison is

$$G(n_j)>G(n_i)\left(1+\frac{3.2961}{(\log t_0{)}^2\log \log t_0}\right).$$

Corollary 3 states that a CA3 source has a later CA2 source with larger $G$; Corollary 4 states the reduction of Robin’s criterion to CA2. These are existing source comparisons and a published test-set reduction, not a new improving-multiplier construction or criterion.

For the current source problem, reuse the Kalyabin endpoint conditions at the same actual integer $N\in U_1$, with $P=P^{+}(N)=p_k$ and $k>4$. They already give $P<\log N<P^{+}$. Thus, when the required CA-prefix identification is present, this source lies in Wu’s CA2 class. The starting condition $P^{+}<\log N$ of Theorem 2 is incompatible with those endpoints. Its improving comparison cannot be used at that source by changing the class label or selecting another CA integer.

This excludes that particular supplier on the eligible source class; it does not prove a Robin sign on CA2. For the selected counterexample-level global maximizer, the linked Kalyabin note separately excludes small supports by the existing seven-smooth theorem and Axler valuation stop, giving $k\ge 6$; this does not extend Wu’s CA3 starting domain. A quantitative comparison on CA3 and the reduction to CA2 do not estimate the selected source’s signed prime-error remainder. The source’s finite tables and checks are cited rather than rerun.