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Statement

Corollary (printed p. 387, unnumbered). Let $P$ be a set of primes with $\sum_{p\in P}1/p\le K$ (display (1.2)). Then the number of squarefree integers up to $x$ divisible by no element of $P$ is at least $cx$, where $c=c(K)>0$.

Source. P. Erdős and I. Z. Ruzsa, On the small sieve. I. Sifting by primes, J. Number Theory 12 (1980), 385–394; the Corollary on printed p. 387 (PDF p. 3). The edition is identified in the source digest.

Read depth. Claims checked: the statement and its one-line derivation were read on the page image.

Proof pointer

The paper obtains it from Theorem 3 applied to the set formed by $P$ and the squares $q^2$ of the primes $q$ outside $P$. That set is pairwise coprime, does not contain $1$, and has reciprocal sum below $K+\sum_q q^{-2}<K+1$; an integer divisible by none of its elements is squarefree and free of primes of $P$.

Dependencies

• Theorem 3.

Bears on

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