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Source. H. Davenport and P. Erdős, On sequences of positive integers, J. Indian Math. Soc. (N.S.) 15 (1951), 19–24, the edition identified on the source card: the unnumbered "specially simple case" stated on p. 19 and proved on pp. 19–20.

Read depth. Claims checked: the hypothesis, the conclusion and the argument were read clause by clause on the print's page images; nothing here is independently reviewed.

Statement

Let $a_1,a_2,\ldots$ be an infinite sequence of distinct natural numbers in increasing order, let $b_1,b_2,\ldots$ be the numbers divisible by at least one $a_j$, and let $A=\lim_{m\to\infty}A(a_1,\ldots,a_m)$ be the limit of the densities of the multiples of the first $m$ terms, as on the main theorem's page.

Remark (p. 19, proved pp. 19–20). If $\sum_n 1/a_n$ converges, the $b$ sequence has a density in the ordinary sense, and it equals $A$. Under this hypothesis the paper writes $A(a_1,a_2,\ldots)$ for $A$.

Proof

The $b$'s up to $x$ that are not multiples of any of $a_1,\ldots,a_m$ are multiples of some $a_n$ with $n>m$, so there are at most $\sum_{n>m}\lfloor x/a_n\rfloor\leq x\sum_{n>m}1/a_n$ of them. Hence, for each $m$, every limit point of the proportion of $b$'s among the integers up to $x$ lies between $A(a_1,\ldots,a_m)$ and $A(a_1,\ldots,a_m)+\sum_{n>m}1/a_n$; the tail tends to $0$ as $m\to\infty$, so the proportion tends to $A$.

Used by

• The main theorem applies the remark to the terms supported on the first $k$ primes, whose reciprocal sum always converges (equations (7) and (9), pp. 21–22).

Bears on

• Problem 26: the claim page crediting this paper starts from the remark. The further step, that this density is below one when every $a_j\geq2$, so that no shift of a set with convergent reciprocal sum has almost all integers as multiples, is the claim page's own; the paper does not state it.