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Statement

Setting (p. 431). $A$ is an infinite sequence of integers $a_1<a_2<\cdots$. A chain is an infinite subsequence $a_{n_1}<a_{n_2}<\cdots$ with $a_{n_i}\mid a_{n_{i+1}}$ for every $i$, and $c_1,c_2,\ldots$ denote positive absolute constants. Condition (1) is

$$\limsup_{x\to+\infty}\frac{1}{\log x}\sum_{a_i<x}\frac{1}{a_i}>0 . \tag{1}$$

The sentence introducing (1) calls it positive lower logarithmic density, but the display is an upper limit, so (1) as printed is positive upper logarithmic density. Under (1) Davenport and Erdős had proved that $A$ contains a chain; the paper sharpens that result.

Theorem 1 (p. 431). If $A$ satisfies (1), then $A$ contains a chain such that, for infinitely many $y$,

$$\sum_{a_{n_i}<y}1>c_1(\log\log y)^{1/2} . \tag{2}$$

Sharpness (pp. 431--432). The exponent $1/2$ cannot be improved. The paper's example: let $m_1<m_2<\cdots$ tend to infinity sufficiently fast, and let $A$ consist of the integers $a$ whose number of distinct prime factors $\nu(a)$ satisfies $\log\log m_i-(\log\log m_i)^{1/2}<\nu(a)<\log\log m_i+(\log\log m_i)^{1/2}$ for some $i\ge1$. The paper says this $A$ satisfies (1), by the methods of Erdős's 1948 paper on integers with exactly $k$ prime factors, and that if the $m_i$ grow fast enough every chain of $A$ has $\sum_{a_{n_i}<x}1<3(\log\log x)^{1/2}$; no range of $x$ is printed with this bound.

Proof pointer

The paper does not prove Theorem 1. It says (p. 431) that the methods of its proof of Theorem 2 can be used, and it only outlines the sharpness example above, calling the computation behind the bound simple.

Read depth

Claims checked: the setting, (1), Theorem 1 and the sharpness outline were read clause by clause on the page images of pp. 431--432 of the print. There is no proof in the paper to follow. Nothing here is independently reviewed.

Dependencies

The chain theorem of Davenport and Erdős, which the paper cites as its reference [1]: Theorem 2 of their Acta Arithmetica paper. The sharpness example relies on the methods of P. Erdős, On the integers having exactly k prime factors, Ann. of Math. 49 (1948), 53--66, the paper's reference [2].

Source. P. Erdős, A. Sárközi and E. Szemerédi, On divisibility properties of sequences of integers, Studia Sci. Math. Hungar. 1 (1966), 431--435; the edition read is named on the source card.

Bears on

• Problem 1217: Theorem 1 gives, under (1), a chain whose count of terms below $y$ exceeds $c_1(\log\log y)^{1/2}$ infinitely often, and the paper's example shows that (1) alone does not give a chain of order $\log\log y$. The problem asks for a chain whose count has upper growth rate against $\log\log x$ at least that of $\sum_{a_n<x}1/(a_n\log a_n)$; Theorem 1 does not address that comparison.