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Statement

Setting (p. 4). Jacobsthal's function $J(m)$ is the least integer $J$ such that every $J$ consecutive integers contain one coprime to $m$; for $m=P(z)=\prod_{p\le z}p$, $J(m)$ is the least $y$ with $S(x,y,z)\ge1$ for all $x$. $\omega(m)$ is the number of distinct prime factors of $m$.

Context the paper recalls on p. 4, citing others and proving none of it: Iwaniec's interval-sieve bound gives $J(P(z))\ll z^2$, so $J(m)\ll(\omega(m)\log\omega(m))^2$ for $m=P(z)$, and Iwaniec deduced that this upper bound holds for all integers $m$; the proof in Ford, Green, Konyagin, Maynard and Tao (the paper's reference [3]) gives, for $m=P(z)$, $J(m)\gg\omega(m)(\log\omega(m))^2\log_3\omega(m)/\log_2\omega(m)$; and its methods suggest the conjecture that the largest $J(m)$ is about $\omega(m)(\log\omega(m))^{3+o(1)}$.

Remark (p. 4). The paper states that its proof of Corollary 2 implies: if there are infinitely many Siegel zeros $\beta$ with $1-\beta<1/(\log q)^B$ for some integer $B\ge1$, then there are integers $m$ with

$$J(m)\gg\omega(m)(\log\omega(m))^B,$$

so the conjecture above is false if $B$ can be taken larger than $3$. The paper adds that this also follows easily from the discussion in Ford's paper on large prime gaps (its reference [4]).

Proof pointer

No separate proof is given. In the proof of Corollary 2 (p. 13) the modulus $P(Z)$ and an interval of length $y$ all of whose integers share a factor with it are constructed, with $Z\sim\log x$ and $y\sim A^{-B}\log x(\log\log x)^{B-1}$; the remark reads this as a lower bound for Jacobsthal's function.

Read depth

Claims checked: the paragraphs on Jacobsthal's function were read clause by clause on the page image of arXiv v1 (p. 4). The deduction from the proof of Corollary 2 is the paper's and was not rederived. Nothing here is independently reviewed.

Dependencies

Corollary 2 and, through it, Proposition 1.

Source. A. Granville, Sieving intervals and Siegel zeros, Acta Arith. 205 (2022), 1--19, doi:10.4064/aa201002-25-6; labels and pages are those of arXiv:2010.01211v1, the edition named on the source card.

Bears on

• Problem 970: the problem's $h(k)$ is the largest $J(m)$ over $m$ with at most $k$ distinct prime factors, so under the remark's hypothesis $h(k)\gg k(\log k)^B$ at the values $k=\omega(m)$ of the integers it supplies. This is far below $k^2$ and decides neither the order of magnitude nor the question $h(k)\ll k^2$; the hypothesis is unproved.