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Claim. For a finite set $A$ of pairwise coprime integers not containing $1$, write $\mu(A)=\sum_{a\in A}1/a$ and let $\sigma_N(A)$ be the proportion of $n\le N$ divisible by no element of $A$. Theorem 1.1 of T. Tao, Sieving by coprime numbers (the third version, dated February 22, 2026, posted 23 February 2026), states that if $\mu(A)\le C$ then

$$\sigma_N(A)\ge\rho\bigl(e^{\mu(A)}\bigr)+o(1)\qquad(N\to\infty),$$

with $\rho$ the Dickman function and the $o(1)$ depending on $C$ but not on $A$. Since the primes between $N^{e^{-C}}$ and $N$ have $\mu=C+o(1)$ by Mertens's theorem and leave $(\rho(e^C)+o(1))N$ integers unsifted by Dickman's theorem, the least number of $m\le N$ not divisible by any element of an admissible $A$ is $(\rho(e^C)+o(1))N$, and Erdős's construction, the largest primes up to $N$ whose reciprocal sum stays within $C$, is optimal up to $o(N)$. The proof reduces to the prime case, Hildebrand's corollary on the sibling page Hildebrand 1987: in any dyadic interval at most $O(\sqrt x)$ elements of a pairwise coprime set are composite, so the large composite elements can be deleted at negligible cost to $\mu$; medium elements are made negligible by dyadic pigeonholing; and the small elements are handled by a Bonferroni truncation of inclusion-exclusion and the log-concavity of $\rho$, for which the note supplies a proof, in the manner of Hildebrand's paper. The note remarks that $\sigma_N$ is Lipschitz in $\mu$, which recovers the case $C\le\log2$ on Chojecki's claim page at once, and that the error term, $O((\log N)^{-c})$ in the prime case, is not determined here.

Submission note. Posted to the site's forum by Terence Tao on 20 February 2026:

It turns out that one can use the existing results of Hildebrand (or Granville-Soundararajan) in the prime case to generalize to the arbitrary pairwise coprime case, as suggested in Section 6.1 of Chojecki's GPT 5.2 text; writeup here. The simplest case is when all the elements of $A$ are large. Then one can basically just move each element of $A$ (with a small number of exceptions) to the nearest available prime, and an analysis of the inclusion-exclusion / Bonferroni inequalities shows that this doesn't affect the size of the sieved set much (nor does it affect the sum of reciprocals by much either). (There is a simple argument (mapping each element of $A$ to its smallest prime factor) showing that $A$ can't be that much more dense than the primes, so it is relatively easy to move (most of) $A$ to be prime without incurring a large transport cost, thanks to the prime number theorem.)

One then has to deal with small and medium elements of $A$. By a standard pigeonholing argument (the "dyadic pigeonholing" trick of Bourgain) one can assume that the medium elements have negligible size. For small elements one can use a sieve: the traditional thing to do here is invoke the fundamental lemma of sieve theory, but for this problem just the pure Brun sieve suffices actually (because we are happy with a qualitative error term $o(1)$). One can then replace the small elements of $A$ with an equivalent set of small primes that has almost the same Euler product.

The upshot is that the minimum size of the sifted set under the condition $\sum_{n \in A} 1/n \leq C$ and pairwise coprimality is indeed $(\rho(e^C)+o(1)) N$, without requiring the elements of $A$ to be prime, so that the Erdos construction is optimal up to $o(N)$ errors.

[The writeup, by the way, was written in the old-fashioned way, with the only AI assistance being autocomplete (which was a non-trivial speedup in this case, as Github Copilot did anticipate several lines of the argument in advance).]

(The site has been updated to address this comment.)

Covers. The whole corrected Statement: the least number of $m\le N$ divisible by no element of an admissible $A$ is $(\rho(e^C)+o(1))N$, attained by Erdős's prime tail, so that tail minimizes up to $o(N)$ and no admissible $A$ does better by more than $o(N)$. The site's wording, which admissible $A$ attains the exact minimum for a given $N$, is not determined here (the problem page's Notes); the author's thread post of 23 February 2026 says the more precise estimate of the minimum remains open. The structure of near-minimizers for $C>\log2$ is the subject of Chojecki's pending claim.

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 Tao has resolved the question, asymptotically at least, with the bound above, and links the third write-up. Not refereed: the author wrote in the thread on 23 February 2026 that the result is essentially a corollary of Hildebrand's and that the author did not plan to publish the preprint. The thread records the checks the write-up received: the first version of 20 February 2026 had a sign error in the treatment of small primes, found by a thread participant who had GPT-5.2 Thinking check the write-up and confirmed by the author, who repaired the argument through Hildebrand's inequality $\rho(u_1)\rho(u_2)\ge\rho(u_1u_2)$ in the version of 22 February, and a third version of 23 February added the reduction to $A\subseteq[2,N]$ and the log-concavity proof; the note's acknowledgment records this help. The author's post of 20 February 2026 says the write-up was written by hand, with autocomplete from GitHub Copilot its only AI assistance. The same day Chojecki posted a rework of the argument produced with GPT-5.2 Pro, https://www.ulam.ai/research/erdos783-tao.pdf, which Chojecki's post describes as the same argument with minor tweaks, with an appendix, also produced with GPT-5.2, on Selberg-sieve weights in place of the Bonferroni truncation as a route to better bounds; it is a later posting of this result, claims nothing new, and has no page. Those are forum checks and restatements, not acceptance, and the acceptance evidence is the curator's.

Depends on. Hildebrand 1987, the prime case, from which the theorem is deduced; the note cites a simplified proof of that case by Granville and Soundararajan.