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Claim. For $0<C\le\log2$ the least proportion of integers $m\le N$ divisible by no element of an admissible $A$ (pairwise coprime, $A\subseteq\{2,\ldots,N\}$, $\sum_{a\in A}1/a\le C$) is $1-C+o(1)$, and the primes in $(y,N]$ with $y=N^{e^{-C}+o(1)}$ attain it up to $o(1)$. This is Theorem 1.1 of P. Chojecki, Extremal coprime coverings under a reciprocal budget and a Dickman-type conjecture (text dated January 23, 2026; the version posted 14 February 2026), which also records (Theorem 1.2) the upper bound $\rho(e^C)+o(1)$ for every $C$ from the prime-tail construction and conjectures that this value is the true minimum. The reason for the threshold: when $C\le\log2$ the prime tail starts above $\sqrt N$, so no $m\le N$ is divisible by two of its primes and the unsifted proportion is exactly $1-\sum_{p\in A}1/p$ up to rounding, while Mertens's theorem matches the budget; the lower bound for every admissible $A$ is the union bound $\sigma_N(A)\ge1-\mu(A)$. Since $\rho(u)=1-\log u$ on $[1,2]$, the value $1-C$ equals $\rho(e^C)$, so the claim is the $C\le\log2$ case of the corrected Statement. The first posting, a note of 23 January 2026 on the Buchstab identity and the Dickman equation, states the same clean regime in its last section. The author's thread post of 23 January 2026 links a GPT-5.2 write-up of the argument, a ChatGPT share at https://chatgpt.com/share/69734b76-4504-8008-817c-dd7844c931c5, and offers the note as a short account of it for readers who prefer a document; the share is a chat transcript rather than a document, so it is cited here and not among the page's links. The post of 14 February 2026 presents the final text as a cleaner write-up with remarks beyond $\log2$ and names no system, and neither PDF names one. Tao's thread post of 20 February 2026 calls the final text Chojecki's GPT-5.2 text, and Tao's write-up credits the $C\le\log2$ case to Chojecki and GPT-5.2.

Submission note. Posted to the site's forum by Przemyslaw Chojecki on 23 January 2026:

For $C \leq \log 2$ the conjecture follows from Mertens' theorem (minimization by consecutive primes). Here's the GPT-5.2 write-up of that argument and identification what's needed for a general $C$ (Buchstab/Dickman machinery). Here's a short note on it, if you prefer that to reading ChatGPT share.

Posted to the site's forum by Przemyslaw Chojecki on 14 February 2026:

BTW, here's a more clean write-up of the argument with some additional remarks beyond $log \ 2$.

Covers. The corrected Statement for $0<C\le\log2$: the minimum unsifted count is $(1-C+o(1))N=(\rho(e^C)+o(1))N$, achieved by the prime tail. Not covered: budgets $C>\log2$, where sifted integers divisible by two elements of $A$ appear, settled asymptotically on Tao's claim page; and the exact minimizer for a given $N$, the site's wording: the site's commentary notes that the literal prime tail is not always the exact minimizer, since small improving perturbations exist, and the note does not determine it.

Acceptance. Reviewed: the site's curator, Thomas Bloom, who is independent of the author, writes in the commentary (page last edited 28 May 2026, accessed 2026-09-05) that Chojecki proved the prime tail extremal when $C\le\log2$, linking the note of 23 January 2026, and thanks the author on the page. The curator's wording credits exact extremality; that is stronger than Theorem 1.1 as the February version states it, the asymptotic form $1-C+o(1)$ attained by the prime tail up to $o(1)$. This page reads the acceptance as covering the asymptotic claim. A thread comment of 24 January 2026 reports an audit of the argument finding it mostly correct with a small modification needed at $C=\log2$; that is a forum check, not acceptance. Not refereed: the notes are posted on the author's research site. Read depth here: the abstract, Theorems 1.1 and 1.2 and the introduction of the February version, and the final section of the January note, were read; the proofs were not read, and nothing is independently reviewed by this project.

Depends on. No page of this wiki.