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Statement

Theorem 2 (§3, p. 150). Let $a_1,a_2,\dots$ be distinct positive integers with

$$\alpha=\varlimsup_{x\to\infty}(\log x)^{-1}\sum_{a_n\le x}a_n^{-1}>0.$$

Then some subsequence $a_{i_1},a_{i_2},\dots$ satisfies $a_{i_k}\mid a_{i_{k+1}}$ for every $k\ge1$.

The hypothesis is an upper limit (a bar over $\lim$ on the page image), that is, positive upper logarithmic density. The introduction (p. 148) states the same result with the same upper limit and adds: "Naturally every sequence of positive lower density satisfies the condition." Page 151 adds that "The condition in Theorem 2 is easily seen to be best possible of its kind, i. e. one can construct sequences $\{a_i\}$ for which $(\log x)^{-1}\sum_{a_n\le x}a_n^{-1}$ tends to zero arbitrarily slowly, but in which no subsequence with the desired property exists."

Source. H. Davenport and P. Erdős, On sequences of positive integers, Acta Arith. 2 (1936), 147--151 (received 10 January 1936); Theorem 2 on printed p. 150 (PDF p. 4 of the retained five-page scan), proof on pp. 150--151 (PDF pp. 4--5), the introduction on p. 148 (PDF p. 2), read on the page images.

Read depth. Claims checked: the statement, the introduction's version and the best-possible remark were read clause by clause on the page images. The half-page proof was read for its structure (below) and not checked step by step.

Proof pointer

Pages 150--151. It suffices to find one $a_i$ with (3) $\varlimsup(\log x)^{-1}\sum_{a_n\le x,\ a_i\mid a_n}a_n^{-1}>0$ (then iterate). Choose $r$ with $\sum_{\nu>r}A_\nu<\alpha$ (4), where the $A_\nu$ are the inclusion-exclusion densities of §1. If (3) failed for all $i\le r$, then $\alpha$ would be at most the upper limit of $(\log x)^{-1}\sum\theta(n)n^{-1}$ over the $n\le x$ divisible by none of $a_1,\dots,a_r$, which by Theorem 1(a) (the logarithmic density of the set of multiples) equals $A-\sum_{\nu\le r}A_\nu=\sum_{\nu>r}A_\nu$, against (4).

Dependencies

Theorem 1(a) of the same paper (§2): the set of multiples of $a_1,a_2,\dots$ has logarithmic density $A=\sum_\nu A_\nu$, proved through the Dirichlet series identity $F(s)=\zeta(s)A(s)$ and a Tauberian theorem of Hardy and Littlewood.

Bears on

• Problem 487: the result the site's commentary records for sets of positive upper logarithmic density. It is context, not a proof of the problem: in a chain $a\mid b\mid c$ the least common multiple of two members is one of them, so a chain alone gives no triple of distinct members with $[a,b]=c$ (an elementary remark made on the problem page).
• Problems 281 and 486, listed on the card, rest on Theorem 1 of the same paper (Section 2, p. 149, the densities of the set of multiples), not on this theorem; their rows with locators are on the card.