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Statement

$C(r)$ is "the maximal length $C(r)$ of a sequence of consecutive integers each divisible by one of $r$ arbitrarily chosen primes" (p. 225). For a sequence $\mathcal A$ of $X$ consecutive integers and the product $Q$ of $r$ primes $q_1,\ldots,q_r$, the sifting function $S(\mathcal A,Q)=\sum_{a\in\mathcal A,(a,Q)=1}1$ counts the elements of $\mathcal A$ coprime to $Q$, and Jacobsthal's problem "is how large $X$ and $r$ must be in order to have $S(\mathcal A,Q)>0$" (p. 225).

Theorem (p. 226, quoted). "There exists an absolute constant $c>0$ such that for arbitrarily chosen primes $q_1,\ldots q_r$, $r>1$ each interval of the length

$$c\Bigl(1-\frac1{q_1}\Bigr)^{-1}\cdots\Bigl(1-\frac1{q_r}\Bigr)^{-1}r^2\log r$$

contains at least $r^2$ integer numbers coprime to $q_1\cdots q_r$."

The paper introduces it with (p. 226): "The aim of this paper is to prove (1) for $C(r)$. Modifying the arguments used in [3] we shall show that slightly more is true", where (1) is $C_0(r)\ll r^2\log^2r$ for the first $r$ primes, credited to the author's 1971 paper on the error term in the linear sieve. The Corollary, $C(r)\ll r^2\log^2r$, follows on the same page.

Source. H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), no. 1, 225--231; the Theorem on printed p. 226 (PDF p. 2 of the publisher's scan) and its proof on pp. 228--230 (PDF pp. 4--6), read on the page images. The edition read is identified in the source digest.

Read depth. Claims checked: the statement, the definition of $C(r)$ and the sieve setting were read clause by clause on the page images. The proof was followed at the level of its displays on the page images of pp. 229--230: the choice of weights, Lemma 2 as quoted, display (7) and the closing parameter choice; Lemma 1 (pp. 227--228) was read for structure only, and Lemma 2 is quoted by the paper from its [3] without proof. Nothing here is independently reviewed.

Proof pointer

§ 3, pp. 228--230, by the shifted sieve of § 2. For $X$ consecutive integers $\bigl||\mathcal A_d|-X/d\bigr|<1$, so the sieve hypothesis (R) holds with $A=B=1$ and $f(d)=d$. Let $z\ge2$, $y\ge z^2$ and $P$ the product of the primes $p\le z$; the lower-bound weights $\lambda_n=\mu(n)$ for $n=p_1\cdots p_u$, $p_1>\cdots>p_u$, with $p_1\cdots p_{2l}<yp_{2l}^{-2}$ for $2l\le u$, and $\lambda_n=0$ otherwise, satisfy the lower-bound condition (-) of Lemma 1. Lemma 2, quoted from [3] for $4\le z^2\le y<z^4$: (5) $\sum_{n\mid P}|\lambda_n|\ll y(\log y)^{-2}$ and (6) $\sum_{n\mid P}\sigma_n/\prod_{p\mid n}(p-1)=2e^\gamma\log(s-1)/s+O(1/\log y)$ with $s=\log y/\log z$. Given primes $q_1<\cdots<q_r$, $r>1$, put $Q=q_1\cdots q_r$, $z=p_r$ and $l(q_i)=p_i$, the $i$-th prime, so that $g(n)=n$ on $n\mid P$ and $f(d)=d$ on $d\mid Q$ satisfy the shift condition (2), $g(l(d))\le f(d)$, because $p_i\le q_i$. Lemma 1 (4) with Lemma 2 gives (7)

$$S(\mathcal A,Q)\ge X\prod_{q\mid Q}\Bigl(1-\frac1q\Bigr) \Bigl\{2e^\gamma\frac{\log(s-1)}s+O\Bigl(\frac1{\log y}\Bigr)\Bigr\} +O\Bigl(\frac y{\log^2y}\Bigr).$$

With $y=Cz^2$ and $X=\prod_{q\mid Q}(1-1/q)^{-1}y/\log z$ for a sufficiently large absolute constant $C$, "the right hand side of (7) is $>y/\log^2z>r^2$", and $z=p_r\asymp r\log r$ makes $X$ of the stated order. Not checked here beyond the displays.

Dependencies

Within the paper: Lemma 1 (p. 227), the shifted sieve inequality, proved on p. 228 in its lower-bound form (4). Outside it: Lemma 2 (p. 229), the two estimates (5) and (6) for the linear-sieve weights, quoted from the author's 1971 paper On the error term in the linear sieve, Acta Arith. 19 (1971), 1--30 (the paper's [3], not held), whose proof the paper calls "very complicated"; the idea of the shift is credited to Halberstam and Richert, Mean value theorems for a class of arithmetic functions, Acta Arith. 18 (1971), 243--256 (the paper's [2], not held).

Bears on

• Problem 970: the source of the Corollary $C(r)\ll r^2\log^2r$, the site's $h(k)\ll(k\log k)^2$, with the explicit factor $\prod_{i\le r}(1-1/q_i)^{-1}$, which is largest for the first $r$ primes.
• Problem 687: at $q_i=p_i$ and $r=\pi(x)$ the interval length is $\asymp\pi(x)^2\log^2\pi(x)\ll x^2$, the source of $Y(x)\ll x^2$ through the Corollary.