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Claim. Corollary 3 of A. Granville, Sieving intervals and Siegel zeros, Acta Arith. 205 (2022), no. 1, 1--19 (arXiv:2010.01211v1, 2 October 2020, the date this page carries): if there are infinitely many Siegel zeros, then there are arbitrarily large $y$ with admissible sets of length $y$, that is, inside $[0,y]$, having $\sim2y/\log y$ elements. The paper introduces the corollary by recalling the belief that the largest admissible set of length $y$ has $\sim y/\log y$ elements and says that "our results show that this belief is untrue if there are Siegel zeros" (arXiv v1, p. 5). The proof takes an interval left unsieved by the primes up to $y^{1-\epsilon}$, of size $\sim2y/\log y$ by the paper's Corollary 1, and deletes for each larger prime up to $y$ its least-populated residue class.

The inversion, an authored one-line step that the Granville card also records: $A(k)\le x-1$ exactly when an admissible $k$-set lies in an interval of $x$ integers. So $k_j=\#A(y_j)$ gives $A(k_j)\le y_j=(\tfrac12+o(1))k_j\log k_j$. With the known lower bound $A(k)\ge(\tfrac12+o(1))k\log k$, $A(k_j)/(k_j\log k_j)\to1/2$, and $A(k)\sim k\log k$, the question of Problem 1204, fails.

Hypothesis. The claim is conditional on the unproved existence of infinitely many Siegel zeros: real zeros $\beta_q$ of the $L$-functions of real primitive characters of conductor $q$ with $(1-\beta_q)\log q\to0$ along a sequence. As it stands the result decides nothing; the Siegel zeros it assumes are believed not to exist, and the OpenAI release's family 003 manuscripts, which are unreviewed, claim results that exclude them.

Acceptance. Refereed: Acta Arithmetica 205 (2022). The site does not cite the paper.

Depends on. No page of this wiki.