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Statement

For a set $A$ of natural numbers, $F(x,A)$ is the number of natural numbers $n\le x$ divisible by no element of $A$ (p. 385). Put

$$H(x,K)=\min F(x,A),$$

the minimum over sets $A$ with

$$\sum_{a\in A}1/a\le K,\qquad 1\notin A$$

(display (1.5), p. 386).

The claim (printed p. 386, unnumbered). The paper says that the condition that the elements of $P$ be primes cannot be omitted, and continues: "In the second part of the paper we shall show that"

$$H(x,K)<x^{\varepsilon},\qquad K>K_0(\varepsilon),$$

and, more exactly, that

$$\lim_{x\to\infty}\frac{\log H(x,K)}{\log x}=e^{1-K}\qquad(K\ge1).$$

It adds that $H(x,1)=o(x)$ follows from Schinzel and Szekeres (1959), who did not state it explicitly.

This paper contains no proof of either display. The second part is I. Z. Ruzsa, On the small sieve. II. Sifting by composite numbers, J. Number Theory 14 (1982), 260–268, whose card records the limit as that paper's Theorem I.

Source. P. Erdős and I. Z. Ruzsa, On the small sieve. I. Sifting by primes, J. Number Theory 12 (1980), 385–394; the definition (1.5) and the claim on printed p. 386 (PDF p. 2). The edition is identified in the source digest.

Read depth. Claims checked: the definition and the claim were read on the page image. There is no proof in this paper to check.

Dependencies

None in this paper.

Bears on

• Problem 784: the quantity of that problem's corrected statement is $H(x,C)$, with $1$ excluded as in (1.5). For fixed $K>1$ the claimed limit makes $H(x,K)$ at most $x^{\theta}$ for some $\theta<1$ and all large $x$, which is eventually below $x/(\log x)^c$ for every $c$, so the claim, if proved, answers the problem negatively for $K>1$. At $K=1$ the limit is $1$ and decides nothing. This paper only announces the claim.