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Statement

$G(x,K)$ is the least number of natural numbers $n\le x$ divisible by no element of $P$, over sets $P$ of primes with $\sum_{p\in P}1/p\le K$ (displays (1.1) and (1.2), p. 385).

Problem 1 (printed p. 386). "Is $G(x,K)$ asymptotically given by the primes in $(x^{e^{-K}},x)$?"

The problem carries the pointer "cf. Erdős [3]", the paper's reference to Erdős, Problem 2, in Number Theory, Colloq. Math. Soc. János Bolyai 2 (1968), p. 232.

The paragraph before it explains the interval. The primes up to $x$ of largest size whose reciprocal sum does not exceed $K$ are, "roughly speaking", those in $(x^{e^{-K}},x)$, and by de Bruijn's result they leave about $xe^{-Ke^K}$ unsifted integers (p. 386). This is far below the expectation $x\prod_{p\in P}(1-1/p)$, printed as $\succ xe^{-K}$ (at least of order $xe^{-K}$), that the Brun and Selberg sieves would give, which they give only when every sifting prime lies below $x^a$ with $a<1$ (p. 385). The paper's best result in the direction of the question is Theorem 1, $G(x,K)\ge e^{-e^{cK}}x$.

Source. P. Erdős and I. Z. Ruzsa, On the small sieve. I. Sifting by primes, J. Number Theory 12 (1980), 385–394; Problem 1 on printed p. 386 (PDF p. 2), with the preceding discussion on pp. 385–386. The edition is identified in the source digest.

Read depth. Claims checked: the question and the surrounding discussion were read on the page images. A question has no proof to check.

Dependencies

None.

Bears on

• Problem 783: a set of primes is pairwise coprime, so Problem 1 is the case of sets of primes of the problem's question, asked up to asymptotic equality. With the asserted (1.12) of Theorem 3, an answer to Problem 1 would carry over to pairwise coprime sets up to $\varepsilon x$. The paper poses the question and does not answer it.