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Statement

$S_x$ is the Schinzel–Szekeres set defined on the page of Lemma 2.1.

Lemma 2.10 (printed p. 265). For all $x$, with a positive constant $c_4<1$,

$$\sum_{a\in S_x}1/a\le1+O(\log^{-c_4}x).$$

With the lower bound $\sum_{a\in S_x}1/a\ge1-\log^{-c_3}x$ recorded on the page of Lemma 2.5, the reciprocal sum of $S_x$ tends to $1$.

Source. I. Z. Ruzsa, On the small sieve. II. Sifting by composite numbers, J. Number Theory 14 (1982), 260–268; Lemma 2.10 on printed p. 265. The edition is identified in the source digest.

Read depth. Claims checked: the statement was read on the page images. The proof was not checked.

Proof pointer

The paper calls it an immediate consequence of Lemmas 2.1, 2.5 and 2.8: $S_x$ has the least-common-multiple property, leaves $\delta\le\log^{-c_3}x$ of the integers up to $x$ unsifted, and Lemma 2.8 then bounds its reciprocal sum by $1+3\sqrt\delta$.

Dependencies

• Lemma 2.1, Lemma 2.5 and Lemma 2.8.

Bears on

• Problem 542: the Schinzel–Szekeres sets, which have the problem's least-common-multiple property, have reciprocal sum $1+O(\log^{-c_4}x)$, below $31/30$ for large $x$; this concerns that family only, not every admissible set.
• It feeds the upper bounds of Theorem I and Theorem II, which bear on Problem 784.